SIMPLICIAL NONPOSITIVE CURVATURE

by TADEUSZ Januszkiewicz^{⋆} and JACEK Świa̧tkowski^{⋆⋆}

ABSTRACT

We introduce a family of conditions on a simplicial complex that we call local k-largeness (k ≥ 6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to nonpositively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

1. Introduction

Spaces of non-positive curvature have been intensively investigated over the past 50 years. More recently non-riemannian metric spaces, for which non-positive or negative curvature is defined by comparison inequalities, the so-called CAT (0) or CAT (−1) spaces, have been studied, mainly in geometric group theory [4].

Many CAT (0) spaces are obtained by combinatorial constructions. These constitute a significant part of small cancellation theory [22], which deals mostly with 2-dimensional complexes. Cubical complexes are the main source of high dimensional CAT (0) spaces. The crucial observation which permits their study is Gromov’s Lemma: a cubical complex with its standard piecewise euclidean metric is CAT (0) if and only if the links of its vertices are flag simplicial complexes. The flag property is an easily checkable, purely combinatorial condition.

It is natural to ask if something similar holds for simplicial complexes:

1. can one formulate the CAT (0) property of the standard piecewise euclidean metric on a simplicial complex in combinatorial terms;

2. is there a simple combinatorial condition implying CAT (0);

3. is there a simple condition implying Gromov hyperbolicity.

We do not answer the first question but we provide a satisfactory answers to the other two. Namely, in Section 2 we introduce the notion of a

⋆ Partially supported by the NSF grant DMS-0405825.

⋆⋆ Partially supported by the Polish State Committee for Scientific Research (Komitet Badań Naukowych), KBN grant 2 P03A 017 25. Partially supported by the NSF grant DMS-0405825.

locally k-large simplicial complex, where k ≥ 4 is an integer. It is defined in terms of links in the complex by very simple combinatorial means. We show in Sections 16 end 17 that, for every n, there is an explicit constant k(n) such that if X^{n} is a locally k(n)-large, n-dimensional simplicial complex, then its standard piecewise euclidean metric is CAT (0). Taking a slightly bigger constant k(n) we conclude that X^{n} admits a CAT (−1) metric. We also show (Section 3) that the universal covers of locally 7-large complexes are Gromov hyperbolic. These facts are well known in dimension 2, where our definition of locally 6- and 7-large coincides with the CAT (0) and, respectively the CAT (−1) property of the standard piecewise constant-curvature metrics.

We claim that “locally 6-large” is the right simplicial analogue of nonpositive curvature. This condition neither implies nor is implied by the CAT (0) property of the standard metric, but shares many of its consequences. We describe some of them later in this introduction. The results are proved using combinatorial (but metrically inspired) concepts. This is very much in the spirit of small cancellation theory. The novelty is that our approach works in any dimension.

Let us point out that the flag condition from Gromov’s Lemma is equivalent to the “4-large” property. Also, Siebenmann’s “flag-no-square” condition appearing in the study of CAT (−1) property of cubical complexes is equivalent to “5-large”.

Finer properties of high dimensional locally 6-large simplicial complexes seem to be fairly different from the properties one sees when studying nonpositively curved manifolds. Manifolds of dimension greater than 2 do not admit locally 6-large triangulations. As we show in [17], the fundamental groups of many aspherical manifolds cannot be embedded into the fundamental groups of locally 6-large complexes. Still, high dimensional locally 6-large spaces abound. We construct a great many very symmetric examples by developing certain simplices of groups. In particular, we can obtain in this way compact orientable locally 6-large pseudomanifolds of any dimension.

We now briefly describe the contents of the paper, which naturally splits into five parts.

In the first part (Sections 2 and 3) we introduce the concepts of a locally k-large simplicial complex, a k-systolic simplicial complex, and a ksystolic group. Here we briefly recall these concepts (see also the remark after Lemma 1). Given an integer k ≥ 4, a simplicial complex is locally k-large if every cycle consisting of less than k edges in any of its links has some two consecutive edges contained in a 2-simplex of this link. A simplicial complex is k-systolic if it is locally k-large, connected and simply connected. A group is k-systolic if it acts simplicially, properly discontinously and cocompactly on a k-systolic complex. A simplicial complex (or a group) is systolic if it is 6-systolic.

In Section 2 we give a useful criterion for k-largeness (k ≥ 6) in terms of links and lengths of homotopically nontrivial loops (Corollary 1). This is done with a simplification argument on simplicial disc diagrams reminiscent of small cancellation theory arguments. Similar reasoning allows us to establish in Section 3 the following result.

1. Theorem. — (See Theorem 13 and Corollary 2 in the text).

1. Let X be a 7-systolic simplicial complex. Then the 1-skeleton X^{(1)} of X with its standard geodesic metric is hyperbolic in the sense of Gromov.

2. Any 7-systolic group is word-hyperbolic.

The main idea exploited in the second part of the paper (Sections 4-7) is that of local convexity. We introduce it in Section 4 under the name of local 3-convexity. It allows us to define “small extensions” (Sections 5 and 5). These may be viewed as an analogue of the exponential map with a builtin divergence property for trajectories. Using small extensions we show the following three results.

2. Theorem. — (See Theorem 14.1 in the text). The universal cover of a finite dimensional connected locally 6-large simplicial complex is contractible. In particular, any finite dimensional systolic simplicial complex is contractible.

This is an analogue of the classical Cartan-Hadamard theorem.

3. Theorem. — (See Theorem 14.2 in the text). Let f : Q → X be a locally 3-convex map of a connected simplicial complex Q to a finite dimensional connected locally 6-large simplicial complex X. Then the induced homomorphism f_{∗} : π_{1}Q → π_{1}X of fundamental groups is injective.

Note that Theorem 3 applies to the inclusion maps of locally 3-convex subcomplexes Q ⊂ X. The analogous statement in riemannian geometry asserts that the fundamental group of a locally geodesically convex subset in a complete non-positively curved manifold injects into the fundamental group of the ambient space (this is also true for locally CAT (0) geodesic metric spaces).

4. Theorem. — (See Theorem 15 in the text). Every connected locally 6-large simplicial complex of groups is developable.

Theorem 4 will be crucial for the constructions in the last part of the paper. It is analogous to the classical result asserting that non-positively curved complexes of groups are developable.

The results in part three of the paper (Sections 8-14) are based on a certain convexity property of balls in systolic complexes, described in Section 8 (Corollary 7). The main result in this part is the following.

5. Theorem. — (See Theorem 16 in the text). Let G be a systolic group, i.e. a group acting simplicially, properly discontinuously and cocompactly on a systolic simplicial complex. Then G is biautomatic.

Many corollaries of Theorem 5 can be obtained by using the welldeveloped theory of biautomatic groups [9]. In particular, systolic groups satisfy quadratic isoperimetric inequalities, their abelian subgroups are undistorted, their solvable subgroups are virtually abelian, etc.

Theorem 5 is the culmination of a series of results concerning systolic complexes, which have independent interest. For example, in Section 9 we define a simplicial analogue of the projection map onto a convex subset. We also show that this map does not increase distances (Fact 6). In Section 10 we introduce the concept of directed geodesics and show their existence and uniqueness (Corollary 12). Finally, we establish in Sections 12-13 the twosided fellow traveller property for directed geodesics, the main ingredient in the proof of Theorem 5.

To prove Theorem 5 one needs, besides properties of directed geodesics, an argument which enables the passage from the space on which the group acts to the group itself, especially in the case where the group action has nontrivial stabilizers. The argument we use in this paper has been expanded and applied in other situations by the second author [23].

Part four of the paper (Sections 15-17) addresses the relationship between the k-systolic and CAT (κ) properties. We have

6. Theorem. — (See Theorem 17 in the text). Let Π be a finite set of isometry classes of metric simplices of constant curvature 1, 0 or −1. Then there is an integer k ≥ 6, depending only on Π, such that:

1. if X is a piecewise spherical k-large complex with Shapes(X) ⊂ Π then X is CAT (1);

2. if X is piecewise euclidean (respectively, piecewise hyperbolic), locally k-large and Shapes(X) ⊂ Π then X is non-positively curved (respectively, has curvature κ ≤ −1);

3. if, in addition to the assumptions of (2), X is simply connected, then it is CAT (0) (respectively, CAT (−1)).

We offer two proofs of Theorem 6. The first one (in Section 15) covers the general case, but the estimates for the systolic constants are not explicit. The second one (Section 16) yields potentially explicit constants, but covers only metrics for which the simplices have all angles acute. In Section 17 we work out explicit estimates for the standard piecewise euclidean metric (based on the second proof), and obtain the following.

7. Theorem. — (See Theorem 19 in the text). Let k be an integer such that

k ≥ \frac{7π\sqrt{2}}{2} · n + 2.

Then any k-systolic simplicial complex X with dim X ≤ n is CAT (0) with respect to the standard piecewise euclidean metric.

The last part of the paper (Sections 18-21) deals with constructions of k-large complexes of high dimensions. The complexes we obtain arise as developments of appropriate simplices of groups. The constructions are based on the second important idea of the paper, the notion of extra-tilability of simplices of groups (Section 19). Extra-tilability matches with local convexity of balls in systolic spaces in an interesting way, and allows us to construct subgroups with large fundamental domains. As a consequence, we obtain large compact quotients of universal covers of simplices of groups, which in turn allows us to use induction in the constructions.

The key result in this part is Theorem 8 below. The technical notions occurring in its statement, which are standard in the theory of complexes of groups, are recalled in Section 18.

8. Theorem. — (See Proposition 21 in the text). Let Δ be a simplex and suppose that for any codimension-1 face s of Δ we are given a finite group A_{s}. Then for any k ≥ 6 there exists a simplex of finite groups G = ({G_{σ}}, {ψ_{στ}}) and a locally injective and surjective morphism m : G → F to a finite group F such that G_{Δ} = {1}, G_{s} = A_{s} for any codimension-1 face s of Δ, and the development D(G, m) associated with the morphism m is a (finite and) k-large simplicial complex.

As an application of Theorems F and H we obtain the following.

9. Theorem. — (See Corollary 23.2,3 in the text).

1. For each natural number n there exists an n-dimensional compact simplicial orientable pseudomanifold whose universal cover is CAT (0) with respect to the standard piecewise euclidean metric.

2. For each natural number n and each real number d > 0 there exists an n-dimensional compact simplicial orientable pseudomanifold whose universal cover is CAT (−1) with respect to the piecewise hyperbolic metric for which the simplices are regular hyperbolic with edge lengths d.

Theorem 9 answers a question of D. Burago and collaborators [5,6], motivated by their investigations of billiards. The result can be extended from simplices to more general domains. We present the exposition of this more general result in [18].

As a step in the proof of Theorem 9 one gets the existence of k-large compact orientable pseudomanifolds of arbitrary dimension n, for any k ≥ 6. It is interesting to compare this with our earlier paper [16], where we establish the existence of hyperbolic Coxeter groups of arbitrary virtual cohomological dimension. The existence of such (right-angled) groups is reduced in [16] to the existence in arbitrary dimension of compact orientable pseudomanifolds which satisfy the flag-no-square condition (they occur as nerves of the corresponding right-angled Coxeter groups). Since the flag-no-square condition is equivalent to 5-largeness, we obtain in the present paper compact orientable pseudomanifolds which satisfy even stronger conditions, with a significantly different construction than that in [16].

The result from [16] mentioned above can also be compared with another result from the present paper, Theorem 10, which can be deduced from Theorem 8.

10. Theorem. — (See Corollary 23.1 in the text). For each natural number n there exists a developable simplex of groups whose fundamental group is Gromov-hyperbolic, virtually torsion-free, and has virtual cohomological dimension n.

A less immediate consequence of Theorem 8, below, answers a question of M. Gromov. Normal simplicial pseudomanifolds occurring in the statement of this result form a natural class containing, among others, all triangulations of manifolds. By a branched covering we mean a simplicial map which is a covering outside the codimension-2 skeleton.

11. Theorem. — (See Theorem 20 in the text). Let X be a compact connected normal simplicial pseudomanifold equipped with a piecewise euclidean (respectively, piecewise hyperbolic) metric. Then X has a compact branched covering Y which is non-positively curved (respectively, has curvature κ ≤ −1) with respect to the induced piecewise constant curvature metric.

We apply the same methods to answer a question of Ian Leary concerning homotopy types of classifying spaces for proper G-bundles of Gromov hyperbolic groups G (see [2, Question 1.24]).

We refer to [21] for the background on the following result.

12. Theorem. — (See Corollary 24 in the text.) Any finite complex K is homotopy equivalent to the classifying space for proper G-bundles of a CAT (−1) (hence Gromov hyperbolic) group G.

We started to work on the present paper in 2000. The initial aim was to construct hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension via 5-large pseudomanifolds. After proving first few results on 6-large spaces, we found a shortcut – retractible and extra retractible complexes of groups – which we eventually used in [16]. Our further study of the subject was motivated by the question of D. Burago et al. on existence of simplicial pseudomanifolds whose standard piecewise flat metrics are nonpositively curved. This led us to the question about the relationship between the k-large and CAT (κ) conditions.

Since 2002 we gave several lectures on the subject (at the conferences in Luminy in 2002, in Durham in 2003, and in several other places). At the Luminy conference, M. Gromov asked the question about ramified covers (see Theorem 11), and gave us significant moral support with the rest of the project. In the Spring of 2003 one of us had first discussions with Dani Wise which were very useful. We did not circulate a preprint, and in late 2003, we have learned that Frederic Haglund has independently obtained some of our results (roughly, those in Sections 2-9 and 18-20). Part of his work is described in [15].

We are grateful to Misha Gromov, Dani Wise and many other colleagues for useful feedback. We would like to thank the referee for careful reading the first version of the paper and to Carrie Schermetzler who pointed out several annoying mistakes.

The first author thanks Max-Planck-Institut für Mathematik for its hospitality in the summer of 2003. We thank the Ohio State University Research Foundation for partial support of the visit of the second author to OSU, during which we completed the paper.

2. k-large and k-systolic simplicial complexes.

In this section we define and study first properties and examples of k-large and k-systolic simplicial complexes.

We allow that simplicial complexes are not locally finite. In Sections 2-4, if not explicitly assumed otherwise, we also allow that they are not finite dimensional, i.e. may contain simplices of arbitrarily large finite dimension. (We call such complexes infinite dimensional.) Starting from Section 5, we assume that simplicial complexes are finite dimensional. It is possible that some of the results in Sections 5-13 extend to infinite dimensional case, but this would require arguments different from ours.

If a simplicial complex is finite dimensional, its topology is induced by the standard piecewise euclidean metric.

We express topological properties of infinite dimensional simplicial complexes in terms of their appropriate finite dimensional skeleta.

We are most interested is in simplicial complexes equipped with proper discontinuous and cocompact group actions by simplicial automorphisms. Those are finite dimensional and locally finite.

Let X be a simplicial complex, and σ a simplex in X. The link of X at σ, denoted X_{σ}, is a subcomplex of X consisting of all simplices that are disjoint from σ and which together with σ span a simplex of X. The residue of σ in X, Res(σ, X), is the union of all simplices of X that contain σ. It is also called the (closed) star of σ. The residue Res(σ, X) is naturally the join of σ and the link X_{σ}.

A subcomplex K in X is called full (in X) if any simplex of X spanned by a set of vertices in K is a simplex of K. If K is full in X, then K_{σ} is full in X_{σ} for any simplex σ in K. A similar property holds also for residues.

A simplicial complex X is flag if any finite set of vertices, which are pairwise connected by edges of X, spans a simplex of X. Clearly, a full subcomplex in a flag complex is flag. Note also that X is flag if and only if for any simplex σ the link X_{σ} is full in X. Flag simplicial complexes arise naturally in the study of CAT (0) property of cubical complexes [14,4].

A cycle in a simplicial complex X is a subcomplex γ of X isomorphic to a triangulation of S^{1}. Denote by |γ| the length of γ, i.e. the number of 1-simplices in γ. A full cycle in X is a cycle that is full as subcomplex of X. Define the systole of X to be

sys(X) = min{|γ| : γ is a full cycle in X}.

In particular, we have sys(X) ≥ 3 for any simplicial complex X, and if there is no full cycle in X, sys(X) = ∞. This definition is somewhat reminiscent of the notion of systole in riemannian geometry, hence the name.

1. Definition. — Given a natural number k ≥ 4, a simplicial complex X is – k-large if sys(X) ≥ k and sys(X_{σ}) ≥ k for each simplex σ of X;

– locally k-large if the residue of every simplex of X is k-large;

– k-systolic if it is connected, simply connected and locally k-large.

(Here we use convention that an infinite dimensional simplicial complex is connected when its 1-skeleton is connected, and it is simply connected when its 2-skeleton is simply connected. To be consistent with the latter, by the universal cover of an infinite dimensional flag complex we mean the flag completion of the universal cover of its 2-skeleton.)

A group acting properly discontinously and cocompactly, by automorphisms, on a k-systolic simplicial complex, is called a k-systolic group.

A 6-systolic complex or a group is called systolic.

6-systolic complexes and groups are the main objects of study in this paper. Since the word “six-systolic” is somewhat hard to pronounce, we abbreviate it to “systolic”.

Some easy properties of the above introduced classes of simplicial complexes are gathered in Fact 1. The proofs are immediate hence we omit them.

1. Fact. —

1. A complex is locally k-large if and only if the link of every nonempty simplex has the systole at least k.

2. A (locally) k-large complex is (locally) m-large for k ≥ m.

3. A full subcomplex in a (locally) k-large complex is (locally) k-large.

4. A simplicial complex is 4-large if and only if it is flag.

5. For k > 4, X is k-large if and only if it is flag and sys(X) ≥ k.

Note that, in view of property (4) above, a simplicial complex X is 5-large if it is a “flag-no-square” complex, or verifies “Siebenmann no square condition”, a condition which arises in the study of CAT (−1) property of cubical complexes [14].

The next result will be used in the proof of Lemma 3 for the purpose analogous to reducing van Kampen diagrams in small cancellation theory.

1. Lemma. — Suppose that X is k-large and S_{m}^{1} denotes the triangulation of S^{1} with m 1-cells. If m < k then any simplicial map f : S_{m}^{1} → X extends to a simplicial map from the disc D^{2}, triangulated so that triangulation on the boundary is S_{m}^{1} and so that there are no interior vertices in D^{2}.

Proof. — We will use induction with respect to m. For m = 3 the statement follows from flagness of X. Suppose m > 3. Then there are some non-consecutive vertices u, w of S_{m}^{1} whose images under f either coincide or are connected with an edge of X. Split S_{m}^{1} into polygonal paths A and B with endpoints u, w. Consider new triangulations of S^{1}, denoted S_{A}^{1} and S_{B}^{1}, obtained by adding the edge (u, w) to A and B, respectively. Note that, by the choice of u and w, restrictions of the map f to A and B extend uniquely to the simplicial maps f_{A} : S_{A}^{1} → X and f_{B} : S_{B}^{1} → X.

Since the vertices u, w are non-consecutive in S_{m}^{1}, the triangulations S_{A}^{1}, S_{B}^{1} consist of fewer than m edges. By the inductive assumption, there are triangulations D_{A}, D_{B} of the 2-disk, and their simplicial maps F_{A}, F_{B} to X extending the maps f_{A}, f_{B}, as required in the lemma. We then get extension of f as required by gluing D_{A} to D_{B} along (u, w) and taking as F the union of the maps F_{A}, F_{B}.

1. Remark. — It follows easily from the above lemma that every cycle of length less than k in a k-large simplicial complex X has some two consecutive edges contained in a common 2-simplex of X. In view of the fact that links of links of a simplicial complex X are themselves links of X, this gives the following characterization of k-systolicity: a simplicial complex X is k-systolic if it is connected, simply connected, and every cycle of length less than k in any link of X has some two consecutive edges contained in a 2-simplex of this link.

There are 4-systolic (respectively 5-systolic) complexes that are not 4large (respectively, 5-large). For example, take two octahedra (respectively, icosahedra), delete the interior of a triangle from each copy and glue the resulting boundaries. However, for k ≥ 6 we have

1. Proposition. — If X is a k-systolic simplicial complex with k ≥ 6 then X is k-large.

Before proving the above proposition, we derive its corollary which will be useful for our later constructions of k-large complexes in Sections 19-20.

A homotopical systole of a simplicial complex X is the minimal length of a cycle that is homotopically nontrivial in X. (If X is infinite dimensional, we say that a cycle is homotopically nontrivial in X if it is homotopically nontrivial in the 2-skeleton of X.) We denote homotopical systole of X by sys_{h}(X).

1. Corollary. — Let k ≥ 6. A simplicial complex X is k-large if and only if it is locally k-large and sys_{h}(X) ≥ k.

Proof. — One of the implications follows from Proposition 1 by noting that if X is locally k-large then there is no full homotopically trivial cycle of length less than k in X (because, by Proposition 1, there is no such cycle in the universal cover of X). The second implication follows by observing that the shortest homotopically nontrivial cycle in any simplicial complex X is full.

Proof (Proposition 1). — We need to show that sys(X) ≥ k. Consider a full cycle γ in X. A filling of γ is a continuous map f : Δ → X such that Δ is the 2-disc and the restriction f|_{∂Δ} is a homeomorphism on γ. Since X is simply connected, there is a filling f_{0} : Δ_{0} → X. Using relative Simplicial Approximation Theorem we can also arrange that Δ_{0} is a simplicial disc and f_{0} is a simplicial map (which is a simplicial homeomorphism on the boundary). Recall that a simplicial map is nondegenerate if it is injective on each simplex of the triangulation.

To proceed with the proof we need two lemmas, the first of which is related to van Kampen Lemma from the small cancellation theory. The elementary proofs of both lemmas are deferred until the end of this section.

2. Lemma. — Let X be a simplicial complex, and γ a homotopically trivial cycle in X. Then there exists a filling f of γ, f : Δ_{1} → X, which is a nondegenerate simplicial map from a simplicial 2-disc Δ_{1}, and which maps the boundary of Δ_{1} isomorphically on γ.

3. Lemma. — Let X, γ, satisfy the assumptions of Lemma 2, and X is locally k-large. Then there exists a nondegenerate simplicial filling f : Δ_{2} → X of γ, such that every interior vertex of Δ_{2} is contained in at least k triangles. Any filling of γ with the minimal number of triangles has this property. If moreover γ is a full subcomplex in X, then every boundary vertex of Δ_{2} is contained in at least two triangles, and there is at least one internal vertex in Δ_{2}.

To conclude the proof of Proposition 1 we use the Gauss-Bonnet Theorem. Let χ(v) denote the number of triangles containing vertex v. Then

1 = χ(Δ_{2}) = \frac{1}{6}[∑_{v∈B}(3 − χ(v)) + ∑_{v∈I}(6 − χ(v))].

where B denotes the set of vertices on the boundary and I the set of vertices in the interior of Δ_{2}. Since the second sum is at most 6 − k, and the terms of the first sum are at most 1, we conclude that |γ| = #B ≥ k, and hence sys(X) ≥ k.

1. Example. — Examples and non-examples of k-large and k-systolic complexes, for k ≥ 6.

1. A graph X is k-large if and only if sys(X) ≥ k. It is k-systolic if and only if it is a tree.

2. Trees are the examples of complexes that are k-large for any k. We will call such complexes ∞-large. Connected ∞-large complexes are necessarily ∞-systolic, since they are easily seen to be simply connected.

3. Let Y be a triangulation of Euclidean or hyperbolic plane by congruent equilateral triangles with angles 2π/m. Then Y is m-systolic. Let X be a simplicial surface obtained as a quotient of Y. If 6 ≤ k ≤ m then X is k-large if and only if sys_{h}(X) ≥ k. By residual finiteness of the automorphism group of Y, this gives lots of k-large surfaces.

4. Let X be the Cayley complex of a group with triangular presentation (i.e. presentation with all relations of length 3). Then X is systolic if and only if the group (presentation) satisfies the C(3) − T (6) small cancellation condition. More generally, a connected and simply connected 2-dimensional simplicial complex X is systolic if and only if it is a so called C(3) − T (6) simplicial complex [22]. For example, buildings of type Ã_{2}, viewed as simplicial complexes, are systolic.

5. Using the combinatorial Gauss-Bonnet theorem, one sees that a triangulation of the 2-sphere is never k-large, for any k ≥ 6. It follows that no triangulation of a manifold M with dim M ≥ 3 is 6-large, since 2spheres occur as links of some simplices in M.

6. As we show later in this paper, for any k ≥ 6 there exist k-large simplicial pseudomanifolds in any dimension (see the next example). Moreover, any finite simplicial pseudomanifold admits a finite k-large branched cover, for any k ≥ 6.

7. We briefly describe an example of a 3-dimensional systolic pseudomanifold, which is the simplest new example obtained with the method presented later in this paper. For any integer m ≥ 3 consider the Coxeter group

W_{m} = ⟨s_{1}, s_{2}, s_{3}|s_{1}^{2}, s_{2}^{2}, s_{3}^{2}, (s_{1}s_{2})^{m}, (s_{1}s_{3})^{m}, (s_{2}s_{3})^{m}⟩

and its Coxeter complex Σ_{m} being the triangulation of the plane with 2m triangles around every vertex. By residual finiteness of W_{m}, there is a torsion free normal subgroup N < W_{m} of finite index, such that the quotient surface N\Σ_{m} is 2m-large. We show that there exists a simply connected 3-dimensional pseudo-manifold X = X(m, N) with all vertex links isomorphic to N\Σ_{m}. It is obtained as the universal development of the 3-simplex of groups with groups Z_{2} at 2-faces, with dihedral groups D_{m} (of order 2m) at edges, and with the quotient groups W_{m}/N at vertices.

Developability of this simplex of groups is addressed later in this paper, see Theorem 4.

8. There is a characterization of finite ∞-large simplicial complexes, due to G. A. Dirac [8]. It says that the class of finite ∞-large simplicial complexes is precisely the smallest class C such that:

(a) single simplex of arbitrary dimension belongs to C;

(b) if X is obtained from some complexes X_{1}, X_{2} ∈ C by gluing them along a single simplex, then X ∈ C. The proof of this fact may be also found in [11].

Proof (of Lemma 2). — We introduce a class of complexes and maps more general than simplicial ones. An almost simplicial 2-complex is a cell complex whose cells are simplices glued to lower dimensional skeleta through nondegenerate maps. More precisely, we allow multiple edges and loops in the 1-skeleton, and we require that the interior of each boundary edge of a 2-cell is glued to the 1-skeleton homeomorphically on the interior of some 1-cell. A simplicial map from an almost simplicial 2-complex to a simplicial complex is determined by its values at the vertices in the same way as an ordinary simplicial map (for example, a loop is necessarily mapped to a vertex).

Suppose γ is a closed embedded (contractible) polygonal curve in a simplicial complex X, and suppose f_{0} : Δ_{0} → X is a simplicial filling of γ. We will first modify it to a nondegenerate simplicial filling f′_{0} : Δ′_{0} → X with Δ′_{0} almost simplicial. This will be done in a sequence of modifications as follows. Suppose e is an edge in Δ_{0} which is mapped by f_{0} to a vertex. Then there are two 2-cells in Δ_{0} adjacent to e. Delete (the interior of the union of) these two cells from Δ_{0} and glue the four resulting free edges in pairs, so that the two distinct vertices of e are identified. This gives an almost simplicial disc Δ′ with the simplicial map f′ to X induced from f_{0} (and is the reason for introducing almost simplicial triangulations).

We wish to repeat the same modification procedure with the new triangulation, but now, due to the fact that the triangulation is almost simplicial, we need to consider two more cases.

The first possibility is that e is a loop. It then bounds a sub-disc D of Δ′. There is also a 2-cell C outside D adjacent to e. If all the edges of C are loops, then we have a nested family of discs bounded by them; take e^{∗} to be the outermost loop and repeat the argument with e^{∗} in place of e. Eventually we arrive at the situation where the two remaining edges of C are embedded. Now delete from Δ′ the interior of the union of D and C, and glue the two resulting free edges to get a new almost simplicial disc Δ′ with the induced simplicial map f′ to X.

The second possibility is that e is adjacent on both sides to the same 2-cell C of Δ′. Then e is not a loop, and plays the role of two out of three boundary edges of C. The remaining third edge is necessarily a loop; thus we are in the situation as in the previous case, and we perform the modification as above.

Since a modification reduces the number of 2-cells in Δ′, we eventually obtain an almost simplicial filling f′_{0} : Δ′_{0} → X which is nondegenerate (since it is nondegenerate on the 1-skeleton of Δ′_{0}).

The next step is to further modify the filling so that it remains nondegenerate but becomes simplicial. Note that, since f′_{0} is nondegenerate, Δ′_{0} has no loop edges. It is then sufficient to eliminate multiple edges (i.e. edges sharing both endpoints), while keeping induced maps to X nondegenerate, as an almost simplicial disc without loops and multiple edges is simplicial.

Consider a pair e_{1}, e_{2} of edges in Δ′_{0} with common endpoints. Their union bounds a subdisc D of Δ′_{0}. Remove the interior of D from Δ′_{0} and glue the resulting two free edges with each other, getting new Δ′_{0} with new nondegenerate simplicial map f′ to X induced from the previous one. Again, the procedure terminates, since the number of 2-cells in Δ′_{0} decreases. The final result f_{1} : Δ_{1} → X is a nondegenerate simplicial filling, as required.

Notice that the procedure we describe does not change the map f on the boundary.

Proof (of Lemma 3). — Take a filling produced in Lemma 2 and suppose v is an interior vertex of Δ_{1} contained in less than k triangles. First we shall construct a filling f′_{1} : Δ′_{1} → X of γ, with Δ′_{1} having one less interior vertex than Δ_{1}. We delete the interior of subdisc Res(v, Δ_{1}), replace it with some triangulation given by Lemma 1, and define f′_{1} so that it coincides with f_{1} on Δ_{1} \ int[Res(v, Δ_{1})].

The resulting filling is in general not nondegenerate, but the triangulation does have fewer simplices. Now we apply to it procedure used in the proof of Lemma 2, which produces a nondegenerate simplicial map with still fewer simplices.

Iteration of this procedure terminates after finitely many steps yielding a simplicial disc Δ_{2} and a map f_{2} : Δ_{2} → X which establishes the first part of Lemma 3.

Now, each boundary vertex is contained in at least 2 triangles and there is at least one interior vertex, since otherwise the boundary ∂Δ_{2} is not full in Δ_{2} and thus γ is not full in X. This completes the proof of Lemma 3.

3. 7-systolic implies hyperbolic.

One of the main themes of this paper is that k-systolic complexes with k ≥ 6 resemble to a large extent CAT (0) spaces, though there are no obvious CAT (0) metrics on them. As a first step in this direction we show in this section that 7-systolic complexes and groups are hyperbolic in the sense of Gromov. This solves a problem pointed out by M. Gromov [13, Remark (a) on p. 176] to find a purely combinatorial condition for simplicial complexes of arbitrary dimension that yields hyperbolicity. For an exposition of the theory of hyperbolic metric spaces and groups see [4,10].

13. Theorem. — Let X be a 7-systolic simplicial complex. Then the 1-skeleton X^{(1)} of X with its standard geodesic metric is hyperbolic in the sense of Gromov. More precisely, any geodesic triangle in X^{(1)}, with vertices at vertices of X, is δ-thin with δ = \frac{5}{2}.

2. Remark. — Note that, strictly speaking, to prove hyperbolicity of a graph one needs to show uniform thinness of all geodesic triangles, not only those with vertices at vertices of the graph. However, uniform thinness of the latter triangles, say with constant δ, easily implies uniform thinness of geodesic m-gons with m ≤ 6 and with vertices at vertices of the graph, with constant 4δ. (By thinness of an m-gon we mean that any of its edges remains close to the union of all other edges.) This is true since we can ”decompose” an m-gon into m−2 ≤ 4 geodesic triangles, without adding new vertices. Since an arbitrary geodesic triangle in a graph determines a geodesic m-gon as above, with edges of length 1 near some vertices of the triangle, it inherits thinness from this m-gon, with constant 4δ + 1. In particular, the 1-skeleton of a 7-systolic complex is δ-hyperbolic with δ = 11. This estimate is by no means optimal.

Since a 7-systolic group is quasi-isometric to (the 1-skeleton of) the corresponding 7-systolic simplicial complex on which it acts, Theorem 13 implies the following.

2. Corollary. — A 7-systolic group is word-hyperbolic

Proof (of Theorem 13). — Take any three vertices x, y, z in X, and join them by three geodesics γ_{xy}, γ_{xz}, γ_{yz} in X^{(1)} to obtain a triangle γ. We need to show that every point on the side γ_{xy} is distance at most \frac{5}{2} in X^{(1)} from the union of the remaining two sides.

Clearly γ_{xy} is embedded. Without loss of generality we can assume that γ is embedded (i.e. geodesics γ_{xy}, γ_{xz}, γ_{yz} intersect only at their endpoints) in view of the following

4. Lemma. — Suppose that two vertices x, y ∈ X are joined by two geodesics γ_{xy}, γ_{xy}^{∗} in X^{(1)}. Then for any vertex a on γ_{xy} there is a vertex a^{∗} on γ_{xy}^{∗}, so that a, a^{∗} are joined by an edge in X. In particular, any point on the geodesic γ_{xy} is distance at most \frac{3}{2} in X^{(1)} from the geodesic γ_{xy}^{∗}.

Proof (of Lemma 4). — Without loss of generality we can assume γ_{xy}, γ_{xy}^{∗} are disjoint (except at the endpoints). Lemma 2 produces a filling in X of the digon formed by γ_{xy}, γ_{xy}^{∗}, so that each vertex on the boundary is contained in at least 2 triangles, possibly with the exception of x, y. Suppose Lemma 4 is false. Then (the filling of) the digon has at least one internal vertex.

Apply the Gauss-Bonnet formula as in the proof of Proposition 1 to the digon. In the first sum at most two terms can be equal 2; the second sum is strictly negative. Thus, if there are k negative terms in the first sum, there are also at least k + 3 positive terms. Hence on one of the geodesics, say γ_{xy}, there are n vertices with negative contribution to the Gauss-Bonnet sum and at least n + 2 vertices with positive contribution. Thus negative vertices cannot separate positive ones, and we have two positive vertices that are either consecutive or separated only by several zero vertices (i.e. vertices with zero contribution to the Gauss-Bonnet sum). But this contradicts the fact that γ_{xy} is a geodesic in X^{(1)}, hence the lemma.

Coming back to the proof of Theorem 13, take a filling of γ in X constructed as in Lemma 2. The domain of the filling map is a disc Δ triangulated so that each vertex in the interior is contained in at least 7 triangles, and each vertex at the boundary, with possible exception of points x, y, z, is contained in at least two triangles.

A vertex on the boundary is called positive, negative or zero vertex if 3−χ(v) is positive, negative or zero respectively. Let p (respectively n) denote the number of positive (respectively negative) vertices other than x, y, z at the boundary ∂Δ. Since γ_{xy} is a geodesic in X^{(1)}, any two positive vertices in the interior of γ_{xy} are separated by a negative one. Apply the Gauss-Bonnet formula to the disc Δ. The three terms in the first sum corresponding to the vertices x, y, z are ≤ 2 and the remaining terms of this sum are at most 1. If the second sum is ≤ −4 then p ≥ n + 4, and hence on one of the sides of the triangle there are at least 2 more positive vertices than negative. Thus there are two positive vertices which are not separated by a negative one, a contradiction. Hence Δ has at most three internal vertices.

We claim that a geodesic triangle γ having a filling Δ as above with at most three internal vertices is \frac{5}{2}-thin To prove this, take first a vertex v on γ_{xy} whose distance from both x, y is bigger than 2. If its distance in Δ^{(1)} from the union of remaining sides is also bigger than 2, there are at least 6 vertices in Δ which are distance 2 from v. Only two of these vertices are on γ_{xy}, so at least four of them are internal in Δ, a contradiction. Thus the distance of v from the remaining two sides is at most 2. It follows easily that the distance in Δ^{(1)} of any point on the side γ_{xy} from the union of remaining two sides is at most \frac{5}{2}.

Triangles in the range of the filling map are thinner than in the source, which concludes the proof.

4. 3-convexity in simplicial complexes.

In this section we introduce the notion of 3-convexity and study its basic properties. It is inspired by the notion of convexity in spheres, or by the notion of π-convexity in CAT (1) spaces (compare Fact 10). The notion will play the key role in our later developments.

Given a simplicial complex X and its subcomplex Q, a cycle in the pair (X, Q) is a polygonal path γ in the 1-skeleton of X with endpoints contained in Q and without self-intersections, except a possible coincidence of the endpoints . A cycle γ as above is full in (X, Q) if its simplicial span in X is contained in the union γ ∪ Q. A subcomplex Q in a simplicial complex X is 3-convex if Q is full in X and every full cycle in (X, Q) of length less than 3 (i.e. consisting of less than 3 edges) is contained in Q.

3. Remark. — 3-convexity can be expressed equivalently as follows. A subcomplex Q in a simplicial complex X is 3-convex if and only if it is full and, given any geodesic γ of length 2 in the 1-skeleton of X, with both endpoints in Q, the mid-point of γ (and thus the whole of γ) is also contained in Q.

2. Example. — Examples of 3-convex subcomplexes.

1. Each simplicial complex X is 3-convex in itself.

2. Let X be a flag simplicial complex. Then any simplex Δ ⊂ X is 3-convex in X. To see this, consider a cycle γ of length 2 in (X, Δ), with its mid-vertex not in Δ. We need to show that γ is not full in (X, Δ). Since both endpoints of γ are in Δ, they are connected with an edge of X. Since X is flag, this implies that the three vertices of γ span a 2simplex of X. Since this 2-simplex is not a face of Δ, the simplicial span of γ is not contained in γ ∪ Δ, which completes the proof.

3. Let X be a 5-large simplicial complex. Then the residue Y = Res(σ, X) of any simplex σ is 3-convex in X. Indeed, Y is a full subcomplex of X, because X is flag. Moreover, any cycle of length 2 in (X, Y), with its mid-vertex not in Y, and with the endpoints connected with an edge of X, is not full in (X, Y) by the argument as in the previous example. Thus it remains to exclude existence of a cycle γ of length 2 in (X, Y), not contained in Y, with the endpoints not connected with an edge of X. Since the endpoints p, q of such γ are at distance 2 and contained in the residue of σ, they both are not in σ. Moreover, there is a vertex w of σ at distance 2 from the mid-vertex m of γ, since otherwise (due to flagness of X) m is in the residue. But then we get a full cycle of length 4 in X, passing through vertices p, m, q, w, p, which contradicts 5-largeness.

4. By a clique in a flag simplicial complex X we mean the subcomplex spanned by any set of vertices that are pairwise connected with edges of X. The clique spanned by a finite set is a simplex, and no other cliques occur in finite dimensional complexes. An infinite dimensional complex may contain an infinite clique, i.e. the one spanned by an infinite vertex set. Clearly, infinite cliques are not simplices, though some of their properties are analogous to those of simplices. For example, the argument as in (2) above shows that any clique in a flag simplicial complex X is 3-convex in X.

Facts 2-4 below follow easily from the definitions.

2. Fact. —

1. The intersection of any family of 3-convex subcomplexes is a 3convex subcomplex.

2. If Q is 3-convex in X and L is 3-convex in Q then L is 3-convex in X.

3. Let X be a flag simplicial complex and Q its 3-convex subcomplex. Then for any simplex σ of Q the link Q_{σ} is 3-convex in the link X_{σ}.

A subcomplex Q is locally 3-convex in X if for every nonempty simplex σ of Q the link Q_{σ} is 3-convex in the link X_{σ}. Note that this definition allows equalities Q_{σ} = X_{σ} at simplices σ of Q.

3. Fact. —

1. Any 3-convex subcomplex in a flag complex X is a locally 3-convex subcomplex in X.

2. The intersection of any family of locally 3-convex subcomplexes is a locally 3-convex subcomplex.

We now turn to convexity properties in k-large and locally k-large complexes. Since a full subcomplex of a k-large complex is k-large, we have

4. Fact. —

1. A 3-convex subcomplex of a k-large simplicial complex is k-large.

2. A locally 3-convex subcomplex in a locally k-large simplicial complex is locally k-large.

A cycle γ is homotopically trivial in (X, Q) if there is a path η in Q connecting the endpoints of γ such that the loop γ ∪ η is contractible in X. In this definition we allow that η reduces to a point (when the endpoints of γ coincide). A cycle is homotopically non-trivial if it is not homotopically trivial. Note that a cycle connecting distinct components of Q is always homotopically non-trivial.

A relative homotopical systole for the pair (X, Q) of a simplicial complex and its subcomplex, denoted sys_{h}(X, Q), is the length of the shortest homotopically non-trivial cycle in (X, Q). The next proposition shows that in locally 6-large simplicial complexes, 3-convexity can be characterized in terms of local 3-convexity and the relative homotopical systole.

2. Proposition. — Let X be a locally 6-large simplicial complex and let Q be a full subcomplex of X.

1. If Q is locally 3-convex in X and sys_{h}(X, Q) ≥ 3 then Q is 3convex.

2. The converse implication holds provided X is flag.

Proof. — To prove (2), take the shortest cycle γ homotopically nontrivial in (X, Q) and note that it intersects Q only at its endpoints. The length |γ| of γ cannot be 1 since Q is full. If |γ| = 2 then γ is not full in (X, Q) due to 3-convexity of Q. Then either the endpoints of γ span an edge not contained in Q, which contradicts the fullness of Q, or otherwise the three vertices of γ span a 2-simplex in X, contradicting the fact that γ is homotopically nontrivial in (X, Q). Hence sys_{h}(X, Q) ≥ 3. Since X is flag and Q is 3-convex, it is also locally 3-convex (Fact 4.1), and part (2) follows.

To prove part (1), suppose we have a length d full cycle γ in (X, Q), intersecting Q only at its endpoints. We have to prove that d ≥ 3. If γ is homotopically nontrivial in (X, Q) we are done, since sys_{h}(X, Q) ≥ 3. We therefore assume that γ is homotopically trivial in (X, Q). Thus there is a polygonal path η contained in Q, with the same endpoints as γ, such that the union γ ∪ η is a contractible loop in X. Moreover, η can be chosen so that it is of minimal length. In particular the closed polygonal path γ ∪ η is embedded in X. Beware that if the endpoints of γ coincide then η reduces to a single vertex.

By Lemma 2, there is a simplicial disc D filling the loop γ ∪ η in X. Among all choices of η and D, we pick one for which D has the smallest number of triangles (that may affect the choice of η). By Lemma 3 the interior vertices of D are contained in at least 6 triangles of D.

Every interior vertex of γ (viewed as the boundary vertex of D) is contained in at least two triangles of D, since γ is full in (X, Q). Every interior vertex v of η (viewed as the boundary vertex of D) is contained in at least 3 triangles of D. Indeed, if v is contained in one triangle of D, (the image of) the triangle is in Q (since Q is full), and η is not of minimal length. If v is contained in two triangles of D, they are both in Q by local 3-convexity and by minimality of η, and then D is not minimal. Finally, initial and terminal vertices of γ (which may coincide) are contained in at least one triangle.

Denote, as in Section 2, by χ(v) the number of triangles in D containing v. Let I, G, E denote the sets of interior vertices in D, γ and η respectively. Suppose that the endpoints of γ do not coincide, and denote them by a, b. Applying the inequalities we just established and the Gauss-Bonnet theorem we get

1 = \frac{1}{6}[ ∑_{v∈I}(6 − χ(v)) + ∑_{v∈G}(3 − χ(v)) + ∑_{v∈E}(3 − χ(v)) + 3 − χ(a) + 3 − χ(b)]

≤ \frac{1}{6}(0 + d − 1 + 0 + 4).

Thus 3 ≤ d as required.

Dealing similarly with the remaining case, in which the endpoints of γ coincide, we get even sharper estimate 4 ≤ d. Hence the Proposition.

Proposition 2 allows to decide inductively if a subcomplex in a 6-large complex is 3-convex, by referring to 3-convexity of its links. The next three results apply this idea and give some criteria for 3-convexity. By diameter of a complex we mean the maximum distance between its vertices in the 1-skeleton of the complex.

5. Lemma. — Let Q be a full locally 3-convex subcomplex in a 6-large complex X and suppose that Q is connected and diam(Q) ≤ sys_{h}(X) − 3. Then Q is 3-convex in X.

Proof. — With Proposition 2, it suffices to prove that sys_{h}(X, Q) ≥ 3. Let γ be a homotopically nontrivial cycle in (X, Q). By the assumptions of the lemma, there is a polygonal path η of length ≤ sys_{h}(X) − 3 contained in Q and with the same endpoints as γ. Moreover, the closed path γ ∪ η is homotopically nontrivial in X, and thus the length of this path is at least sys_{h}(X) by Corollary 1. But this means that the length of γ is at least 3, which finishes the proof.

6. Lemma. — Let Q be a full connected finite dimensional subcomplex in a 6-large simplicial complex X. Suppose that diam(Q) ≤ sys_{h}(X) − 3 and that for each simplex σ of Q either Q_{σ} = X_{σ} or Q_{σ} is connected with diam(Q_{σ}) ≤ 3. Then Q is 3-convex in X.

Proof. — Induction over the dimension of Q using Lemma 5.

Lemmas 5 and 6 immediately imply the following corollary in which a subcomplex Q may have infinite dimension.

3. Corollary. — Let Q be a full connected subcomplex in a 6-large simplicial complex X. Suppose that diam(Q) ≤ sys_{h}(X) − 3 and that for each simplex σ of Q either Q_{σ} = X_{σ} or Q_{σ} is a finite dimensional connected complex with diameter diam(Q_{σ}) ≤ 3. Then Q is 3-convex in X.

5. Locally 3-convex maps and their applications.

In this section we introduce the concept of a locally 3-convex map. We also use this concept to prove that finite dimensional 6-large simplicial complexes are aspherical, or equivalently, finite dimensional systolic complexes are contractible Theorem 14.1, or Theorem 2 of Introduction). This result may be viewed as an analogue (for simplicial nonpositive curvature) of the Cartan-Hadamard theorem.

Although the notion of a locally 3-convex map makes sense in infinite dimensional case, its application in our arguments requires the additional assumption about finite dimension.

Given a nondegenerate simplicial map f : Q → X and a simplex σ ∈ Q, the induced map on links f_{σ} : Q_{σ} → X_{f(σ)} is the map obtained by restricting f to the link Q_{σ} (the image of this restriction is necessarily contained in the link X_{f(σ)}). We will say that a nondegenerate simplicial map f : Q → X is locally injective, if for any simplex σ ⊂ Q the induced map f_{σ} is injective. Let X be a locally 6-large simplicial complex and Q an arbitrary simplicial complex. A nondegenerate locally injective simplicial map f : Q → X is locally 3-convex, if for each simplex σ ⊂ Q the image f_{σ}(Q_{σ})) is 3-convex in X_{f(σ)} (in particular, f_{σ}(Q_{σ}) may be the whole of X_{f(σ)}). Note that if Q ⊂ X is a locally 3-convex subcomplex then the inclusion map is clearly locally 3-convex.

14. Theorem. — Let X be a finite dimensional connected locally 6-large simplicial complex.

1. The universal cover X̃ of X is contractible. In particular, any finite dimensional systolic simplicial complex is contractible.

2. Suppose that Q is a connected simplicial complex and f : Q → X is a locally 3-convex simplicial map. Then the induced homomorphism f_{∗} : π_{1}Q → π_{1}X of fundamental groups is injective.

To prove Theorem 14 we will use the fact that locally 3-convex maps can be extended to covering maps. We formulate this fact more precisely as Proposition 3 below, and then show how it implies the theorem. The proof of Proposition 3 occupies the last part of this section and it uses a technical result, Lemma 7, the proof of which we defer until Section 6.

3. Proposition. — Suppose that X is a finite dimensional locally 6large simplicial complex and let f : Q → X be a locally 3-convex map. Then f extends to a covering map f_{e} : Q_{e} → X in such a way that Q is a deformation retract of Q_{e}.

Proof (of Theorem 14). — A function f : {v} → X that sends a vertex v to a vertex of X is clearly locally 3-convex. By Proposition 3, it extends to a covering map f_{e} : Y → X, where Y is contractible. This proves part (1).

To prove (2), note that by Proposition 3 the map f extends to a covering map f_{e} : Q_{e} → X such that the inclusion Q ⊂ Q_{e} is a homotopy equivalence. Since a covering map induces a monomorphism of fundamental groups, the theorem follows.

The proof of Proposition 3 requires some preparations. Given a locally 3-convex map f : Q → X, define

∂_{f}Q := {σ ∈ Q | f_{σ} : Q_{σ} → X_{f(σ)} is not an isomorphism},

and observe that ∂_{f}Q is a simplicial subcomplex of Q. ∂_{f}Q can be thought of as a kind of boundary of Q relative to f, hence the notation. For a subcomplex K of a simplicial complex L, denote by N_{L}(K) the subcomplex of L being the union of all (closed) simplices that intersect K.

A small extension of a locally 3-convex map f : Q → X is a map Ef : EQ → X satisfying the following conditions:

(E1) is a simplicial complex containing Q as a subcomplex and N_{EQ}(Q) = EQ;

(E2) is a nondegenerate simplicial map that extends f;

(E3) for each simplex τ ∈ EQ that intersects Q the map (Ef)_{τ} : (EQ)_{τ} → X_{f(τ)} is an isomorphism;

(E4) is locally 3-convex;

(E5) is a deformation retract in EQ.

7. Lemma. — Every locally 3-convex map f : Q → X to a finite dimensional locally 6-large simplicial complex X admits a small extension.

We defer the proof of the lemma until Section 6 but show now how it implies Proposition 3.

Proof (of Proposition 3). — Put E^{0}f = f and E^{0}Q = Q. Define recursively a sequence of small extensions E^{j}f : E^{j}Q → X by E^{j+1}Q = E(E^{j}Q) and E^{j+1}f = E(E^{j}f). Put Q_{e} := ⋃_{j=0}^{∞} E^{j}Q and f_{e} := ⋃_{j=0}^{∞} E^{j}f, thus getting a map f_{e} : Q_{e} → X. Since by property (E3) of a small extension the induced map (f_{e})_{τ} : (Q_{e})_{τ} → X_{f_{e}(τ)} is an isomorphism for each simplex τ ∈ Q_{e}, it follows that f_{e} is a covering map. By property (E5), Q is contained in Q_{e} as a deformation retract, hence the proposition.

6. Existence of small extensions.

This section is entirely devoted to the proof of Lemma 7.

We start with some definitions and notation. Given a finite dimensional locally 6-large simplicial complex X and a locally 3-convex map f : Q → X, define the following family of pairs of simplices

E_{f} := {(σ, τ) ∈ ∂_{f}Q × X : τ ⊂ X_{f(σ)}, τ ∩ f_{σ}(Q_{σ}) = ∅}.

The motivation for considering the family E_{f} is as follows. Suppose we are given a small extension Ef : EQ → X of f. Then to any pair (σ, τ) ∈ E_{f} there corresponds a simplex (Ef)_{σ}^{−1}(τ) ∈ (EQ)_{σ} ⊂ EQ, which we denote shortly τ^{σ}. Moreover, we have Ef(τ^{σ}) = τ. This shows that pairs from E_{f} represent “germs” of the extension of f to Ef. In fact, we will construct a small extension Ef making use of the set E_{f}. For this we also need the smaller family

E_{f}^{max} := {(σ, τ) ∈ E_{f} : there is no ρ ⊃ σ with (ρ, τ) ∈ E_{f}}.

As we will see later, the elements of the set E_{f}^{max} will correspond bijectively, through the map (σ, τ) → τ^{σ}, to the simplices disjoint with Q in the constructed small extension domain EQ.

The next lemma collects basic properties of the families E_{f} and E_{f}^{max}.

8. Lemma. —

1. If (σ, τ) ∈ E_{f} and ρ ⊂ σ then (ρ, τ) ∈ E_{f}.

2. If (σ_{i}, τ) ∈ E_{f} for i = 1, 2 and σ_{1} ∩ σ_{2} ≠ ∅ then there is σ ∈ Q containing both σ_{1} and σ_{2} such that (σ, τ) ∈ E_{f}.

3. If (σ_{i}, τ) ∈ E_{f}^{max} for i = 1, 2 and if σ_{1} ≠ σ_{2} then σ_{1} ∩ σ_{2} = ∅.

4. Given (σ, τ) ∈ E_{f}, there exists a unique simplex π_{σ,τ} ⊂ ∂_{f}Q such that σ ⊂ π_{σ,τ} and (π_{σ,τ}, τ) ∈ E_{f}^{max}.

5. If (σ, τ) ∈ E_{f} and ρ ⊂ σ then π_{ρ,τ} = π_{σ,τ}.

In the proofs of Lemma 8 and of the remaining results in this section we will often use the following. Notation.

1. Given a simplex σ and its face ρ, we denote by σ − ρ the face of σ spanned by all the vertices not contained in ρ.

2. Given simplices σ, τ in a simplicial complex K, denote by σ ∗ τ the simplex of K spanned by the union of the vertex sets of σ and τ. Note that in general such a simplex in K may not exist. We will speak of simplices of this form only when they exist.

Proof (of Lemma 8). — To prove (1), consider first the case when τ is a 0-simplex (i.e. a vertex). Let (σ, v) ∈ E_{f}, where v is a vertex, and let ρ ⊂ σ. If (ρ, v) ∉ E_{f}, it follows that v ∈ f_{ρ}(Q_{ρ}). We also have σ − ρ ⊂ f_{ρ}(Q_{ρ}), because σ ⊂ Q. On the other hand, the simplex f(σ − ρ) ∗ v ⊂ X_{f(ρ)} is not contained in f_{ρ}(Q_{ρ}), because the simplex f(σ) ∗ v ⊂ X is not contained in f(σ) ∗ f_{σ}(Q_{σ}). This contradicts fullness of f_{ρ}(Q_{ρ}) ⊂ X_{f(ρ)} (which holds by local 3-convexity of f). Thus the assertion follows in this case.

To deal with the other cases, suppose now that (σ, τ) ∈ E_{f} and dim τ ≥ 1. For any vertex v of τ we clearly have (σ, v) ∈ E_{f}. It follows from what we have just proved for vertices that if ρ ⊂ σ then v ∉ f_{ρ}(Q_{ρ}) for any vertex v ∈ τ. Then clearly τ ∩ f_{ρ}(Q_{ρ}) = ∅ and thus (ρ, τ) ∈ E_{f}. This finishes the proof of (1).

To prove (2), we first show that the union of the vertices of σ_{1} and σ_{2} spans a simplex of Q. Put ρ = σ_{1} ∩ σ_{2}. Since Q_{ρ} is flag (because the isomorphic complex f_{ρ}(Q_{ρ}) is 3-convex, and hence full, in X_{f(ρ)} which is 6large and hence flag), it is sufficient to show that there is an edge in Q_{ρ} between any two vertices v_{1} ∈ σ_{1} − ρ and v_{2} ∈ σ_{2} − ρ. For an arbitrary vertex t ∈ τ we get polygonal path f(v_{1})tf(v_{2}) in (X_{f(ρ)}, f_{ρ}(Q_{ρ})), intersecting f_{ρ}(Q_{ρ}) only at its endpoints. By 3-convexity of f_{ρ}(Q_{ρ}) in X_{f(ρ)}, this curve cannot be full in (X_{f(ρ)}, f_{ρ}(Q_{ρ})), and hence there is an edge in X_{f(ρ)} between f(v_{1}) and f(v_{2}). By the fact that f_{ρ}(Q_{ρ}) is full in X_{f(ρ)}, this edge is in f_{ρ}(Q_{ρ}), and thus v_{1}v_{2} is an edge in Q_{ρ}.

Let σ be the simplex of Q spanned by the union of σ_{1} and σ_{2}. We now show that τ ∈ X_{f(σ)} or equivalently that f(σ) and τ span a simplex of X. For this it is sufficient to show that the three simplices τ, f(σ − σ_{1}) and f(σ − σ_{2}) span a simplex of X_{f(ρ)}. The latter follows from the fact that X_{f(ρ)} is flag (since X is locally 6-large) and from the easy observation that the three simplices span the simplices of X_{f(σ_{1}∩σ_{2})} pairwise.

It remains to show that τ ∩ f_{σ}(Q_{σ}) = ∅, but this follows from the inclusion f_{σ}(Q_{σ}) ⊂ f_{σ_{1}}(Q_{σ_{1}}) and the assumption that (σ_{1}, τ) ∈ E_{f}. Thus we get (σ, τ) ∈ E_{f}, which completes the proof of (2).

Part (3) is a direct consequence of part (2). In view of the assumption that X is finite dimensional, (4) and (5) follow easily from (3).

We now start the construction of a small extension. Together with verification of conditions (E1)-(E5) from the definition, this construction occupies the rest of this section.

Simplicial complex EQ. As the vertex set of EQ take the (disjoint) union of the vertex set of Q and the set {(σ, v) ∈ E_{f}^{max} : v is a vertex}. For any pair (σ, τ) ∈ E_{f} let δ_{σ,τ} be the simplex spanned by the set consisting of all vertices in σ and all vertices of form (π_{σ,t}, t), where t is a vertex of τ. Define EQ to be the union of Q and the simplices δ_{σ,τ} for all (σ, τ) ∈ E_{f}.

It is immediate from the above description that Q ⊂ EQ and N_{EQ}(Q) = EQ, i.e. that the constructed complex EQ satisfies condition (E1) in the definition of a small extension. The next fact collects some more detailed properties of the complex EQ, useful for later arguments in this section.

5. Fact. —

1. The simplices of EQ with all vertices in Q are exactly the simplices of Q. In other words, Q is a full subcomplex in EQ.

2. The simplices of EQ with part of vertices in Q and part of vertices outside Q are exactly the simplices δ_{σ,τ} : (σ, τ) ∈ E_{f}. Moreover, for distinct pairs (σ, τ) ∈ E_{f} the corresponding simplices δ_{σ,τ} are distinct.

3. The simplices of EQ disjoint with Q are exactly the simplices δ_{σ,τ} − σ : (σ, τ) ∈ E_{f}^{max}.

4. If σ_{1} ⊂ σ_{2} and (σ_{i}, τ) ∈ E_{f} for i = 1, 2 then the corresponding simplices δ_{σ_{i},τ} − σ_{i} coincide.

5. For distinct pairs (σ, τ) ∈ E_{f}^{max} the corresponding simplices δ_{σ,τ} − σ are distinct. Moreover, if (σ_{i}, τ) ∈ E_{f}^{max} for i = 1, 2 and σ_{1} ≠ σ_{2} (which by Lemma 8.3 means that these simplices σ_{i} are disjoint) then the corresponding simplices δ_{σ_{i},τ} are also disjoint.

6. Complex EQ is the union of Q and the family of (closed) simplices δ_{σ,τ} : (σ, τ) ∈ E_{f}^{max}.

Proof. — All parts except (5) follow easily from the description of EQ. To prove (5), suppose that (σ_{i}, τ_{i}) : i = 1, 2 are distinct pairs from E_{f}^{max}. If τ_{1} ≠ τ_{2} then the sets of vertices of the simplices δ_{σ_{i},τ_{i}} − σ_{i} : i = 1, 2 are easily seen to be distinct. If τ_{1} = τ_{2} then σ_{1} ≠ σ_{2}, and we are in the assumptions of the second statement in (5). Since we know that then σ_{1} ∩ σ_{2} = ∅, it is sufficient to show that the simplices δ_{σ_{i},τ_{i}} − σ_{i} are disjoint for i = 1, 2. For brevity, put τ := τ_{1} = τ_{2}, and let t ∈ τ be a vertex. We will show that the vertex (π_{σ_{2},t}, t) ∈ δ_{σ_{2},τ} − σ_{2} is not a vertex of the simplex δ_{σ_{1},τ} − σ_{1}, which is clearly sufficient for completing the proof of (5). The vertices in δ_{σ_{1},τ_{1}} − σ_{1} other than (π_{σ_{1},t}, t) are distinct from (π_{σ_{2},t}, t), since their projections to X differ from t. It thus remains to show that (π_{σ_{1},t}, t) ≠ (π_{σ_{2},t}, t), i.e. that π_{σ_{1},t} ≠ π_{σ_{2},t}. Suppose that the latter is not true and π_{σ_{1},t} = π_{σ_{2},t}. Then σ_{1} ∗σ_{2} is a simplex of ∂_{f}Q, since both σ_{1} and σ_{2} are contained in π_{σ_{1},t}. We then have f(σ_{1}) ∗ (τ − t) ⊂ X_{t}, f(σ_{2}) ∗ (τ − t) ⊂ X_{t} and f(σ_{1}) ∗ f(σ_{2}) = f(σ_{1} ∗ σ_{2}) ⊂ X_{t}. Since the link X_{t} is flag (because X is locally 6-large), it follows that f(σ_{1} ∗ σ_{2}) ∗ (τ − t) ⊂ X_{t}, and hence (σ_{1} ∗ σ_{2}, τ) ∈ E_{f}. This contradicts any of the assumptions (σ_{i}, τ_{i}) ∈ E_{f}^{max} thus completing the proof.

Simplicial map Ef : EQ → X. Define Ef by putting first Ef|_{Q} = f and Ef((σ, v)) = v for all vertices (σ, v), and then extending simplicially. Observe that since in this way the vertices of any simplex δ_{σ,τ} are mapped bijectively to the vertices of the simplex f(σ)∗τ ⊂ X, the simplicial map Ef : EQ → X is both well defined and nondegenerate, hence it fulfills condition (E2) of a small extension.

Passing to condition (E3), note that if g : K → L is a nondegenerate simplicial map, and if for some vertex v ∈ K the induced map g_{v} : K_{v} → L_{g(v)} is an isomorphism, then for any simplex σ ⊂ K containing v the map g_{σ} : K_{σ} → L_{g(σ)} is also an isomorphism. It is then sufficient to prove that (Ef)_{v} : (EQ)_{v} → X_{f(v)} is an isomorphism for any vertex v ∈ Q. This fact is immediate for all vertices v of Q not contained in ∂_{f}Q, since for them we have (EQ)_{v} = Q_{v} and (Ef)_{v} = f_{v}. It remains to prove this fact for vertices v ∈ ∂_{f}Q.

A nondegenerate simplicial map is an isomorphism if it is bijective on the vertex sets and surjective. We now check those two properties for the map (Ef)_{v} with any vertex v ∈ ∂_{f}Q.

Given v ∈ ∂_{f}Q, the simplices of EQ that contain v are either contained in Q or have a form δ_{σ,τ} with (σ, τ) ∈ E_{f} and v ∈ σ. Thus, the vertices of (EQ)_{v} are either contained in Q_{v} or are the vertices other than v in 1simplices δ_{v,w} (for all (v, w) ∈ E_{f} with w a vertex). The latter vertices are the vertices (π_{v,w}, w) ∈ E_{f}^{max}. Vertices of Q_{v} are mapped by (Ef)_{v} bijectively on the vertices of f_{v}(Q_{v}), while the vertices (π_{v,w}, w) are mapped bijectively to the vertices w ∈ X_{f(v)} not contained in f_{v}(Q_{v}). Thus the map (Ef)_{v} : (EQ)_{v} → X_{f(v)} is bijective on the vertex sets.

To prove surjectivity of the map (Ef)_{v}, choose any simplex ρ in the link X_{f(v)}. We need to show that ρ is in the image of (Ef)_{v}. If ρ ⊂ f_{v}(Q_{v}), there is nothing to show. Otherwise, put ρ_{0} := ρ ∩ f_{v}(Q_{v}). Since, by local 3convexity of f, f_{v}(Q_{v}) is a full subcomplex of X_{f(v)}, ρ_{0} is either empty or a single proper face of ρ. We then clearly have (v, ρ − ρ_{0}) ∈ E_{f}, and we deduce that (v ∗ f^{−1}(ρ_{0}), ρ − ρ_{0}) ∈ E_{f}. Since clearly Ef((δ_{v∗f^{−1}(ρ_{0}),ρ−ρ_{0}})) = ρ ∗ f(v), it follows that ρ is in the image of (Ef)_{v} as required.

Local 3-convexity of Ef. Since, according to (E3), the map (Ef)_{δ} : (EQ)_{δ} → X_{f(δ)} is an isomorphism for any simplex δ ⊂ EQ that intersects Q, the local 3-convexity condition for Ef is fulfilled at such simplices. Thus to establish (E4), it remains to check that for any simplex ρ in EQ disjoint with Q the induced map (Ef)_{ρ} : (EQ)_{ρ} → X_{f(ρ)} is injective and the subcomplex (Ef)_{ρ}((EQ)_{ρ}) is 3-convex in the link X_{Ef(ρ)}. For this we need the following.

9. Lemma. — Given a simplex ρ in EQ disjoint with Q, let ρ = δ_{σ,τ} −σ for the appropriate (σ, τ) ∈ E_{f}^{max} as in Fact 5.3). Then σ ⊂ (EQ)_{ρ} and N_{(EQ)_{ρ}}(σ) = (EQ)_{ρ}.

The proof of Lemma 9 requires the following. 5.4 Claim. Under assumptions of Lemma 9, the residue Res(ρ, EQ) is equal to the union U of the simplices δ_{σ_{0},τ_{0}} such that (σ_{0}, τ_{0}) ∈ E_{f}, σ_{0} ⊂ σ and τ ⊂ τ_{0}.

Proof. — The inclusion U ⊂ Res(ρ, EQ) is easy in view of Fact 5.4. To get the converse inclusion, denote by π an arbitrary simplex in EQ that contains ρ. By the construction of EQ, π is contained in a simplex δ_{σ′,τ′} for some (σ′, τ′) ∈ E_{f}. Looking at vertices not contained in Q in δ_{σ,τ} and δ_{σ′,τ′}, we conclude that τ ⊂ τ′. Then (σ′, τ) ∈ E_{f} and consequently (π_{σ′,τ}, τ) ∈ E_{f}^{max}. Since we have also (σ, τ) ∈ E_{f}^{max}, Lemma 8.3 implies that either π_{σ′,τ} = σ or π_{σ′,τ} ∩ σ = ∅. In the first of these two cases we have σ′ ⊂ π_{σ′,τ} = σ and thus ρ ⊂ δ_{σ′,τ′}, (σ′, τ′) ∈ E_{f}, τ ⊂ τ′ and σ′ ⊂ σ. Hence π ⊂ U. The case of π_{σ′,τ} ∩ σ = ∅ is in fact impossible, since if it holds then the argument as in the proof of the second statement in Fact 5.3 shows that the simplices δ_{σ,τ} and δ_{π_{σ′,τ} ,τ} are disjoint, and thus cannot both contain ρ. Hence the claim.

For later application in Section 8 we state here an observation immediately implied by Claim 5.4.

4. Corollary. — Under assumptions of Lemma 9, the intersection Res(ρ, EQ)∩ Q is equal to σ.

Proof (of Lemma 9). — A simplex δ_{σ_{0},τ_{0}} as in the claim determines the simplex δ_{σ_{0},τ_{0}} − ρ in the link (EQ)_{ρ}. The claim implies that (EQ)_{ρ} is the union of such simplices δ_{σ_{0},τ_{0}} − ρ. Since any such simplex shares a face with the simplex σ, namely the face σ_{0}, it follows that N_{(EQ)_{ρ}}(σ) = (EQ)_{ρ}, as expected.

We are now ready to prove that the map Ef is locally injective, a first step in showing its local 3-convexity. The next lemma establishes much stronger local property of Ef which will be referred to in later parts of the paper.

4. Proposition. — Given a simplex ρ = δ_{σ,τ} − σ with (σ, τ) ∈ E_{f}^{max}, the induced map (Ef)_{ρ} maps the link (EQ)_{ρ} isomorphically onto the subcomplex N_{X_{Ef(ρ)}}(f(σ)) in the link X_{Ef(ρ)}. In particular, this map is injective.

Proof. — The proof relies on the following general observation which we state without proof.

Claim. Let K be a simplicial complex, π ⊂ K a simplex, and suppose that N_{K}(π) = K. Furthermore, let L be a flag simplicial complex and h : K → L a nondegenerate simplicial map. If for any simplex α ⊂ π the induced map h_{α} : K_{α} → L_{h(α)} is an isomorphism, then h maps K isomorphically on the subcomplex N_{L}(h(π)).

We now check that putting K = (EQ)_{ρ}, π = σ, L = X_{Ef(ρ)} and h = (Ef)_{ρ}, all the assumptions in the claim are satisfied. The fact that N_{K}(π) = K follows from Lemma 9. The map h = (Ef)_{ρ} is nondegenerate because, by condition (E2), so is Ef. The complex L = X_{Ef(ρ)} is flag because X is locally 6-large. It remains to check the properties of the induced maps h_{α} = ((Ef)_{ρ})_{α} : ((EQ)_{ρ})_{α} → (X_{Ef(ρ)})_{(Ef)_{ρ}(α)}.

Observe that we have the identifications ((EQ)_{ρ})_{α} = (EQ)_{ρ∗α}, (X_{Ef(ρ)})_{(Ef)_{ρ}(α)} = X_{Ef(ρ∗α)} and ((Ef)_{ρ})_{α} = (Ef)_{ρ∗α}. The fact that ((Ef)_{ρ})_{α} is an isomorphism follows then from the already proved property (E3) for Ef, by realizing that the simplex ρ ∗ α intersects Q at α. Thus, by applying the claim, the proposition follows.

In order to prove that the map Ef is locally 3-convex it now remains to prove that, under notation of Proposition 4, the image complex (Ef)_{ρ}((EQ)_{ρ}) is 3-convex in the link X_{f(ρ)}. We do this by referring to Lemma 6. The proposition implies that the subcomplex (Ef)_{ρ}((EQ)_{ρ}) is connected and that diam[(Ef)_{ρ}((EQ)_{ρ})] ≤ 3. Since the links of the complex (Ef)_{ρ}((EQ)_{ρ}) are isomorphic to the complexes (Ef)_{ρ′}((EQ)_{ρ′}) for appropriate simplices ρ′ ⊃ ρ, it follows that (Ef)_{ρ}((EQ)_{ρ}) satisfies the assumptions of Lemma 6, which completes the proof of property (E4) for Ef.

Deformation retraction. Put

Q^{i} := Q ∪ ⋃{δ_{σ,τ} : (σ, τ) ∈ E_{f}^{max}, dim τ < i}.

Let dim X = n. Then the dimension of any simplex τ such that (σ, τ) ∈ E_{f}^{max} is not greater than n − 1. Consequently, by Fact 5.5, we get

Q = Q^{0} ⊂ Q^{1} ⊂ ⋯ ⊂ Q^{n} = EQ.

We will show that Q^{i} is a deformation retract of Q^{i+1} for i = 0, 1, …, n − 1, which clearly implies that Q is a deformation retract of EQ.

10. Lemma. — Let (σ, τ) ∈ E_{f}^{max} and dim τ = i. Then, denoting τ^{σ} = δ_{σ,τ} − σ, we have

1. δ_{σ,τ} ∩ Q^{i} = σ ∗ ∂τ^{σ}, where ∂τ^{σ} is the ordinary boundary subcomplex of the simplex τ^{σ};

2. δ_{σ,τ} \ Q^{i} is a connected component in Q^{i+1} \ Q^{i}.

Proof. — By definition, Q^{i} is a subcomplex of EQ consisting of all those simplices of EQ which have at most i vertices outside Q. Thus δ_{σ,τ} ∩Q^{i} consists of those faces of δ_{σ,τ} which have at most i vertices outside Q. Since δ_{σ,τ} = σ ∗ τ^{σ}, δ_{σ,τ} ∩ Q = σ and dim τ^{σ} = dim τ = i, this easily implies (1).

To prove (2), it is sufficient to show that for any (σ′, τ′) ∈ E_{f}^{max} with dim τ′ = i, distinct from (σ, τ), we have (δ_{σ,τ} \ Q^{i}) ∩ (δ_{σ′,τ′} \ Q^{i}) = ∅. Suppose this is not true and consequently δ_{σ,τ} ∩ δ_{σ′,τ′} is not contained in σ ∗ ∂τ^{σ}. Then τ^{σ} ⊂ δ_{σ,τ} ∩ δ_{σ′,τ′}, and in fact τ^{σ} has to be a face in (τ′)^{σ′}, because the vertices of τ^{σ} are all outside Q. Since dim τ^{σ} = dim(τ′)^{σ′} (they are both equal to i), we have τ^{σ} = (τ′)^{σ′}. In view of Fact 5.3 this implies that (σ, τ) = (σ′, τ′), which contradicts the assumption that these pairs are distinct. Thus the lemma follows.

To finish the proof that Q^{i} is a deformation retract of Q^{i+1} observe that, in view of Lemma 10.2, deformation retraction of Q^{i+1} onto Q^{i} can be composed out of independently performed deformation retractions of simplices δ_{σ,τ} (for (σ, τ) ∈ E_{f}^{max} and dim τ = i) onto their intersections with Q^{i}. The existence of the latter deformation retractions is implied by Lemma 10.1 and the elementary fact that σ ∗ ∂τ is a deformation retract of σ ∗ τ. Since this gives the last condition (E5) from the definition of a small extension, the proof of Lemma 7 is completed.

7. Locally 6-large simplicial complexes of groups.

In this section we sketch the necessary background for and the proof of the following.

15. Theorem. — Every connected, locally 6-large, finite dimensional simplicial complex of groups is developable.

Theorem 15 allows to construct locally 6-large simplicial complexes by means of complexes of groups. We will extensively exploit this possibility in our constructions in Sections 19-21.

The proof of Theorem 15 is based on a version of Proposition 3 for locally 3-convex maps to locally 6-large simplicial complexes of groups, and it is very similar to the proof of Theorem 14.1.

We refer the reader to [4] for details related to the notion of a complex of groups. We refer also to Section 18 of this paper for an easier exposition of the special case of simplex of groups.

For a simplicial complex X, let X be the scwol (small category without loops, as defined in [4, p. 520]) related to the barycentric subdivision of X, defined as follows. A vertex set V = V(X) of X consists of simplices σ of X and a set E = E(X) of directed edges of X consists of pairs a = (τ, σ) such that σ is a proper face of τ (i.e. σ ⊂ τ and σ ≠ τ).

A complex of groups G(X) = ({G_{σ}}, {ψ_{στ}}, {g_{στρ}}) over a simplicial complex X is given by the following data (cf. [4, p. 535, Definition 2.1]):

1. for each σ ∈ V a group G_{σ} called the local group at σ;

2. for each (τ, σ) ∈ E an injective homomorphism ψ_{στ} : G_{τ} → G_{σ};

3. for each triple σ ⊂ τ ⊂ ρ of simplices with σ ≠ τ ≠ ρ a twisting element g_{στρ} ∈ G_{σ} with the following compatibility conditions:

(i)  Ad(g_{στρ})ψ_{σρ} = ψ_{στ}ψ_{τρ},

where Ad(g_{στρ}) is the conjugation by g_{στρ} in G_{σ}, and

(ii)  ψ_{στ}(g_{τρπ})g_{στπ} = g_{στρ}g_{σρπ}.

for each σ ⊂ τ ⊂ ρ ⊂ π with σ ≠ τ ≠ ρ ≠ π.

4. Remark. — For many purposes (e.g. for our considerations in Sections 18-21) it is sufficient to deal with the so called simple complexes of groups, for which all the twisting elements are trivial. We may then speak of a complex of groups G(X) = ({G_{σ}}, {ψ_{στ}}) consisting of local groups G_{σ} and injective homomorphisms ψ_{στ}. Since the compatibility condition (i) reads then as ψ_{σρ} = ψ_{στ}ψ_{τρ}, we may view the homomorphisms ψ_{στ} as inclusions of subgroups.

Let G(X) be a complex of groups over a simplicial complex X, and let σ be a simplex of X. For any simplex τ ∈ X_{σ} put G_{τ}^{σ} := ψ_{σ(τ∗σ)}(G_{τ∗σ}) ⊂ G_{σ}. A link of G(X) at σ, denoted L(G(X), σ) is a complex defined by

L(G(X), σ) := [ ⋃_{τ∈X_{σ}} τ × (G_{σ}/G_{τ}^{σ})]/ ∼,

where the equivalence relation ∼ is determined by the maps (τ_{1}, g_{1}G_{τ_{1}}^{σ}) → (τ_{2}, g_{2}G_{τ_{2}}^{σ}) induced by inclusions on first coordinates, for all simplices τ_{1} ⊂ τ_{2} ∈ X_{σ} and for all g_{1}, g_{2} ∈ G_{σ} such that g_{1}G_{τ_{1}}^{σ} = g_{2}g_{σ(τ_{1}∗σ)(τ_{2}∗σ)}^{−1}G_{τ_{1}}^{σ} (cf. [4, p. 564, Section 4.20]).

5. Remark. — Simplices (τ, gG_{τ}^{σ}) map injectively into the link L(G(X), σ). Nevertheless, L(G(X), σ) needn’t be a simplicial complex in the strict sense, since it may contain double edges.

Link L(G(X), σ) carries a natural action of the group G_{σ}, defined by g(x, g′G_{τ}^{σ}) = (x, gg′G_{τ}^{σ}). There is a G_{σ}-invariant map p_{σ} : L(G(X), σ) → X_{σ} defined by p_{σ}(x, gG_{τ}^{σ}) = x, which is nondegenerate (i.e. injective on each simplex) and induces an isomorphism G_{σ}\L(G(X), σ) → X_{σ}.

2. Definition. — A complex of groups G(X) over a simplicial complex X is locally 6-large, if for each simplex σ of X the link L(G(X), σ) is a 6-large simplicial complex.

The above definition makes the statement of Theorem 15 precise. Our method of proof requires the notion of a locally 3-convex map to a locally 6-large complex of groups.

Let Q be a simplicial complex and G(X) a locally 6-large complex of groups over a simplicial complex X. A map of Q to G(X) consists of a nondegenerate simplicial map f : Q → X (which induces in the obvious way maps V(Q) → V(X) and E(Q) → E(X), denoted also by f, for the associated scwols Q and X), and a family ϕ(τ, σ) : (τ, σ) ∈ E(Q) of elements ϕ(τ, σ) ∈ G_{f(σ)}, such that

ϕ(ρ, σ) = ϕ(τ, σ)ψ_{f(σ)f(τ)}(ϕ(ρ, τ))g_{f(σ)f(τ)f(ρ)} for σ ⊂ τ ⊂ ρ.

6. Remark. — The above notion of map to a simplicial complex of groups is a special case of the notion of morphism for complexes of groups, cf. [4, p.536, Definition 2.4]. It is obtained by viewing a simplicial complex Q as the trivial simplicial complex of groups over Q (i.e. a complex with trivial local groups, homomorphisms and twisting elements).

For any simplex σ ∈ Q a map (f, ϕ) : Q → G(X) induces the map (f, ϕ)_{σ} : Q_{σ} → L(G(X), f(σ)) of links, defined by

(f, ϕ)_{σ}(τ) = (f(τ), ϕ(σ, σ ∗ τ)G_{f(τ)}^{f(σ)})

(compare [4, p. 565, Proposition 4.23]).

3. Definition. — Let G(X) be a locally 6-large simplicial complex of groups. A map (f, ϕ) : Q → G(X) is locally 3-convex if for each simplex σ ∈ Q the induced map (f, ϕ)_{σ} : Q_{σ} → L(G(X), f(σ)) is injective and the image (f, ϕ)_{σ}(Q_{σ}) is 3-convex in the link L(G(X), f(σ)). A map (f, ϕ) : Q → G(X) is a covering, if for each simplex σ ∈ Q the induced map (f, ϕ)_{σ} : Q_{σ} → L(G(X), f(σ)) is an isomorphism.

We now state a result that generalizes Proposition 3 to the case of locally 3-convex maps to simplicial complexes of groups.

5. Proposition. — Let (f, ϕ) : Q → G(X) be a locally 3-convex map of a simplicial complex Q to a locally 6-large finite dimensional simplicial complex of groups G(X). Then (f, ϕ) extends to a covering map (f_{e}, ϕ_{e}) : Q_{e} → G(X) in such a way that Q is a deformation retract of Q_{e}.

The proof of the above proposition goes along the same lines as the proof of Proposition 3. The objects ∂_{f}Q and E_{f} occurring in the latter proof (especially in the construction of a small extension for a convex map f in Section 7) have to be replaced by the objects ∂_{(f,ϕ)}Q and E_{(f,ϕ)} defined in an analogous way as follows. ∂_{(f,ϕ)}Q is the subcomplex of Q consisting of all those simplices σ ⊂ Q for which the induced map (f, ϕ)_{σ} : Q_{σ} → L(G(X), f(σ)) is not an isomorphism. E_{(f,ϕ)} is the set of all pairs (σ, τ) such that σ ⊂ ∂_{(f,ϕ)}Q, τ ⊂ L(G(X), f(σ)) and τ ∩(f, ϕ)_{σ}(Q_{σ}) = ∅. We omit details.

Proof (of Theorem 15). — Let G(X) be a simplicial complex of groups over a connected finite dimensional simplicial complex X and suppose it is locally 6-large. We have to show that G(X) is developable.

Denote by {v} the simplicial complex consisting of a single vertex v. A map i : {v} → X that sends v to any vertex of X may be viewed as a convex map of {v} to G(X) (the family ϕ(τ, σ) : (τ, σ) ∈ E({v}) is then empty). By Proposition 3, the map i extends to a covering map (h, ψ) : Y → G(X), with Y that retracts on v and thus is contractible. In particular, (h, ψ) is the universal covering of G(X).

Let Γ be the group of deck-transformations of the covering (h, ψ). The elements of Γ are the simplicial automorphisms γ : Y → Y which satisfy the following two conditions:

1. the map h ∘ γ : Y → X and the family ψ ∘ γ(τ, σ) : (τ, σ) ∈ E(Y) describe a well defined map (h ∘ γ, ψ ∘ γ) from Y to G(X);

2. γ preserves the projection h, i.e. h ∘ γ = h.

By the properties of the universal covering, G(X) is isomorphic to the complex of groups associated to the action of Γ on Y and hence it is developable. This finishes the proof.

8. Systolic complexes and their convex subcomplexes.

Recall that a simplicial complex X is systolic if it is locally 6-large, connected and simply connected. In this section, as in the remaining part of the paper, we assume that X is finite dimensional. We start the systematic study of systolic complexes, by introducing the notions of convexity and strong convexity, and deriving their basic properties.

4. Definition. — A subcomplex Q in a systolic complex X is convex if it is connected and locally 3-convex.

Note that, by Fact 3.1, any connected 3-convex subcomplex of X is convex. In particular, any simplex and any residue in a systolic complex is convex (see Example 2.2 and 2.3).

11. Lemma. — Let Q be a convex subcomplex of a systolic complex X. Then

1. Q is contractible;

2. Q is full in X;

3. Q is 3-convex in X;

4. if X is k-systolic (for some k ≥ 6) then Q is k-systolic.

Proof. — In view of contractibility of X (Theorem 14.1), (1) follows from Proposition 3 applied to the inclusion map Q → X. By Proposition 3 (and its proof), X is isomorphic to the complex Q_{e} obtained from Q by the infinite sequence of small extensions. Moreover, the full simplicial span of Q in X is clearly contained in EQ. Together with Fact 5.1 (which says that Q is full in EQ), this implies (2). By contractibility of X and connectedness of Q, there is no homotopically nontrivial cycle in (X, Q) and thus sys_{h}(X, Q) = ∞. Together with Proposition 2.1, this implies (3). In view of Fact 1.2, part (4) follows from (1) and (2).

The next lemma describes small extensions of (the inclusion maps of) convex subcomplexes.

12. Lemma. — Let f : Q → X be the inclusion map of a convex subcomplex Q in a systolic complex X. Then any small extension Ef : EQ → X maps EQ isomorphically to the subcomplex N_{X}(Q) ⊂ X. Thus EQ can be identified with the subcomplex N_{X}(Q) and Ef with the inclusion map N_{X}(Q) → X.

Proof. — According to Proposition 3 and its proof, a small extension Ef : EQ → X can be further extended to a covering map f̃ : Y → X, in such a way that Q is a deformation retract of Y. One easily observes that then EQ = N_{Y} (Q) and Ef = f̃|_{EQ}. Since X is simply connected and Y connected, the covering map f̃ is an isomorphism. Hence the lemma.

5. Corollary. — Let Q be a convex subcomplex in a systolic complex X. Then

1. the subcomplex N_{X}(Q) is also convex in X;

2. Q is a deformation retract of the neighborhood N_{X}(Q).

Proof. — In view of Lemma 12, it follows from condition (E4) of a small extension that the neighborhood N_{X}(Q) is locally 3-convex in X. Since it is also connected, part (1) of the corollary follows. Part (2) follows similarly from condition (E5).

Given a convex subcomplex Q in a systolic complex X, define a system B_{n} = B_{n}(Q, X) of combinatorial balls in X of radii n centered at Q as B_{0} := Q and B_{n+1} := N_{X}(B_{n}) for n ≥ 1. ¿From Corollary 5 and Lemma 11 we get

6. Corollary. — Let Q be a convex subcomplex in a systolic complex X. Then for any natural n the ball B_{n}(Q, X) is a convex subcomplex in X. Any ball B_{n}(Q, X) is full in X and contractible. It is also a deformation retract of the ball B_{m}(Q, X) for any m > n. Finally, if X is k-systolic (for some k ≥ 6) then the ball B_{n}(Q, X) is also k-systolic.

For n ≥ 1, the sphere of radius n centered at a convex subcomplex Q is the full subcomplex S_{n}(Q, X) in X spanned by the vertices at combinatorial distance n from Q.

For a convex subcomplex Q ⊂ X the boundary ∂Q is a subcomplex consisting of all simplices σ ⊂ Q with Q_{σ} ≠ X_{σ}. If f : Q → X denotes the inclusion map, we have ∂Q = ∂_{f}Q.

13. Lemma. — Let Q be a convex subcomplex Q in a systolic complex X and let n ≥ 1 be a natural number. Then

1. B_{n}(Q, X) is the full subcomplex of X spanned by the set of all vertices of X at combinatorial distance ≤ n from Q;

2. S_{n}(Q, X) ⊂ B_{n}(Q, X);

3. S_{n}(Q, X) is equal to the union of those simplices in the ball B_{n}(Q, X) which are disjoint with B_{n−1}(Q, X);

4. ∂B_{n}(Q, X) ⊂ S_{n}(Q, X).

Proof. — Observe that, by definition of balls, the vertex set of the ball B_{n}(Q, X) is exactly the set of all vertices of X at combinatorial distance ≤ n from Q. Since, by Corollary 6, the ball B_{n}(Q, X) is full in X, this proves (1). Parts (2) and (3) follow from (1). In view of Lemma 12, part (4) follows from property (E3) of small extension.

The next result will be often used in later sections, especially in establishing properties of projection maps (onto convex subsets) and directed geodesics.

14. Lemma. — For any convex subcomplex Q in a systolic complex X and for any simplex σ ⊂ N_{X}(Q) disjoint from Q, the intersection Q ∩ Res(σ, X) is a single simplex of Q. Moreover, if σ′ is a face of σ, then Q ∩ Res(σ, X) is a face of Q ∩ Res(σ′, X).

Proof. — In view of Lemma 12, it follows from Corollary 4 that the intersection Q∩Res(σ, N_{X}(Q)) is a single simplex of Q. Since it is clear from the definition of the neighborhood that Res(σ, X) ∩ Q = Res(σ, N_{X}(Q)) ∩ Q, the first assertion follows. The second assertion is clear due to reversed inclusion between residues of a simplex and its face.

15. Lemma. — Let Q ⊂ X be a convex subcomplex and let ρ be a simplex in N_{X}(Q) disjoint with Q. Let σ = Q ∩ Res(σ, X) be the simplex as in Lemma 14. Then the link of the subcomplex N_{X}(Q) at ρ has the form of a simplicial neighborhood of a single simplex, namely [N_{X}(Q)]_{ρ} = N_{X_{ρ}}(σ).

Proof. — In view of Lemma 12, it follows from Lemma 9 that [N_{X}(Q)]_{ρ} = N_{[N_{X}(Q)]_{ρ}}(σ). But, since σ ⊂ Q, we have N_{X}(σ) = N_{N_{X}(Q)}(σ) and hence also N_{[N_{X}(Q)]_{ρ}}(σ) = N_{X_{ρ}}(σ), which finishes the proof.

For later applications, we state the specializations of Lemmas 14 and 15 to the case of balls.

7. Corollary. — Let X be a systolic simplicial complex, Q a convex subcomplex in X, and n ≥ 1 a natural number. Then for any simplex ρ of the sphere S_{n}(Q, X)

1. the intersection σ = B_{n−1}(Q, X) ∩ Res(ρ, X) is a single (nonempty) simplex of X;

2. if σ = B_{n−1}(Q, X) ∩ Res(ρ, X) is the simplex as in (1), we have [B_{n}(Q, X)]_{ρ} = N_{X_{ρ}}(σ).

We now turn to the notion of strong convexity, a sharper variant of convexity. This notion is intimately related with the concept of extra-tilability playing a central role in our constructions in Sections 19- 20. The definition is inspired by the property of neighborhoods of convex subcomplexes described in Lemma 15.

5. Definition. — A subcomplex Q in a systolic simplicial complex X is strongly convex if it is connected and for any simplex ρ of Q the link Q_{ρ} coincides either with the whole link X_{ρ} or with the neighborhood N_{X_{ρ}}(σ) of some simplex σ ⊂ X_{ρ}.

Since the neighborhoods of simplices have diameters ≤ 3, Lemma 6 implies that links of a strongly convex subcomplex are 3-convex in the corresponding links of X. In particular, this gives the following.

8. Corollary. — Every strongly convex subcomplex is convex.

Next corollary is an immediate consequence of Lemmas 13.3 and 15, and of Corollary 6.

9. Corollary. — Let Q be a convex subcomplex in a systolic simplicial complex X. Then the neighborhood N_{X}(Q) is strongly convex in X. Moreover, for any natural n ≥ 1 the ball B_{n}(Q, X) is strongly convex in X.

9. Projections onto convex subcomplexes.

In this section we define and study a natural map from a finite dimensional systolic complex to its convex subcomplex, which we call projection. This map resembles the projection of a CAT (0) space to its convex subset along the shortest geodesics connecting points of the space with the subset. We introduce also projection rays which are analogues of the above geodesics.

Given a simplicial complex K, we denote by K′ its first barycentric subdivision. For a simplex σ ⊂ K, we denote by b_{σ} the barycenter of σ, a vertex in K′. We denote by dist_{K} the combinatorial distance (in the 1skeleton of K) between the vertices of K. We also use a simplified notation B_{n}Q, S_{n}Q for balls B_{n}(Q, X) and spheres S_{n}(Q, X), X being fixed throughout the whole section. In particular, under this convention, the neighborhood N_{X}(Q) is denoted by B_{1}Q.

Given a convex subcomplex Q in a systolic complex X, define an elementary projection π_{Q} : (B_{1}Q)′ → Q′ between the barycentrically subdivided complexes by putting

π_{Q}(b_{σ}) = \begin{cases}b_{σ∩Q} & if σ ∩ Q ≠ ∅ \\ b_{τ} & if σ ∩ Q = ∅, where τ = Res(σ, X) ∩ Q \end{cases}

and extending simplicially. By Lemmas 11.3 and 14, π_{Q} is a well defined simplicial map. It is also clear that π_{Q} restricted to Q′ is the identity on Q′, i.e. π_{Q} is a retraction.

7. Remark. — One verifies that, viewing B_{1}Q as a small extension domain EQ for the inclusion map Q → X, the elementary projection π_{Q} coincides with the deformation retraction EQ → Q constructed in Section 6.

16. Lemma. — Let Q be a convex subcomplex in a systolic complex X, and let σ ⊂ (B_{1}Q)′ be a simplex not contained in Q′. Then π_{Q}(σ) ⊂ (∂Q)′.

Proof. — Since (∂Q)′ is a full subcomplex in Q′, it is sufficient to prove the lemma for vertices. A vertex in (B_{1}Q)′ not contained in Q′ has the form b_{τ} for some simplex τ ⊂ B_{1}Q not contained in Q. Let ρ ⊂ Q be the simplex given by π_{Q}(b_{τ}) = b_{ρ}. By the definition of π_{Q}, if τ intersects Q then τ − ρ ∈ X_{ρ} and if τ is disjoint with Q then τ ∈ X_{ρ}. In any case it follows that Q_{ρ} ≠ X_{ρ}, hence ρ ⊂ ∂Q and b_{ρ} ∈ (∂Q)′.

Denote by P_{Q}^{n} : (B_{n}Q)′ → Q′ the composition map π_{B_{n−1}Q} ∘ π_{B_{n−2}Q} ∘ … π_{B_{1}Q} ∘ π_{Q} and observe that P_{Q}^{n} extends P_{Q}^{m} if n > m. Denote then by P_{Q} : X′ → Q′ the union ⋃_{n} P_{Q}^{n} and call it the projection to Q. The first two parts of the next fact follow from the properties of elementary projections. Part (3) is true for any simplicial map between two simplicial complexes.

6. Fact. — The projection P_{Q} satisfies the following properties:

1. P_{Q}|_{Q′} = id_{Q′};

2. if σ is a simplex of X′ not contained in Q′ then P_{Q}(σ) ⊂ ∂Q;

3. dist_{Q′}(P_{Q}(v), P_{Q}(w)) ≤ dist_{X′}(v, w) for any vertices v, w ∈ X′.

We turn to defining projection rays. To do this, we need the following variants of projection maps, between face posets rather than barycentric subdivisions. For any convex subcomplex Q ⊂ X, and any simplex σ of X, put P̂_{Q}(σ) = τ if and only if P_{Q}(b_{σ}) = b_{τ}, and similarly put π̂_{Q}(σ) = τ if and only if π_{Q}(b_{σ}) = b_{τ}.

Let Q ⊂ X be a convex subcomplex and let σ ⊂ S_{n}Q. The projection ray from σ to Q is the sequence σ = σ_{0}, σ_{1}, …, σ_{n} of simplices in X given by σ_{k} = π̂_{B_{n−k+1}Q}(σ_{k−1}) for k = 1, …, n. Equivalently, this sequence is given by σ_{k} = P̂_{B_{n−k}Q}(σ_{0}).

Now we list obvious properties of projection rays.

7. Fact. —

1. Any two consecutive simplices σ_{k}, σ_{k+1} in a projection ray are disjoint and span a simplex of X.

2. If σ_{k} and σ_{m} are simplices in a projection ray then for any vertices v ∈ σ_{k} and w ∈ σ_{m} we have dist_{X}(v, w) = |k − m|.

3. If σ_{0}, …, σ_{n} is a projection ray on Q then σ_{k}, σ_{k+1}, …, σ_{n}, for any 0 < k < n, is also a projection ray on Q.

A less obvious property, giving an intrinsic characterization of projection rays, is

17. Lemma. — If σ_{0}, …, σ_{n} is a projection ray on Q then σ_{0}, …, σ_{k}, for any 0 < k ≤ n, is a projection ray on σ_{k}, where we view σ_{k} as a convex subcomplex of X.

Proof. — Note first that B_{m}σ_{k} ⊂ B_{n−k+m}Q for any 0 ≤ m < k. Since Res(σ_{k−m−1}, X) contains σ_{k−m}, we have

σ_{k−m} ⊂ Res(σ_{k−m−1}, X) ∩ B_{m}σ_{k} ⊂ Res(σ_{k−m−1}, X) ∩ B_{n−k+m}Q = σ_{k−m}.

Thus all the inclusions above are equalities, so in particular

Res(σ_{k−m−1}, X) ∩ B_{m}σ_{k} = σ_{k−m},

hence the lemma.

10. Corollary. — A projection ray in a systolic complex is uniquely determined by its initial and final simplex.

18. Lemma. — Let σ and τ be two simplices in a systolic complex X such that dist_{X}(v, w) = n for all vertices v ∈ σ and w ∈ τ. Then there is a face ρ ⊂ τ such that σ is connected to ρ by a projection ray of form σ, σ_{1}, …, σ_{n−1}, ρ.

Proof. — the required projection ray corresponds to the projection P̂_{τ} on the subcomplex τ ⊂ X, with ρ = P̂_{τ}(σ).

10. Directed geodesisc.

In this section we introduce the notion of a directed geodesic in a locally 6-large simplicial complex. It is a sequence of simplices satisfying certain condition that involves triples of consecutive simplices in the sequence, and is in this sense local. The adjective “directed” tells that a directed geodesic is in general not symmetric, i.e. it fails to be a directed geodesics after reversing the order. We show that in systolic complexes the local notion of a directed geodesic coinsides with the global notion of a projection ray (introduced in Section 8). In Sections 11-13 we study further global properties of directed geodesics in systolic complexes.

6. Definition. — A sequence (σ_{n}) of simplices in a locally 6-large simplicial complex X is a directed geodesic if it satisfies the following properties:

1. any two consecutive simplices σ_{i}, σ_{i+1} in the sequence are disjoint and span a simplex of X;

2. for any three consecutive simplices σ_{i}, σ_{i+1}, σ_{i+2} in the sequence we have

Res(σ_{i}, X_{σ_{i+1}}) ∩ B_{1}(σ_{i+2}, X_{σ_{i+1}}) = ∅.

Observe the lack of symmetry in condition (2), and the local nature of the definition. It is clear that images of directed geodesics under covering maps, or their lifts under such maps, are again directed geodesics. The next lemma shows an alternative and simpler way to define directed geodesics in systolic complexes.

19. Lemma. — If X is a systolic complex then condition (2) in the definition of a directed geodesic (Definition 6) can be replaced with the following condition:

(2′)  Res(σ_{i}, X) ∩ B_{1}(σ_{i+2}, X) = σ_{i+1}.

Proof. — Since Res(σ_{i}, X_{σ_{i+1}}) = Res(σ_{i}, X)∩X_{σ_{i+1}} and B_{1}(σ_{i+2}, X_{σ_{i+1}}) = B_{1}(σ_{i+2}, X) ∩ X_{σ_{i+1}}, we get the inclusion

σ_{i+1} ∗ [Res(σ_{i}, X_{σ_{i+1}}) ∩ B_{1}(σ_{i+2}, X_{σ_{i+1}})] ⊂ Res(σ_{i}, X) ∩ B_{1}(σ_{i+2}, X)

(where σ∗∅ = σ). Hence (2’) implies (2). To prove the converse, suppose that Res(σ_{i}, X) ∩ B_{1}(σ_{i+2}, X) contains a vertex v not in σ_{i+1}. Then (2) implies that v is not in the link X_{σ_{i+1}}, and hence also not in the ball B_{1}(σ_{i+1}, X). Moreover, both σ_{i} and σ_{i+2} are contained in Res(v, X) ∩ B_{1}(σ_{i+1}, X), which is a simplex according to Lemma 14. Thus σ_{i} and σ_{i+1} span a simplex, but this is impossible due to condition (2) and the fact that B_{1}(σ_{i+1}, X) is full in X.

Existence of many finite directed geodesics is implied by the following two results. We will not deal with infinite directed geodesics in this paper.

20. Lemma. — Each projection ray in a systolic simplicial complex is a directed geodesic.

Proof. — By Fact 7.1, a projection ray σ_{0}, …, σ_{n} satisfies condition (1) of Definition 6. In view of Lemma 19, it is now sufficient to check condition (2’) from this lemma. To do this, note that any subsequence σ_{i}, σ_{i+1}, σ_{i+2} is a projection ray on σ_{i+2} (see Fact 7.3 and Lemma 17). By the definition of a projection ray and by Lemma 14 we get Res(σ_{i}, X) ∩ B_{1}(σ_{i+2}, X) = σ_{i+1}, and the lemma follows.

11. Corollary. — Any sequence of simplices in a locally 6-large complex X that lifts to a projection ray in the universal cover of X is a directed geodesic.

We now turn to proving that (lifts of) directed geodesics coincide with projection rays. We start with a preparatory result.

21. Lemma. — Let X be a systolic complex, Q its convex subcomplex, and suppose that σ is a simplex in the sphere S_{1}(Q, X). Denote by τ the simplex Res(σ, X) ∩ Q. Then [B_{1}(τ, X)]_{σ} = [B_{1}(X, Q)]_{σ}.

Proof. — Since τ ⊂ Q, it is clear that [B_{1}(τ, X)]_{σ} ⊂ [B_{1}(X, Q)]_{σ}. To prove the converse inclusion, note that since all the involved complexes are full in X, it is sufficient to show that if v is a vertex in [B_{1}(X, Q)]_{σ} then v ∈ [B_{1}(τ, X)]_{σ}. Let v be any vertex of [B_{1}(X, Q)]_{σ}. If v ∈ Q then v ∈ Res(σ, X) ∩ Q = τ, and hence v ∈ [B_{1}(τ, X)]_{σ}. If v ∉ Q then σ ∗ v ⊂ S_{1}(Q, X) and thus Res(σ ∗ v, X) ∩ Q ≠ ∅. Moreover, by Lemma 14 we have Res(σ ∗ v, X)∩Q ⊂ Res(σ, X)∩Q = τ, and hence v ∈ B_{1}(τ, X). Since the ball B_{1}(τ, X) is full in X, we get that σ ∗ v ⊂ B_{1}(τ, X) and thus again v ∈ [B_{1}(τ, X)]_{σ}, hence the lemma.

6. Proposition. — A directed geodesic σ_{0}, …, σ_{n} in a systolic complex is a projection ray on its final simplex σ_{n}.

Proof. — By Lemma 19 we have Res(σ_{n−2}, X) ∩ B_{1}(σ_{n}, X) = σ_{n−1}, so σ_{n−2}, σ_{n−1}, σ_{n} is a projection ray on σ_{n}. Suppose inductively that for some 1 ≤ k ≤ n−2 the sequence σ_{k}, σ_{k+1}, …, σ_{n} is a projection ray on σ_{n}. We will prove that the sequence σ_{k−1}, σ_{k}, …, σ_{n} is also a projection ray on σ_{n}. To do this, we need to show that (1) σ_{k−1} is disjoint with the ball B_{n−k}(σ_{n}, X) and (2) Res(σ_{k−1}, X) ∩ B_{n−k}(σ_{n}, X) = σ_{k}.

By Lemma 21 we have

[B_{1}(σ_{k+1}, X)]_{σ_{k}} = [B_{1}(B_{n−k−1}(σ_{n}, X), X)]_{σ_{k}} = [B_{n−k}(σ_{n}, X)]_{σ_{k}}.

We then get

σ_{k−1} ∩ [B_{n−k}(σ_{n}, X)]_{σ_{k}} = σ_{k−1} ∩ [B_{1}(σ_{k+1}, X)]_{σ_{k}} = ∅,

where the last equality follows from the definition of directed geodesic applied to the simplices σ_{k−1}, σ_{k}, σ_{k+1}. Since the ball B_{n−k}(σ_{n}, X) is full in X and σ_{k−1} ∗ σ_{k} is a simplex of X, this implies (1). Moreover, since X is flag and balls in X are full, we get

(9.6.1) B_{1}(σ_{k}, X)∩B_{1}(σ_{k+1}, X) = σ_{k} ∗[B_{1}(σ_{k+1}, X)]_{σ_{k}} = σ_{k} ∗[B_{n−k}(σ_{n}, X)]_{σ_{k}} =

= B_{1}(σ_{k}, X) ∩ B_{n−k}(σ_{n}, X).

By Lemma 14, the intersection Res(σ_{k−1}, X) ∩ B_{n−k}(σ_{n}, X) is a simplex containing σ_{k}, so in particular this intersection is contained in the ball B_{1}(σ_{k}, X). Consequently, by applying (9.6.1) we have

Res(σ_{k−1}, X) ∩ B_{n−k}(σ_{n}, X) = Res(σ_{k−1}, X) ∩ B_{n−k}(σ_{n}, X) ∩ B_{1}(σ_{k}, X) =

Res(σ_{k−1}, X) ∩ B_{1}(σ_{k+1}, X) ∩ B_{1}(σ_{k}, X) = σ_{k},

where the last equality follows from Lemma 19. This shows that σ_{k−1}, σ_{k}, …, σ_{n} is a projection ray on σ_{n}, hence the proposition.

Proposition 6 and Lemma 20 show that the sets of finite directed geodesics and of projection rays coincide. As a consequence of Corollary 10 and Lemma 18 we obtain therefore the following.

12. Corollary. — Given vertices v, w in a systolic complex there is exactly one directed geodesic from v to w.

As an easy consequence of Fact 7.2 we get also the following.

13. Corollary. — Let v, w be two vertices in a systolic complex X such that dist_{X}(v, w) = n. Then the directed geodesic from v to w consists of n + 1 simplices.

11. Directed geodesics and convexity.

In this section we study the behavior of directed geodesics with respect to convex subcomplexes in systolic complexes. We also obtain several more properties of convex subcomplexes.

In the proofs in this section we will often use (without referring explicitly to) both assertions of Lemma 14.

22. Lemma. — Let Q be a convex subcomplex in a systolic complex X. Let B_{n} = B_{n}(Q, X) and S_{n} = S_{n}(Q, X) be the systems of balls and spheres in X centered at Q. For any directed geodesic σ_{1}, σ_{2}, σ_{3} and for any n ≥ 0:

1. if σ_{1} ⊂ B_{n} and σ_{2} ⊂ S_{n+1} then σ_{3} ⊂ S_{n+2};

2. if σ_{1} ⊂ B_{n} and σ_{2} intersects B_{n} but is not contained in B_{n}, then σ_{3} ∩ B_{n} = ∅;

3. if σ_{1} intersects B_{n} but is not contained in B_{n}, and if σ_{2} ∩ B_{n} = ∅, then σ_{3} is not contained in B_{n+1}.

Proof. — To prove (1), observe that σ_{3} ∩ B_{n} = ∅, since otherwise both simplices σ_{3} ∩ B_{n} and σ_{1} are faces of the simplex Res(σ_{2}, X) ∩ B_{n}, and this contradicts condition (2’) from Lemma 19. Suppose that τ = σ_{3} ∩ S_{n+1} is not empty. It is a face of σ since, by Lemma 13.2, the sphere S_{n+1} is full in X. Note that both simplices σ_{1} and Res(σ_{2} ∗ τ, X) ∩ B_{n} are faces of the simplex Res(σ_{2}, X)∩B_{n}. It follows that the intersection B_{1}(τ, X)∩Res(σ_{1}, X) contains the join σ_{2} ∗ [Res(σ_{2} ∗ τ, X) ∩ B_{n}], and hence is larger than σ_{2} (here we use flagness of X). Thus the same is true for the intersection B_{1}(σ_{3}, X)∩ Res(σ_{1}, X), contradicting condition (2’) of Lemma 19. This implies that σ_{3} is disjoint with both B_{n} and S_{n+1}, hence it is contained in S_{n+2}.

To prove (2), suppose that the intersection τ := σ_{3} ∩ B_{n} is not empty. It is then a face of σ_{3} (because B_{n} is full) and we denote it by τ. Similarly, using the fact that spheres are full in X, denote by ρ the simplex S_{n+1} ∩ σ_{2}. Observe that both σ_{1} and τ are faces of the simplex Res(ρ, X) ∩ B_{n}, which clearly contradicts condition (2’) of Lemma 19.

To prove (3), note that by the assumptions we get that σ_{2} ⊂ S_{n+1}. If τ = σ_{3} ∩ B_{n} ≠ ∅ then both τ and σ_{1} are the faces of the simplex

Res(σ_{2}, X) ∩ B_{n}, contradicting condition (2’) of Lemma 19. If σ_{3} ⊂ S_{n+1} then the simplex Res(σ_{2} ∗ σ_{3}, X) ∩ B_{n} and the simplex σ_{1} ∩ B_{n} are faces of the simplex Res(σ_{2}, X) ∩ B_{n}, which again contradicts (2’).

Since σ_{3} is disjoint with B_{n} and not contained in S_{n+1}, it is not contained in B_{n+1}, hence the lemma.

8. Remark. — The following uniform interpretation of the three parts of Lemma 22 provides the idea for proving the next result. A simplex σ_{1} is closer to Q than a simplex σ_{2} if any of the assumptions from parts (1)-(3) is satisfied. The lemma says that if σ_{1} is closer than σ_{2} then σ_{2} is closer than σ_{3}.

7. Proposition. — Let Q be a convex subcomplex in a systolic complex X, and let σ_{0}, …, σ_{n} be a directed geodesic in X such that σ_{0} ⊂ Q and σ_{n} ⊂ Q. Then for each 0 < i < n we have σ_{i} ⊂ Q.

Proof. — Suppose that some of the simplices in the directed geodesic is not contained in Q. Then there is i such that σ_{i} ⊂ Q and σ_{i+1} is not contained in Q. Applying Lemma 22 inductively, we get that σ_{k} is not contained in Q for all k > i. This contradicts the assumption that σ_{n} ⊂ Q, hence the proposition.

23. Lemma. — The intersection of any family of convex subcomplexes in a given systolic complex is a convex subcomplex.

Proof. — Since any convex subcomplex is locally 3-convex, it follows from Fact 4.2 that the intersection of convex subcomplexes is locally 3convex. It remains to show that this intersection is connected.

Let v, w be any two vertices in the intersection. By Lemma 18, these vertices are connected by a projection ray. Since, according to Lemma 20, this projection ray is a directed geodesic, it follows from Proposition 7 that all its simplices are contained in the intersection. Consequently, since the intersection of full subcomplexes is full, there is a path in (the 1-skeleton of) the intersection between v and w, hence connectivity.

24. Lemma. — For each subcomplex K of a systolic complex X there is the smallest convex subcomplex conv(K) in X that contains K (we will call it the convex hull of K in X). Moreover, if K is bounded (with respect to the combinatorial distance), its convex hull is also bounded.

Proof. — Since K is contained in at least one convex subcomplex of X, namely in X itself, we define conv(K) to be the intersection of all convex subcomplexes in X containing K. According to Lemma 23, this intersection is convex. If K is bounded, it is contained in some ball in X centered at a vertex. Since, by Corollary 6, this ball is convex, the convex hull of K is clearly bounded.

12. Fellow traveller property.

In this section we prove that directed geodesics in a systolic complex satisfy fellow traveller property. We show this property in a setting suitable for applications in Section 14, where we prove that systolic groups are biautomatic.

Let X be a systolic simplicial complex and let v, w be vertices in X. An allowable geodesic from v to w in the 1-skeleton X^{(1)} is an infinite sequence (u_{i})_{i=0}^{∞} of vertices of X such that if v = σ_{0}, σ_{1}, …, σ_{n} = w is the directed geodesic in X from v to w then

1. u_{i} ∈ σ_{i} for 0 ≤ i ≤ n (in particular, u_{0} = v and u_{n} = w);

2. u_{i} = u_{n} = w for i > n.

Fact 7.1 (together with Proposition 6) implies that the sequence of vertices in an allowable geodesic, before it becomes constant, forms a polygonal path in the 1-skeleton X^{(1)}. Moreover, Fact 7.2 implies the following.

8. Fact. — If dist_{X}(v, w) = n and (u_{i})_{i=0}^{∞} is an allowable geodesic from v to w, then for 0 ≤ j < k ≤ n we have dist_{X}(u_{j}, u_{k}) = k − j, i.e. the subsequence (u_{i})_{i=0}^{n} determines a geodesic in X^{(1)}.

We will prove the following variant of the fellow traveller property.

8. Proposition. — Let X be a systolic complex and suppose that (u_{i})_{i=0}^{∞} and (t_{i})_{i=0}^{∞} are allowable geodesics in X^{(1)} from v to w and from p to q respectively. Then for each i ≥ 0 we have

dist_{X}(u_{i}, t_{i}) ≤ 3 · max[dist_{X}(v, p), dist_{X}(w, q)] + 1.

9. Remark. — Note that the fellow traveller property does not in general hold for arbitrary geodesics in the 1-skeleton of a systolic complex, as can be easily observed for example in the triangulation of the euclidean plane by congruent equilateral triangles.

The proof of Proposition 8 is based on Lemma 25, the first part of which we prove at the end of this section, and the second in Section 13. In this lemma we use a convention that if σ_{0}, …, σ_{n} is a directed geodesic then it extends to the infinite sequence (σ_{i})_{i=0}^{∞} by putting σ_{i} = σ_{n} for i > n. We denote by X′ the first barycentric subdivision of a simplicial complex X and by b_{σ} the barycenter of a simplex σ ⊂ X (which is a vertex in X′).

25. Lemma. — Let X be a systolic complex and let (σ_{i})_{i=0}^{n}, (τ_{i})_{i=0}^{m} be directed geodesics in X.

1. If σ_{n} = τ_{m} then dist_{X′}(b_{σ_{i}}, b_{τ_{i}}) ≤ 2 · dist_{X′}(b_{σ_{0}}, b_{τ_{0}}) for each i ≥ 0.

2. If σ_{0} = τ_{0} is a vertex then dist_{X′}(b_{σ_{i}}, b_{τ_{i}}) ≤ dist_{X′}(b_{σ_{n}}, b_{τ_{m}}) for each i ≥ 0.

Proof (of Proposition 8). — (assuming Lemma 25): Let (σ_{i})_{i=0}^{n}, (τ_{i})_{i=0}^{m} and (ρ_{i})_{i=0}^{l} be the directed geodesics in X from v to w, from p to q and from p to w respectively. By Lemma 25, for each i ≥ 0 we have dist_{X′}(b_{σ_{i}}, b_{ρ_{i}}) ≤ 2 · dist_{X′}(b_{σ_{0}}, bρ_{0}) and dist_{X′}(b_{ρ_{i}}, b_{τ_{i}}) ≤ dist_{X′}(b_{ρ_{l}}, b_{τ_{m}}). It clearly implies that for each i ≥ 0

dist_{X′}(b_{σ_{i}}, b_{τ_{i}}) ≤ 3 · max[dist_{X′}(b_{σ_{0}}, b_{τ_{0}}), dist_{X′}(b_{σ_{n}}, b_{τ_{m}})].

Since for any vertices x, y belonging to simplices α, β in X respectively we have

2 · dist_{X}(x, y) ≤ dist_{X′}(b_{α}, b_{β}) + 2 and dist_{X′}(x, y) = 2 · dist_{X}(x, y),

the proposition follows.

Proof (of Lemma 25.1). — Under assumptions of the lemma, σ_{0} ∈ S_{n}(σ_{n}, X) and τ_{0} ∈ S_{m}(τ_{m}, X) = S_{m}(σ_{n}, X). Suppose n ≥ m. Then dist_{X′}(b_{σ_{0}}, b_{τ_{0}}) ≥ 2n − 2m. On the other hand, applying Fact 6 to the convex subcomplex Q = B_{m−i}(σ_{n}, X) = B_{m−i}(τ_{m}, X) (or Q = σ_{n} = τ_{m} if i > m) we get dist_{X′}(b_{σ_{i+n−m}}, b_{τ_{i}}) ≤ dist_{X′}(b_{σ_{0}}, b_{τ_{0}}). This implies the following estimate:

dist_{X′}(b_{σ_{i}}, b_{τ_{i}}) ≤ dist_{X′}(b_{σ_{i}}, b_{σ_{i+n−m}}) + dist_{X′}(b_{σ_{i+n−m}}, b_{τ_{i}}) ≤

dist_{X′}(b_{σ_{0}}, b_{τ_{0}}) + (2n − 2m) ≤ 2 · dist_{X′}(b_{σ_{0}}, b_{τ_{0}}),

which finishes the proof.

13. Inverse fellow traveller property.

In this section we study properties of the family of all directed geodesics started at a fixed vertex p in a finite dimensional systolic complex X. In particular, we obtain the proof of Lemma 25.2.

Let X be a systolic complex and let p be a vertex in X. We say that a simplex τ ⊂ X is accessible from p if there exists a directed geodesic from p to τ. By Fact 7.2, to be accessible from p, a simplex τ must be contained in some sphere S_{n}(p, X). Not all simplices from such spheres are accessible from p. However, it follows from Corollary 12 that every vertex in X distinct from p is accessible from p. Let τ ⊂ S_{n+1}(σ, X) be a simplex accessible from p. Denote by c_{p}(τ) the simplex that precedes τ in the directed geodesic from p to τ. More precisely, if σ_{0}, σ_{1}, …, σ_{n}, σ_{n+1} is the directed geodesic from p to τ (i.e. σ_{0} = p and σ_{n+1} = τ) then we put c_{p}(τ) := σ_{n}.

We use the notation concerning barycentric subdivisions as in the previous section.

9. Proposition. — For any systolic complex X, any vertex p in X and any n ≥ 0 there is a simplicial map c_{p}^{n} : [B_{n+1}(p, X)]′ → [B_{n}(p, X)]′ satisfying the following properties:

1. c_{p}^{n} restricted to [B_{n}(p, X)]′ is the identity;

2. c_{p}^{n}(b_{τ}) = b_{c_{p}(τ)} for any simplex τ ⊂ S_{n+1}(σ, X) that is accessible from p.

The proof of Proposition 9 requires several preparatory results. Before getting to them we first give the proof of Lemma 25.2 based on the proposition.

Proof (of Lemma 25.2). — Let (σ_{i}) and (τ_{i}) be the sequences as in the lemma. Consider the maps C_{σ_{0}}^{i} : X′ → [B_{i}(σ_{0}, X)]′ given by

C_{σ_{0}}^{i} := ⋃_{k=1}^{∞} c_{σ_{0}}^{i+k} ∘ c_{σ_{0}}^{i+k−1} ∘ ⋯ ∘ c_{σ_{0}}^{i}

and note that we have C_{σ_{0}}^{i}(b_{σ_{n}}) = b_{σ_{i}} and C_{σ_{0}}^{i}(b_{τ_{m}}) = b_{τ_{i}}. Since the maps C_{σ_{0}}^{i} are simplicial map, they do not increase combinatorial distances, hence the lemma.

The next series of results prepares the background for proving Proposition 9.

26. Lemma. — If τ is accessible from p and ϱ is a face of τ, then ϱ is accessible from p and c_{p}(ϱ) = c_{p}(τ).

Proof. — Let p, σ_{1}, …, σ_{n−1}, τ be the directed geodesic from p to τ. It is sufficient to show that p, σ_{1}, …, σ_{n−1}, ϱ is also a directed geodesic. To do this, we only need to check the condition for directed geodesic at the final triple σ_{n−2}, σ_{n−1}, ϱ. It follows easily by observing that B_{1}(ϱ, X_{σ_{n−1}}) ⊂ B_{1}(τ, X_{σ_{n−1}}).

14. Corollary. — Let τ_{1}, τ_{2} be simplices that are accessible from p and suppose they intersect. Then their corresponding directed geodesics from p coincide except at the last simplices. In other words, we then have c_{p}(τ_{1}) = c_{p}(τ_{2}).

27. Lemma. — Suppose e = (v_{1}, v_{2}) is a 1-simplex in S_{n}(p, X) not accessible from p and denote by σ_{0} the last common simplex in the directed geodesics from p to v_{1} and v_{2}. Denote by σ_{0}, σ_{1}^{1}, σ_{2}^{1}, …, σ_{n−1}^{1}, v_{1} and σ_{0}, σ_{1}^{2}, σ_{2}^{2}, …, σ_{n−1}^{2}, v_{2} the directed geodesics from σ_{0} to v_{1} and v_{2} (which are parts of the corresponding geodesics from p). Suppose that the projection ray from σ_{0} on e terminates at v_{1} (it terminates at some vertex of e since e is not accessible from p, and hence also not accessible from σ_{0}). Then (1) σ_{1}^{2} ⊂ σ_{1}^{1}, (2) σ_{2}^{1} ∩ σ_{2}^{2} = ∅ and

(3) σ_{2}^{1}, σ_{2}^{2} span a simplex of X.

Proof. — Note that by our assumptions the directed geodesic σ_{0}, σ_{1}^{1}, σ_{2}^{1}, …, σ_{n−1}^{1}, v_{1} is the projection ray from σ_{0} on e. Therefore we have

σ_{1}^{1} = Res(σ_{0}, X) ∩ B_{n−1}(v_{1}, X) = Res(σ_{0}, X) ∩ B_{n−1}(e, X) and

σ_{1}^{2} = Res(σ_{0}, X) ∩ B_{n−1}(v_{2}, X) ⊂ Res(σ_{0}, X) ∩ B_{n−1}(e, X),

hence (1).

To prove (2), suppose that σ_{2}^{1}∩σ_{2}^{2} = α ≠ ∅. Then, according to Lemma 26, σ_{0}, σ_{1}^{1}, α and σ_{0}, σ_{1}^{2}, α are directed geodesics. Moreover, these geodesics are distinct because σ_{1}^{1} ≠ σ_{1}^{2}, which contradicts uniqueness (Corollary 10 and Proposition 6).

To prove (3), note that in view of (1) we have

σ_{2}^{1} = Res(σ_{1}^{1}, X) ∩ B_{n−2}(e, X) ⊂ Res(σ_{1}^{2}, X) ∩ B_{n−2}(e, X) and

σ_{2}^{2} = Res(σ_{1}^{2}, X) ∩ B_{n−2}(v_{2}, X) ⊂ Res(σ_{1}^{2}, X) ∩ B_{n−2}(e, X),

where the first inclusion follows from (1) and second from the fact that v_{2} ⊂ e. By Lemma 14, the intersection β = Res(σ_{1}^{2}, X) ∩ B_{n−2}(e, X) is a simplex in X, and since we have σ_{2}^{1}, σ_{2}^{2} ⊂ β, the lemma follows.

28. Lemma. — Suppose e = (v_{1}, v_{2}) is a 1-simplex in S_{n}(p, X) not accessible from p. Then the simplices c_{p}(v_{1}) and c_{p}(v_{2}) span a simplex of X.

Proof. — As in the statement of Lemma 27, denote by σ_{0} the last common simplex in the directed geodesics from p to v_{1} and v_{2}. Denote also by σ_{0}, σ_{1}^{1}, σ_{2}^{1}, …, σ_{n−1}^{1}, v_{1} and σ_{0}, σ_{1}^{2}, σ_{2}^{2}, …, σ_{n−1}^{2}, v_{2} the directed geodesics from σ_{0} to v_{1} and v_{2} (which are parts of the corresponding geodesics from p), and assume (without loss of generality) that the first of them is the projection ray from σ_{0} on e. Claim 1. Let σ_{1}^{1} − σ_{1}^{2} be the face of the simplex σ_{1}^{1} spanned by the vertices not contained in σ_{1}^{2}. Then σ_{1}^{1} − σ_{1}^{2} ⊂ S_{n}(v_{2}, X).

To prove Claim 1, note that for any vertex u ∈ σ_{1}^{1} − σ_{1}^{2} we have the estimate

dist_{X}(u, v_{2}) ≤ dist_{X}(u, σ_{1}^{2}) + dist_{X}(σ_{1}^{2}, v_{2}) = 1 + (n − 1) = n.

If dist_{X}(u, v_{2}) = n−1 for some u ∈ σ_{1}^{1}−σ_{1}^{2}, then u ∈ Res(σ_{0}, X)∩B_{n−1}(v_{2}, X) = σ_{1}^{2}, a contradiction.

A similar argument based on Claim 1 and the fact that σ_{1}^{1} − σ_{1}^{2} and σ_{2}^{1} span a simplex in X gives the following. Claim 2. For k = 2, 3, …, n − 1 we have σ_{k}^{1} ⊂ S_{n−k+1}(v_{2}, X).

Returning to the proof of Lemma 28, we will show that for k = 1, 2, …, n− 1 the simplices σ_{k}^{1}, σ_{k}^{2} span a simplex of X. The assertion holds for k = 1, 2 due to Lemma 27. Suppose, by induction, that σ_{k}^{1}, σ_{k}^{2} span a simplex. Then both σ_{k+1}^{1} and σ_{k}^{2} are contained in the intersection Res(σ_{k}^{1}) ∩ B_{n−k}(v_{2}, X) which is a simplex of X (the first inclusion is provided by Claim 2). Consequently, both simplices σ_{k+1}^{1} and σ_{k+1}^{2} are contained in Res(σ_{k}^{2}, X)∩B_{n−k−1}(e, X), which is also a simplex of X, hence σ_{k+1}^{1} and σ_{k+1}^{2} span a simplex.

This shows that the simplices c_{p}(v_{1}) = σ_{n−1}^{1} and c_{p}(v_{2}) = σ_{n−1}^{2} span a simplex of X, as required.

29. Lemma. — For any simplex τ ⊂ S_{n}(p, X) the family {c_{p}(v) : v is a vertex of τ} of simplices spans a simplex in S_{n−1}(p, X).

Proof. — Observe first that any two simplices c_{p}(v_{1}), c_{p}(v_{2}) from the family span a simplex. If (v_{1}, v_{2}) is a 1-simplex not accessible from p, this is due to Lemma 28. Otherwise this follows from the equality c_{p}(v_{1}) = c_{p}(v_{2}) implied by Lemma 26.

Since the complex X is flag and the sphere S_{n−1}(p, X) is a full subcomplex, the above observation implies that the whole family spans a simplex of this sphere.

For a simplex τ ⊂ S_{n}(p, X) not accessible from p let c_{p}(τ) be the simplicial span of the family of simplices {c_{p}(v) : v is a vertex of τ}. By Lemma 26, if τ is a simplex accessible from σ then the simplicial span of the set {c_{p}(v) : v is a vertex of τ} equals c_{p}(τ) (as defined at the beginning of the section). Thus the above definition of c_{p}(τ) applies to all simplices τ ⊂ S_{n}(p, X). In particular, this implies the following.

15. Corollary. — If τ ⊂ S_{n}(p, X) is a simplex and ρ is a face of τ then c_{p}(ρ) is a face of c_{p}(τ).

30. Lemma. — If v ∈ S_{n−1}(p, X) and w ∈ S_{n}(p, X) are vertices that span a 1-simplex e then the simplex c_{p}(w) and the vertex v span a simplex in S_{n−1}(p, X).

Proof. — Since the intersection Res(w, X) ∩ S_{n−1}(p, X) is a simplex (Lemma 14), the lemma follows by observing that both c_{p}(w) and v are contained in this intersection.

An argument similar to that in Lemma 29 gives the following.

16. Corollary. — If τ ⊂ B_{n}(p, X) is any simplex then the family of simplices {c_{p}(v) : v is a vertex of τ ∩ S_{n}(p, X)} ∪ {τ ∩ B_{n−1}(p, X)} spans a simplex in B_{n−1}(p, X).

Proof (of Proposition 9). — In view of Corollary 16, for any simplex τ ⊂ B_{n+1}(p, X) the simplicial span of the family {c_{p}(v) : v is a vertex of τ ∩ S_{n+1}(p, X)} ∪ {τ ∩ B_{n}(p, X)} is a simplex. Denote this simplex by ĉ_{p}^{n}(τ) and note that it is contained in B_{n}(p, X). Put c_{p}^{n}(b_{τ}) to be the barycenter of ĉ_{p}^{n}(τ), and note that this defines the simplicial map [B_{n+1}(p, X)]′ → B_{n}(p, X) (between the first barycentric subdivisions) which we also denote c_{p}^{n}. It follows directly from the definition, that c_{p}^{n} satisfies assertions (1) and (2) of the proposition.

14. Systolic groups are biautomatic.

We refer the reader to [9] for the background on biautomatic groups. Biautomaticity implies various algorithmic and geometric properties of a group, in particular semihyperbolicity [1] and its consequences.

16. Theorem. — Let G be a group acting simplicially, properly discontinuously and cocompactly on a systolic simplicial complex X. Then G is biautomatic.

Proof. — The proof is based on the fact that directed geodesics in X are recognizable in local terms and satisfy fellow traveller property. Specifically, we will construct a finite symmetric subset A ⊂ G generating G as a semigroup, and a language L over A (whose strings are closely related to some directed geodesics in X) such that

1. L is regular;

2. the canonical map L → G is surjective;

3. L satisfies the 2-sided fellow traveller property.

To prove that L is regular, we shall construct a nondeterministic finite state automaton for which L is the accepted language.

Given a systolic group G acting on the corresponding complex X, put K = G\X′, where X′ is the barycentric subdivision of X. Since G acts on X′ without inversions (i.e. if an element g ∈ G fixes a simplex of X′ then it fixes all vertices in this simplex), K is a multisimplicial complex (simplices are embedded in K but a set of vertices may span more than one simplex). Moreover, since the action of G on X is cocompact, K is finite.

Generating set A. Choose a set of representatives V_{0} for the family of G-orbits in the vertex set V (X′) (with respect to the induced action of G on this set). For a vertex v ∈ V (X′) we shall denote by v̄ ∈ V_{0} the representative of its G-orbit. For any v ∈ V (X′) define the set Λ_{v} := {g ∈ G : v = gv̄} and call it the set of labels of v.

9. Fact. — Λ_{v} = g · Gv̄ = G_{v} · g for any g ∈ Λ_{v}, where Gv̄ and G_{v} are the stabilizers of the corresponding vertices in G.

Let E(X′) be the set of all pairs (v, w) ∈ V (X′) × V (X′) such that v, w span a 1-simplex of X′. For any pair (v, w) ∈ E(X′) put Λ_{v,w} := Λ_{v}^{−1} · Λ_{w}. Call the family Λ := {Λ_{v,w} : (v, w) ∈ E(X′)} the multilabelling on E(X′).

31. Lemma. —

1. Λ_{w,v} = Λ_{v,w}^{−1}.

2. Multilabelling Λ on E(X′) is G-invariant, i.e. Λ_{gv,gw} = Λ_{v,w} for any (v, w) ∈ E(X′) and any g ∈ G.

3. For a fixed v_{0} ∈ V_{0} the set

A := [⋃{Λ_{v,w} : (v, w) ∈ E(X′)} ∪ G_{v_{0}}] \ {1}

(where 1 is the unit of G) is a finite symmetric set generating G as a semigroup.

Proof. — Parts (1) and (2) are obvious. To prove (3), observe that by G-invariance multilabelling Λ on E(X′) induces the multilabelling on the set E_{K} of pairs of vertices that span a 1-simplex of K (we will denote this induced labelling also by Λ). Thus the finiteness of A follows from finiteness of K and from finiteness of the label sets Λ_{v,w}, as well as from finiteness of the stabilizers of vertices in X′ (implied by proper discontinuity of the action of G on X). The fact that A is symmetric follows from part (1). It remains to prove that A generates G as a semigroup.

Let g ∈ G be arbitrary. Let v_{0}, v_{1}, …, v_{n} = gv_{0} be the sequence of vertices in a polygonal path in the 1-skeleton of X′. For each v_{i} choose a label g_{i} ∈ Λ_{v_{i}} with the only restriction that g_{n} = g. Put λ_{i} := g_{i−1}^{−1}g_{i} for i = 1, …, n and note that g = g_{0}λ_{1}λ_{2} … λ_{n}. Since g_{0} ∈ Λ_{v_{0}} = G_{v_{0}} and λ_{i} ∈ Λ_{v_{i−1},v_{i}}, the lemma follows.

Language L. Fix V_{0} as above, a vertex v_{0} ∈ V_{0}, and take A to be the generating set as in Lemma 31.3. We define a language L over the alphabet A by describing, for arbitrary g ∈ G, the set of all strings in L that are mapped to g through the evaluation map L → G.

Let σ_{0}, σ_{1}, …, σ_{n−1}, σ_{n} be the directed geodesic in X from v_{0} to gv_{0}. It induces the sequence

σ_{0}, σ_{0} ∗ σ_{1}, σ_{1}, σ_{1} ∗ σ_{2}, …, σ_{n−1}, σ_{n−1} ∗ σ_{n}, σ_{n}

of simplices, and consequently the sequence b_{0}, b_{1}, …, b_{2n} of vertices in X′ being the barycenters of the simplices in the previous sequence. Clearly, this sequence corresponds to a polygonal path connecting v_{0} to gv_{0} in the 1skeleton of X′. Consider all strings over A defined in terms of the path b_{0}, b_{1}, …, b_{2n} as follows. For i = 0, 1, …, 2n choose a label g_{i} ∈ Λ_{b_{i}} arbitrarily, with the only restriction that g_{2n} = g. For i = 1, 2, …, n put λ_{i} := g_{i−1}^{−1}g_{i} and take the string g_{0}λ_{1} … λ_{2n} with omitted occurrences of the unit element of G. Note that for g = 1 this construction gives only the nullstring ε. Take as L the set of all such strings, for all g ∈ G.

It is clear from the description of L and from the existence of directed geodesics in X between any two vertices, that the evaluation map L → G is surjective. To prove fellow traveller property for L, consider the map φ : G → X given by φ(g) := gv_{0} and note that it is a quasi-isometry. Note also that paths in the Cayley graph C(G, A) corresponding to the strings of L are, by definition, mapped through φ uniformly close to the appropriate directed geodesics in X, where the distance is controlled by the diameter of the (finite due to cocompactness of G) set V_{0}. Thus the language L inherits the 2-sided fellow traveller property from the set of directed geodesics in X (Proposition 8). We omit straightforward details of this argument.

To get the fact that G is biautomatic, it remains to prove that the language L is regular.

Finite state automaton. Consider a nondeterministic finite state automaton M defined as follows. The unique start state in M is the vertex v ∈ K corresponding to the vertex v_{0} ∈ X′ through the quotient map X′ → K. Other states are the pairs (v, h) : h ∈ G_{v_{0}} and the triples (u, w, λ) : (u, w) ∈ E(K), λ ∈ Λ_{u,w}. The accept states in M are the state (v, 1) and the states of form (u, w, λ) with w = v.

There are three kinds of arrows in M.

1. For each h ∈ G_{v_{0}} there is an arrow labelled h from the start state v to the state (v, h).

2. For each u ∈ V (K) such that (v, u) ∈ E(K) and for each λ ∈ Λ_{v,u} there is an arrow labelled λ from each of the states (v, h) to the state (v, u, λ).

3. The third kind of arrows requires longer description. Suppose u, w, y are the vertices of K such that (u, w) ∈ E(K) and (w, y) ∈ E(K), and suppose λ ∈ Λ_{u,w} and μ ∈ Λ_{w,y}. Let ū, w̄, ȳ be the representatives in V_{0} of the G-orbits of these vertices. Note that then we have (λ^{−1}ū, w̄) ∈ E(X′) and (w̄, μȳ) ∈ E(X′). Denote by ρ, σ, τ respectively the simplices in X whose barycenters are λ^{−1}ū, w̄, μȳ. There is an arrow labelled μ from the state (u, w, λ) to the state (w, y, μ) if and only if one of the following two conditions holds:

(a) ρ and τ are disjoint and span σ;

(b) σ is a proper face in both ρ and τ and

Res(ρ_{σ}, X_{σ}) ∩ B_{1}(τ_{σ}, X_{σ}) = ∅.

Denote by L_{M} the language accepted by the automaton M. The fact that L ⊂ L_{M} follows easily from the description of strings in L. To prove the converse inclusion, consider any path of arrows in the automaton M that gives an accepted string of the language L_{M}. This path is uniquely determined by the corresponding sequence of states, and we denote this sequence by

u_{0}, (u_{0}, g_{0}), (u_{0}, u_{1}, λ_{1}), …, (u_{n−1}, u_{n}, λ_{n}),

where u_{0} = u_{n} = v, (u_{i−1}, u_{i}) ∈ E(K), g_{0} ∈ G_{v_{0}} and λ_{i} ∈ Λ_{u_{i−1},u_{i}} for 1 ≤ i ≤ n. A string in L_{M} obtained from this path is g_{0}λ_{1} … λ_{n}, where the occurrences of the unit 1 ∈ G are omitted.

For each 0 ≤ i ≤ n denote by g_{i} ∈ G the product g_{0}λ_{i} … λ_{i} and by ū_{i} the vertex in V_{0} representing the G-orbit in V (X′) corresponding to u_{i}. For each such i put b_{i} := g_{i}ū_{i} and denote by σ_{i} the simplex of X with barycenter b_{i}. Observe that any triple b_{i−1}, b_{i}, b_{i+1} can be expressed as

g_{i}λ_{i}^{−1}ū_{i−1}, g_{i}ū_{i}, g_{i}λ_{i+1}ū_{i+1}

and thus the triple ρ, σ, τ of simplices with barycenters λ_{i}^{−1}ū_{i−1}, ū_{i}, λ_{i+1}ū_{i+1} is mapped by g_{i} to the triple σ_{i−1}, σ_{i}, σ_{i+1}. By the description of arrows in M, and by the facts that σ_{0} = v_{0} and σ_{n} = g_{n}v_{0} are the vertices and that each g_{i} is a simplicial automorphism of X, we get that if i is odd then σ_{i−1} and σ_{i+1} are disjoint and span σ_{i}, while if i > 0 is even then σ_{i} is a proper face in both σ_{i−1} and σ_{i+1} and

Res((σ_{i−1})_{σ_{i}}, X_{σ_{i}}) ∩ B_{1}((σ_{i+1})_{σ_{i}}, X_{σ_{i}}) = ∅.

In particular, it follows that n is even and that for any even 0 < i < n we have

Res(σ_{i−2}, X_{σ_{i}}) ∩ B_{1}(σ_{i+2}, X_{σ_{i}}) = ∅.

Thus the sequence σ_{0}, σ_{2}, σ_{4}, …, σ_{n} is a directed geodesic in X from v_{0} to g_{n}v_{0}, and σ_{2i+1} = σ_{2i} ∗ σ_{2i+2} for any 0 ≤ i < n/2. Since we also have g_{i} ∈ Λ_{b_{i}} for 0 ≤ i ≤ n, the string g_{0}λ_{1} … λ_{n} (with occurrences of 1 deleted) has the form as in the description of the language L, i.e. it belongs to L. This proves the regularity of L.

15. Systolic versus CAT (κ).

In this section we discuss the relationship between k-systolic conditions and comparison CAT (κ) conditions for various metrics on simplicial complexes. As a main reference on CAT (κ) spaces we use [4].

We start with few remarks concerning the standard piecewise euclidean metrics on simplicial complexes. In these metrics each simplex is isometric with the regular euclidean simplex of the same dimension with side lengths equal 1. An easy observation shows that in dimension 2 a simplicial complex X is systolic if and only if it is CAT (0) with respect to the standard piecewise euclidean metric. A local version of this observation says that X is locally 6-large if and only if it is nonpositively curved. example in dimension 3, recall that the angle α in the regular 3-simplex between a 2-face and a 1-face meeting at a vertex is less than π/3. Consider a simplicial complex X which is the union of six 3-simplices defined as follows. Consider vertices v_{i} and 1-simplices e_{i} with i ∈ Z/3Z and a 2-complex K given as

K = ⋃_{i∈Z/3Z}(v_{i} ∗ e_{i} ∪ e_{i} ∗ v_{i+1}).

Take X to be the simplicial cone over K. X is easily seen to be 6-systolic, and on the other hand it is not CAT (0) since the spherical link of X at the cone vertex contains a closed geodesic of length 6α, which is less than 2π. Similar counterexamples can be constructed in any dimension n ≥ 3. This shows that 6-systolic complexes are not necessarily CAT (0) for the standard piecewise euclidean metric.

The converse implication between the two conditions is also not true in higher dimensions. Consider the n-dimensional simplicial complex Y_{n} equal to the simplicial join of an (n − 2)-dimensional simplex σ and the 1-dimensional cycle consisting of five edges. Clearly, Y_{n} is not 6-systolic, as its link at σ shows. On the other hand, the dihedral angle β_{n} in the regular n-simplex (between the faces of codimension 1) grows to π/2 as n grows to infinity. In fact, β_{n} > 2π/5 for all n ≥ 4. This implies that Y_{n} is CAT (0) if n ≥ 4, so a CAT (0) complex is not necessarily 6-systolic in these dimensions.

A much more subtle question is whether a 6-systolic complex admits any piecewise euclidean metric for which it is CAT (0). We do not have the answer to this question, but suspect it is negative.

An important problem that we study in the remaining part of this section is whether the stronger systolic conditions, i.e. k-systolicity for sufficiently large k, imply CAT (0) or even CAT (−1) condition for piecewise euclidean or piecewise hyperbolic metrics. Given a metric simplicial complex X, denote by Shapes(X) the set of isometry classes of the faces of X. Our main result in this section is the following.

17. Theorem. — Let Π be a finite set of isometry classes of metric simplices of constant curvature 1, 0 or −1. Then there is a natural number k ≥ 6, depending only on Π, such that:

1. if X is a piecewise spherical k-large complex with Shapes(X) ⊂ Π then X is CAT (1);

2. if X is piecewise euclidean (respectively, piecewise hyperbolic), locally k-large and Shapes(X) ⊂ Π then X is nonpositively curved (respectively, has curvature κ ≤ −1);

3. if, in addition to the assumptions of (2), X is simply connected then it is CAT (0) (respectively, CAT (−1)).

10. Remark. —

1. The above theorem, combined with the constructions of k-systolic complexes in Sections 19 and 20, provides large class of new interesting examples of CAT (1), CAT (0) and CAT (−1) spaces.

2. The proof of Theorem 17 given below does not lead to effective estimates for the number k. In Section 17 we explicitly estimate k for regular piecewise euclidean metrics.

Observe that parts (2) and (3) of the theorem follow directly from part (1) in view of characterization of the curvature bounds in terms of CAT (1) property for spherical links of a complex [4, Theorems 5.2 and 5.4, p. 206]. We thus concentrate on the proof of part (1).

We start with two auxiliary results for which we need some preparation. Given a closed geodesic γ in a piecewise spherical simplicial complex X with Shapes(X) finite, the size of γ is the number of maximal nontrivial subsegments in γ contained in a single simplex of X. Note that this number is always finite since any local geodesic of finite length in X is the concatenation of a finite number of segments, each contained in a simplex ([4, Corollary 7.29, p. 110]). The following result is a reformulation of [4, Theorem 7.28, p. 109] or [3, Lemma 1].

18. Theorem. — Given a finite set S of isometry classes of spherical simplices, there is a natural number N (depending on S) such that if a local geodesic γ in a piecewise spherical simplicial complex X with Shapes(X) ⊂ S has length less than 2π then its size is less than N.

Recall that a simplicial complex is ∞-large if it is k-large for any natural k. Using Definition 1 we can also characterize ∞-large simplicial complexes as those which are flag and contain no full cycle. In the proof of Theorem 17 we need the following.

10. Proposition. — Let X be a piecewise spherical ∞-large simplicial complex with Shapes(X) finite. Then X contains no closed local geodesic.

11. Remark. — Note that the above proposition implies that any piecewise spherical (with constant curvature 1) ∞-large simplicial complex is CAT (1). The straightforward argument for this uses the following two facts:

1. a piecewise spherical complex is CAT (1) if and only if neither this complex nor any of its (spherical) links contains a closed geodesic of length less than 2π (compare [4, Theorem 5.4(7), p. 206]);

2. links of an ∞-large simplicial complex are ∞-large.

Proposition 10 is the direct consequence of the following.

32. Lemma. — Let X be a piecewise spherical ∞-large simplicial complex with Shapes(X) finite. Then any local geodesic γ in X connecting two point of some simplex of X is contained in this simplex.

Proof. — Let K be the full subcomplex of X spanned by those simplices whose interiors are intersected by γ. Since γ has finite size, the subcomplex K is finite. Moreover, γ is a local geodesic in K and K is still ∞-large. We argue by induction with respect to the number of maximal simplices in K. When K is a single simplex the assertion is obvious. Otherwise, by Dirac’s characterization of finite ∞-large complexes (see Example 1.8(8)), K is obtained from some ∞-large complexes K_{1}, K_{2} by gluing along a single simplex σ. Note that γ has to be contained in K_{1} or K_{2}, since otherwise K_{1} or K_{2} contains some subsegment of γ with both endpoints in σ but not contained in σ, and this is impossible by inductive assumption. If γ is contained in K_{1} or K_{2}, the same inductive assumption implies the assertion, and the lemma follows.

Proof (of Theorem 17). — As mentioned before, it is sufficient to prove part (1) of the theorem, i.e. the case of piecewise spherical complexes.

Let S be the link completion of Π, i.e. the union of Π and the set of isometry classes of all links in simplices representing all classes from Π. Since Π is finite, so is S. Consider all closed geodesics γ of length less than 2π in all piecewise spherical flag simplicial complexes X with Shapes(X) ⊂ S. For each such geodesic denote by K_{γ} the full subcomplex in the corresponding complex X spanned by the union of all simplices of X whose interior is intersected by γ. There are only finitely many combinatorial types of complexes K_{γ} as above because, due to Theorem 18, the number of vertices in any such complex is bounded by a universal constant (e.g. by the product of a constant N from Theorem 18 for the set S and the maximal dimension of a simplex with isometry class in S).

Since each of the complexes K_{γ} contains a closed geodesic, it follows from Proposition 10 that it is not ∞-large. In particular, the systole sys(K_{γ}) of any such complex is finite. Put

k = max{sys(K_{γ}) : K_{γ} as above} + 1

and note that the maximum is taken over a finite set (due to finiteness of combinatorial types of complexes K_{γ}).

We claim that any k-large piecewise spherical simplicial complex Y with Shapes(Y) ⊂ Π is CAT (1). To prove this, observe that due to the definition of k, neither Y nor any of its links contains a closed geodesic of length less than 2π (this implies that Y is CAT (1), as already mentioned before; see [4, Theorem 5.4(7), p. 206]). If this were not the case, we would have the corresponding subcomplex K_{γ} with sys(K_{γ}) < k in a complex Z isomorphic either to Y or to some link of Y. Since Shapes(Z) ⊂ S, we would have K_{γ} containing a full cycle of length less than k. But, since K_{γ} is a full subcomplex in Z, the same cycle would be full in the complex Z, contradicting the fact that Y is k-large. This completes the proof.

12. Remark. — Theorem 17 applies in particular to finite dimensional simplicial complexes equipped with the standard piecewise euclidean metrics. Note however that for these metrics the number k in the assertion of the theorem necessarily grows to infinity as the dimension of a complex grows. To see this, recall that if σ is the regular spherical (2n − 1)-simplex with side lengths π/3 (i.e. the simplex occurring as the spherical link of the regular euclidean 2n-simplex at a vertex) then the distance d_{n} between the barycenters of opposite (n − 1)-faces in σ converges to 0 as n grows. In fact d_{n} = arccos(\frac{n}{n+1}). For any m ≥ 3 consider the simplicial complex X_{m}^{2n} of dimension 2n defined as the simplicial cone over the complex ∪_{i∈Z/mZ}τ_{i} ∗ τ_{i+1}, where τ_{i} is an (n − 1)-simplex for any i ∈ Z/mZ. Clearly, X_{m}^{2n} is an msystolic simplicial complex. If we equip it with the standard piecewise euclidean metric, its spherical link at the cone vertex obviously contains a closed geodesic of length m·d_{n}. A necessary condition for X_{m}^{2n} to be CAT (0) is that m · d_{n} ≥ 2π, i.e. that m ≥ 2π/d_{n}, which justifies our observation.

16. Acute angled complexes.

In this section we present another proof of Theorem 17, for the restricted case of acute angled complexes. Despite being less general, the proof has two advantages. First, its conclusion in the spherical case is stronger, namely that there is no homotopically trivial closed local geodesic both in the complex and in any of its links. Second, the proof in this section allows explicit and realistic estimates for the number k in the assertion. In Section 17 we give such estimates for standard piecewise euclidean metrics on complexes of any dimension.

A constant curvature simplex (spherical, euclidean or hyperbolic) is acute angled if all its dihedral angles (between codimension 1 faces) are less than π/2. A constant curvature metric simplicial complex is acute angled if all its faces are acute angled. Observe that if σ is an acute angled simplex then its links σ_{τ} at all faces τ are acute angled spherical simplices. Thus, all links of an acute angled complex are acute angled spherical complexes. Hence, as in Section 15, it is clearly sufficient to prove the theorem for (acute angled) spherical complexes.

We start with a few definitions. A small ball in a systolic simplicial complex X is a subcomplex of form B_{i}(σ, X) for some simplex σ of X and for some i ∈ {0, 1, 2}. Given a real number r > 0, we say that a subset A in a geodesic metric space X is r-convex if for any two points in A at distance in X less than r, any geodesic in X connecting these two points is contained in A. The proof of Theorem 17 presented in this section relies on the following.

11. Proposition. — Let X be a systolic piecewise spherical acute angled simplicial complex and suppose that

1. the set Shapes(X) is finite;

2. all links of X are CAT (1);

3. all the small balls in the links of X are π-convex.

Then X does not admit a closed local geodesic. Moreover, for any simplex ν in X any ball B = B_{m}(ν, X) is local-geodesically convex (i.e. any local geodesic segment in X with its endpoints in B is contained in B).

Before giving a proof we present two useful corollaries to Proposition 11. Note that, by combining assumption (1) and the first assertion of the proposition we get that X as above is CAT (1). This observation is refined in the first corollary below. The girth of the complex X, denoted girth(X), is the infimum of the lengths of homotopically nontrivial paths in X.

17. Corollary. — Let X be a locally 6-large piecewise spherical acute angled simplicial complex satisfying assumptions (0), (1) and (2) in Proposition 11, and suppose that girth(X) ≥ 2π. Then X is CAT (1).

Proof. — Recall that if all links of a piecewise spherical complex X are CAT (1) and if there is no closed geodesic in X of length less than 2π then X is CAT (1). It remains to check the second assumption in the above statement. By applying Proposition 11 to the universal covering of X we conclude that there are no closed homotopically trivial geodesics in X. On the other hand, the length of each homotopically nontrivial closed geodesic in X is not less then girth(X) ≥ 2π, and the corollary follows.

To prove Theorem 17 we will need another result easily implied by Proposition 11.

18. Corollary. — Let X be as in Corollary 17. Put

δ := max{diam(σ) : σ ∈ Shapes(X)}.

Suppose also that girth(X) ≥ π + 5δ. Then any small ball in X is π-convex.

Proof. — Fix a small ball B in X. It is sufficient to prove that any geodesic segment in X intersecting B only at its endpoints has length ≥ π. This is true if X is simply connected since it follows from the last assertion of Proposition 11 that there is no geodesic segment in X intersecting B only at its endpoints. Thus, in the general case, such a geodesic segment has to be homotopically nontrivial in X/B, and hence its length l can be estimated by

l ≥ girth(X) − diam(B) ≥ girth(X) − 5δ ≥ (π + 5δ) − 5δ = π.

This finishes the proof.

To prove Proposition 11 we need four preparatory results.

10. Fact. — Let K be a connected subcomplex in a CAT (1) piecewise spherical complex S. Suppose that links of K are π-convex in the corresponding links of S and that diam(K) < π. Then (1) K is π-convex in S and (2) K is CAT (1).

Proof. — Since diam(K) < π, any two points of K are connected by a geodesic segment in K of length less than π. Since K is locally π-convex in S, this segment is a local geodesic in S (compare [4, Remark 5.7, p. 60] or [7, Lemma 1.6.5]). Since S is CAT (1), this segment is a geodesic in S ([4, Proposition 1.4(2), p. 160]) and, since S is π-uniquely geodesic (condition (6) in [4, Theorem 5.4, p. 206]), it is the unique geodesic in S connecting these two points, hence (1). The same argument shows that K is π-uniquely geodesic, hence (2) (by equivalence of (5) and (6) in [4, Theorem 5.4, p. 206]).

33. Lemma. — Let X_{0} = σ ∗ X and Y_{0} = σ ∗ Y, where Y is a subcomplex in a simplicial complex X, σ is a simplex and ∗ denotes the simplicial join. Suppose that X_{0} is equipped with a piecewise spherical metric with all simplices acute angled, and that the spherical link (X_{0})_{σ} is CAT (1) while the spherical link (Y_{0})_{σ} is π-convex in (X_{0})_{σ}. Then X_{0} is CAT (1) and Y_{0} is π-convex in X_{0}.

Proof. — Denote by ∗_{s} the operation of spherical join for piecewise spherical complexes. Viewing the simplex σ as embedded in the sphere S^{n} of dimension n = dim σ, we can consider the embedding i : X_{0} → S^{n} ∗_{s} (X_{0})_{σ} which is isometric on simplices of X_{0}. For an appropriate choice of a piecewise spherical simplicial structure on S^{n} ∗_{s} (X_{0})_{σ}, the map i identifies X_{0} as a subcomplex in S^{n} ∗_{s} (X_{0})_{σ}.

We will prove simultaneously the following three statements by induction on k = dim(X_{0}):

1. X_{0} is π-convex in S^{n} ∗_{s} (X_{0})_{σ} (here we identify X_{0} with its image i(X_{0}) through the embedding i);

2. X_{0} is CAT (1);

3. Y_{0} is π-convex in X_{0}.

The statements are clearly true if k = 1. The inductive step will be based on the observation that the assumptions in 15.5 are inherited by pairs of spherical links (X_{0})_{τ}, (Y_{0})_{τ} for any simplex τ of Y_{0}. More precisely, denote by σ + τ the smallest simplex in X_{0} containing both σ and τ, and by σ − τ the maximal face of σ disjoint with τ (empty, if σ ⊂ τ). Then, for any τ in X_{0} we have (X_{0})_{τ} = (σ − τ) ∗ (X_{0})_{σ+τ}. Moreover, if τ is contained in Y_{0}, we also have (Y_{0})_{τ} = (σ − τ) ∗ (Y_{0})_{σ+τ}. The metric assumptions of the lemma are satisfied for these links because both CAT (1) and π-convexity are the properties inherited by links.

Fix the pair X_{0}, Y_{0} as in the lemma and suppose inductively that the assertions (1)-(3) are satisfied by the pairs of links as above. View X_{0} as a subset in the spherical join S^{n} ∗_{s} (X_{0})_{σ}. If τ is a simplex of X_{0} containing σ, then the metric links (X_{0})_{τ} and [S^{n} ∗_{s} (X_{0})_{σ}]_{τ} coincide. Otherwise, the inclusion (X_{0})τ ⊂ [S^{n} ∗_{s} (X_{0})_{σ}]_{τ} has the same form as the inclusion X_{0} ⊂ S^{n} ∗_{s} (X_{0})_{σ}. More precisely, the link [S^{n} ∗ (X_{0})_{σ}]_{τ} canonically identifies with the spherical join S^{m} ∗_{s} (X_{0})_{σ+τ}, where S^{m} is the sphere of dimension m = dim(σ+τ)_{τ}. Moreover, (X_{0})_{τ} has the form as X_{0}, with the simplex (σ+τ)_{τ} = σ − τ playing the role of σ, and with [(X_{0})_{τ}]_{σ−τ} = (X_{0})_{σ+τ} (metrically). An inclusion of σ − τ in S^{m} determines then the inclusion of (X_{0})_{τ} in S^{m} ∗_{s} (X_{0})_{σ+τ} which coincides with the inclusion of the metric links at τ of X_{0} and S^{n} ∗_{s} (X_{0})_{σ}.

By combining the above observation with assertion (1) in the inductive assumption we conclude that links of X_{0} are π-convex in the corresponding links of the simplicial join S^{n} ∗_{s} (X_{0})_{σ}. Since this join is CAT (1) (because spherical joins of CAT (1) spaces are CAT (1)) and diam(X_{0}) < π (due to acute angleness), assertions (1) and (2) for X_{0} follow from assertions (1) and (2) of Fact 10.

To prove that Y_{0} is π-convex in X_{0}, observe that links of Y_{0} are πconvex in the corresponding links of X_{0}. In view of the above described forms of links of X_{0} and Y_{0} this follows from the statement (3) in the inductive assumption. Since X_{0} is already proved to be CAT (1), and the diameter of Y_{0} is less than π, Fact 10.1 implies statement (3) for the pair X_{0}, Y_{0}, and the lemma follows.

Next two preparatory results concern combinatorial properties (related to convexity) of balls in systolic simplicial complexes. We fix the following assumptions and notation for these two results. Let X be a systolic simplicial complex with dim(X) = n and let ν be a simplex of X. For a ball B = B_{m}(ν, X) in X with m ≥ 1 and with the sphere S = S_{m}(ν, m) (as defined in Section 8) consider the sequence B = B^{0} ⊂ B^{1} ⊂ B^{2} ⊂ ⋯ ⊂ B^{n} = N_{X}(B) of subcomplexes in X defined recursively by

B^{i} = B^{i−1} ∪ ⋃{Res(σ, X) : σ ⊂ S, dim(σ) = n − i}.

Clearly, we then have B^{n} = B_{1}(B, X) = B_{m+1}(ν, X).

34. Lemma. — Let σ ⊂ S be a simplex of dimension n − i. Then

1. the link (B^{i−1})_{σ} is a small ball in X_{σ};

2. B^{i−1} ∩ Res(σ, X) = Res(σ, B^{i−1}).

Proof. — Recall that, by Corollary 7, B_{σ} = B_{1}(ρ, X_{σ}) for some simplex ρ ⊂ X_{σ}. Moreover, from the definition of B^{i−1} it follows that (B^{i−1})_{σ} = B_{1}(B_{σ}, X_{σ}), hence (1).

To prove (2), suppose that τ is a simplex in B^{i−1} ∩Res(σ, X). Let τ_{1}, τ_{2} be the maximal faces of τ disjoint from B and contained in B respectively. The latter is well defined since B is full in X (Corollary 6 and Lemma 11.2). For the same reason τ_{1}, τ_{2} span τ and that τ_{1}, σ span a simplex of B. Since τ_{2} ⊂ B^{i−1}, there is a simplex ρ ⊂ S of dimension at least n − i + 1 such that τ_{2} ⊂ Res(ρ, X). On the other hand, by Lemma 14, Res(τ_{2}, X) ∩ B is a single simplex, and since it contains ρ, its dimension is at least n − i + 1. It follows that [Res(τ_{2}, X)∩B]∗τ_{2} is a simplex in B^{i−1}. But Res(τ_{2}, X)∩B also contains σ, hence σ and τ = τ_{1} ∗ τ_{2} span a simplex of B^{i−1}. This gives the inclusion B^{i−1} ∩ Res(σ, X) ⊂ Res(σ, B^{i−1}), and since the converse inclusion is obvious, the lemma follows.

35. Lemma. — If σ_{1}, σ_{2} are two distinct simplices of dimension n − i in S then Res(σ_{1}, X) ∩ Res(σ_{2}, X) ⊂ B^{i−1}.

Proof. — Let τ ⊂ Res(σ_{1}, X) ∩ Res(σ_{2}, X) and suppose that, contrary to the assertion, τ is not contained in B^{i−1}. Since B is full in X (see Corollary 6 and Lemma 11.2) and B ⊂ B^{i−1}, by passing to a face of τ if necessary, we may (and will) assume that τ is disjoint with B. By convexity of B we know that the intersection Res(τ, X) ∩ B, which contains both σ_{1} and σ_{2}, is then a single simplex (Lemma 14) which is contained in S and which we denote by σ. It follows that dim σ > dim σ_{1} = dim σ_{2} = n − i, and hence Res(σ, X) ⊂ B^{i−1}. But τ is clearly contained in Res(σ, X), and hence also in B^{i−1}, a contradiction. Hence the lemma.

Proof (of Proposition 11). —

It is sufficient to prove the last assertion in the statement of the proposition, i.e. that balls in X are local-geodesically convex: if there is a closed local geodesic γ in X then any ball intersecting γ and not containing it is not local-geodesically convex.

By the assumption that Shapes(X) is finite we know that a local geodesic in X of finite length is the concatenation of a finite number of segments, each contained in a simplex of X ([4, Corollary 7.29, p. 110]). Thus, to prove the proposition, it is sufficient to apply recursively the following Claim. A local geodesic γ in X that leaves a ball B = B_{m}(ν, X) does not return to B before leaving the ball B_{m+1}(ν, X).

Suppose that dim(X) = n. To get the claim it is sufficient to show that, for any 1 ≤ i ≤ n, if a local geodesic γ leaves B^{i−1} then it does not return to B^{i−1} before leaving B^{i}.

Let γ be a local geodesic that leaves B^{i−1}. We may assume that γ is a local geodesic ray in X starting at a point p ∈ B^{i−1} and locally near p intersecting B^{i−1} only at p. It may happen that γ leaves B_{i} at the same moment, i.e. that locally near p it intersects B^{i} only at p. Then our assertion holds. We will then consider the opposite case, when γ remains in B^{i} near p.

Note that, due to Lemma 35, the sets Res(σ, X) \ B^{i−1} for all simplices σ ⊂ S with dim σ = n − i are pairwise disjoint. Thus, leaving B^{i−1}, γ enters exactly one of them. Again by Lemma 35, it is sufficient to show that γ does not return to B^{i−1} before leaving Res(σ, X).

Now we make use of Lemma 33. Put X_{0} = Res(σ, X) and Y_{0} = B^{i−1} ∩ X_{0}. We then have X_{0} = σ ∗ X_{σ} and, by Lemma 34, Y_{0} = σ ∗ (B^{i−1})_{σ} (simplicially). Since, by Lemma 34.1, (B^{i−1})_{σ} is a small ball in X_{σ}, it follows from assumptions of Proposition 11 that the pair X_{0}, Y_{0} satisfies both combinatorial and metric assumptions of Lemma 33. Thus X_{0} is CAT (1) while Y_{0} is π-convex in X_{0}.

Any part of the local geodesic γ passing through X_{0} is clearly a local geodesic in X_{0}. Moreover, since a local geodesic of length less than π in a CAT (1) space is a geodesic, and since diam(X_{0}) < π, any local geodesic in X_{0} has length less than π. By π-convexity of Y_{0}, the maximal initial segment γ_{0} of γ contained in X_{0} (which has length less than π) intersects Y_{0} only at the initial point p, and hence it intersects B^{i−1} only at p. Thus, γ does not return to B^{i−1} before leaving X_{0} = Res(σ, X), which completes the proof.

Proof (of Theorem 17 (for acute angled piecewise spherical complexes)). —

Note first that the theorem clearly holds for complexes X with dim X ≤ 1. Moreover, the number k can be chosen so large that additionally the small balls in those complexes are all π-convex. We will prove theorem together with the additional property of π-injectivity for all small balls in X, using induction with respect to n = dim X.

Suppose that the theorem and the assertion that all small balls in X are π-convex holds for all complexes X with dim X ≤ n. Let Π be a finite set of (isometry classes of) acute angled spherical simplices, and denote by L(Π) the set of (isometry classes of) all links of simplices from Π. Then L(Π) is also finite. Let k_{1} be a natural number as prescribed by the inductive assumption for complexes X with Shapes(X) ⊂ L(Π) and dim X ≤ n. Let X be a k_{1}-large complex with Shapes(X) ⊂ Π and with dim(X) = n + 1. Then, by the inductive assumption, the links of X are CAT (1) and all small balls in those links are π-convex. Thus X satisfies the assumptions of Proposition 11, and hence also the assumptions of Corollaries 17 and 18 except perhaps those concerning girth. To get the inductive step, note that by requiring that sys_{h}(X) ≥ k for sufficiently large k ≥ k_{1} we can assure that girth(X) is as large as we wish. In particular, we can assure that girth(X) ≥ max(2π, π + 5δ), where δ = max{diamΔ : Δ ∈ Π}. It follows that if X is klarge (which implies that links of X are k_{1}-large and sys_{h}(X) ≥ k) then X is CAT (1) (Corollary 17) and the small balls in X are π-convex (Corollary 18). This finishes the inductive proof.

13. Remark. — In the next section we give explicit estimates of girth(X) in terms of sys_{h}(X) for piecewise spherical complexes occurring as links in complexes with the standard piecewise euclidean metric. In view of the last part of the above proof, this gives explicit constants k in Theorem 17, depending only on dimension, for complexes with the standard piecewise euclidean metric. In principle, such explicit estimates for constants k can be obtained for other finite sets of acute angled shapes as well.

17. Explicit constants.

In this section we prove more explicit version of Theorem 17.3, for complexes with standard piecewise euclidean metrics. It is obtained by referring to the arguments from Section 16. A large part of the section deals with more general metrics and the obtained results can be used to derive explicit estimates for other classes of piecewise constant curvature acute angled complexes. In the case that we study in detail we get the following.

19. Theorem. — Let k be a natural number such that

k ≥ \frac{7π\sqrt{2}}{2} · n + 2.

Then any k-systolic simplicial complex X with dim X ≤ n is CAT (0) with respect to the standard piecewise euclidean metric.

14. Remark. — The estimate for k in the above theorem is obviously not optimal. It gives k ≥ 34 for n = 2, while k ≥ 6 is clearly sufficient. For n = 3 the theorem gives n ≥ 49, while a careful application of our methods allows to get k ≥ 11. The estimate seems also to be far from optimal asymptotically, as k → ∞. We expect that k ≥ C · \sqrt{n} for some constant C is sufficient. Note that, since in Remark 12 we have d_{n} ∼ 1/\sqrt{n}, the latter prediction coincides with the necessary condition observed in this remark. Also, our choice of functions φ_{L,j}^{n} (just before Lemma 13.7, below) is clearly not optimal, and we see potential for improvement of the estimate in making this choice more carefully.

To prove Theorem 19 we need a few preparatory results. We formulate first three of them in the framework of piecewise riemannian simplicial complexes (though we are interested in piecewise spherical case only), because it exhibits better the essence of our arguments.

Recall that a riemannian simplex is a simplex equipped with a smooth riemannian metric. A piecewise riemannian simplicial complex is obtained from riemannian simplices by gluing them together along some of their faces through diffeomorphisms that preserve riemannian metrics restricted to those faces. A piecewise riemannian simplicial complex having the set Shapes(X) of its riemannian simplices finite, is equipped with the metric given by minimizing lengths of piecewise smooth paths. Our aim is the following.

12. Proposition. — Let S be a finite set of isometry classes of riemannian simplices. Then there exists a constant D_{S} > 0 such that if X is a metric simplicial complex with Shapes(X) ⊂ S then girth(X) ≥ D_{S} · (sys_{h}(X) − 2).

In the proof of the above proposition we will need to estimate distances in complexes X in terms of gradients of some piecewise smooth functions. A function f : X → R is piecewise smooth if its restriction f|_{σ} to any simplex σ ⊂ X is smooth. Given such a function, put

M_{f} := sup{max{||∇(f|_{σ})(x)|| : x ∈ σ} : σ ⊂ X},

where ∇ denotes gradient and || · || denotes length (for vectors tangent to σ) with respect to the riemannian metric on σ. One of the well known properties of gradient is the following.

36. Lemma. — Let f : X → R be a piecewise smooth function on a connected metric (riemannian) simplicial complex X. Then for any points p, q ∈ X we have |f(p) − f(q)| ≤ M_{f} · d_{X}(p, q). In particular, if the supremum M_{f} is finite then

d_{X}(p, q) ≥ \frac{1}{M_{f}} · |f(p) − f(q)|.

Given a connected simplicial complex X and a simplex σ ⊂ X, a distance-like function for (X, σ) is a piecewise smooth function f : X → R such that S_{i}(σ, X) ⊂ f^{−1}(i). (Recall that S_{i}(σ, X) is a subcomplex of X spanned by the set of all vertices in X at polygonal distance i from σ.)

37. Lemma. — Given a finite set S of isometry classes of riemannian simplices, there is a constant 0 < M_{S} < ∞ with the following property: For any connected metric simplicial complex X with Shapes(X) ⊂ S and for any simplex σ ⊂ X there is a distance-like function f for (X, σ) with M_{f} ≤ M_{S}.

Proof. — We will show that distance-like functions for complexes X with Shapes(X) finite can be constructed out of an essentially finite collection F_{S} of functions on the simplices from Shapes(X). (By saying that F_{S} is essentially finite we mean that it is obtained from some finite sub-collection F_{S}^{0} by adding constants.) This clearly implies the lemma since for such functions the supremum M_{f} is taken essentially over a subset in a finite set of numbers, namely the set of maxima of gradient lengths for functions in F_{S}. A collection F_{S} as above can be constructed as follows.

For each 1-simplex E in S consider all combinations of values 0 and 1 at the vertices of E. For each such combination take a smooth function φ : E → R compatible with the prescribed values at vertices and such that φ is constant if the two values at vertices are equal. Further, for each 2-simplex Δ in S consider all combinations of values 0 and 1 at the vertices. Given such a combination, for each boundary face of Δ consider the already defined function on the simplex in S isometric to this face, respecting the prescribed values at vertices. Extend the so obtained function on the boundary of Δ to a smooth function on Δ so that it is a constant function if the prescribed values at the vertices are all equal. By applying this procedure gradually to the simplices in S of all dimensions we get a finite collection F_{S}^{0} of functions. As F_{S} take the set of all functions obtained from the functions in F_{S}^{0} by adding constants from the set of natural numbers (including 0).

For any complex X with Shapes(X) ⊂ S and for any simplex σ ⊂ X one can construct a distance-like function f for (X, σ) simplex-wise, out of the functions from F_{S}, as follows. As values of f at the vertices of X take their polygonal distances from σ. Next, observe that for any simplex τ in X one of the following two cases holds:

1. the values of f at the vertices of τ are all equal;

2. the set of values of f at the vertices of τ consists of two natural numbers that differ by 1.

This observation shows that we can extend f gradually to higher dimensional skeleta of X, using the functions from F_{S}.

By the construction of the functions in F_{S} we know that if for some simplex τ in X the above case (1) holds then a function f obtained as above is constant at τ. This implies that f is also constant at the spheres S_{i}(σ, X), with values i, and thus it is a distance-like function for (X, σ), as required. This finishes the proof.

Proof (of Proposition 12). — Let X̃ be the universal cover of X with the lifted metric. Then girth(X) is equal to the infimum of the distances dX̃(p_{1}, p_{2}) over all points p ∈ X and all pairs p_{1}, p_{2} of distinct lifts of p to X̃. Fix a pair p_{1}, p_{2} as above, and let σ be a simplex of X̃ containing p_{1}. Observe that, if m = sys_{h}(X), then p_{2} lies outside the ball B_{m−2}(σ, X̃). It follows that

dX̃(p_{1}, p_{2}) ≥ inf{dX̃(p_{1}, q) : q ∈ S_{m−2}(σ, X̃)}.  (1)

Let M_{S} be as in Lemma 13.4. Since Shapes(X̃) ⊂ S, the same lemma implies that there is a distance-like function f for (X̃, σ) with M_{f} ≤ M_{S}. We clearly have f(p_{1}) = 0 and f(q) = m − 2 for any q ∈ S_{m−2}(σ, X̃). Applying Lemma 13.3 we get

dX̃(p_{1}, q) ≥ \frac{1}{M_{f}} · (m − 2) ≥ \frac{1}{M_{S}} · (m − 2) = \frac{1}{M_{S}} · (sys_{h}(X) − 2).

Combining this with the inequality (16.2.1) we get the proposition for D_{S} = 1/M_{S}.

We now shift our attention to piecewise constant curvature acute angled complexes. We will apply Proposition 12 together with the results and ideas of Section 16 to get the following.

13. Proposition. — Let S_{0} be a finite set of isometry classes of acute angled spherical simplices, and denote by S its link completion, i.e. the union of S_{0} and the set of isometry classes of links at all faces for all simplices in S_{0}. Let D_{S} be a constant as in Proposition 12, and k a natural number such that

k ≥ max[6, \frac{7π}{2D_{S}} + 2].

If X is a k-large piecewise spherical complex with Shapes(X) ⊂ S_{0} then X is CAT (1).

By applying the characterization of the CAT (0) and CAT (−1) conditions in terms of the CAT (1) condition for links (see condition (4) in [4, Theorem 5.4, p. 206]), Proposition 13 implies the following.

19. Corollary. — Let T be a finite set of isometry classes of acute angled euclidean (respectively hyperbolic) simplices, and denote by S the set of isometry classes of links at all faces for all simplices in T . Let D_{S} be a constant as in Proposition 12, and k a natural number such that

k ≥ max[6, \frac{7π}{2D_{S}} + 2].

If X is a k-systolic piecewise euclidean (respectively piecewise hyperbolic) complex with Shapes(X) ⊂ T then X is CAT (0) (respectively CAT (−1)).

Proof (of Proposition 13). — First note that if X is k-large then sys_{h}(X) ≥ k and sys_{h}(X_{σ}) ≥ k for all links X_{σ} of X. It follows then from Proposition 12 that girth(X) ≥ 7π/2 and girth(X_{σ}) ≥ 7π/2 for all links X_{σ}. Moreover, by acuteness, diameters of all simplices in X and in all links X_{σ} are less than π/2, so if δ is as in Corollary 18 for X or for X_{σ} respectively, we get

girth(X) ≥ \frac{7π}{2} ≥ π + 5δ and girth(X_{σ}) ≥ \frac{7π}{2} ≥ π + 5δ.

Now, using induction with respect to the dimension of complexes, based on Corollaries 17, 18 and on the above inequalities, we get that all links X_{σ} in X are CAT (1) and all small balls in them are π-convex. In the end of this inductive proof we get that X is CAT (1), hence the proposition.

In the next series of preparatory results we study piecewise spherical complexes composed of regular simplices with fixed side lengths. Such complexes occur as links in complexes with standard piecewise euclidean metrics. We define and study some functions on the regular spherical simplices. These functions allow to construct appropriate distance-like functions on the complexes as above and to calculate explicitly the constants M_{S} as in Lemma 13.3 in the situations under our interest.

Let Σ_{L}^{n} be the n-dimensional spherical (with constant curvature 1) regular simplex with side lengths L. This makes sense for 0 < L < 2π/3, but we will be interested in the cases when π/3 ≤ L < π/2. Let S^{n} be the sphere of radius 1 canonically embedded in the euclidean space E^{n+1}, and suppose that Σ_{L}^{n} is embedded in S^{n}. Denote by Δ_{L}^{n} the simplex in E^{n+1} affinely spanned by the vertices v_{1}, …, v_{n+1} of Σ_{L}^{n}, with the induced regular euclidean metric in which the sides of Δ_{L}^{n} have lengths 2 sin(L/2). Consider also the radial projection P_{L}^{n} : Σ_{L}^{n} → Δ_{L}^{n}, in the direction of the center of S^{n}, which is clearly a diffeomorphism. For j = 0, 1, …, n let λ_{L,j}^{n} be the linear function on the simplex Δ_{L}^{n} with values 1 at the vertices v_{1}, …, v_{j} and 0 at the remaining vertices. Finally, define functions φ_{L,j}^{n} : Σ_{L}^{n} → R by putting φ_{L,j}^{n} := λ_{L,j}^{n} ∘ P_{L}^{n}.

38. Lemma. — For j = 1, …, n let H_{L,j}^{n} be the distance in Δ_{L}^{n} between the barycenters of opposite faces of dimensions j − 1 and n − j. Denote also by β_{L}^{n} the distance in the simplex Σ_{L}^{n} between its barycenter and any of its vertices. Then

max\{||∇φ_{L,j}^{n}(x)|| : x ∈ Σ_{L}^{n}\} ≤ \frac{1}{H_{L,j}^{n} · cos β_{L}^{n}}

for j = 1, …, n.

Proof. — Note that, since the function λ_{L}^{n} is linear and takes the values 0 and 1 at the opposite faces of dimensions j − 1 and n − j respectively, we have

(16.7.1)  ||∇λ_{L,j}^{n}(y)|| = \frac{1}{H_{L,j}^{n}} for each y ∈ Δ_{L}^{n}.

Since φ_{L,j}^{n} = λ_{L,j}^{n} ∘ P_{L}^{n}, we may use the following estimate for gradient length of a pulled back function, which follows directly from the chain rule.

11. Fact. — Let M_{1}, M_{2} be riemannian manifolds, f : M_{1} → R a smooth function and q : M_{2} → M_{1} a smooth map. Then for any x ∈ M_{2} we have

(16.7.2)  ||∇(f ∘ q)(x)|| ≤ ||∇f(q(x))|| · ||dq_{x}||,

where ||dq_{x}|| is the norm of the differential dq_{x} : T_{x}M_{2} → T_{f(x)}M_{1} with respect to riemannian norms at tangent spaces.

To apply the above fact in our proof we need to estimate the norms ||(dφ_{L,j}^{n})_{x}|| for x ∈ Σ_{L}^{n}. View again Σ_{L}^{n} as embedded in S^{n} ⊂ E^{n+1}, and Δ_{L}^{n} as affinely spanned in E^{n+1} by the vertices of Σ_{L}^{n}. The riemannian lengths of vectors tangent to Σ_{L}^{n} and Δ_{L}^{n} coincide then with the ordinary euclidean lengths of these vectors in E^{n+1}. Fix any x ∈ Σ_{L}^{n} and any vector V tangent to σ_{L}^{n} at x. Put y = P_{L}^{n}(x) ∈ Δ_{L}^{n} and note that the differential (dP_{L,j}^{n})_{x} : T_{x}Σ_{L}^{n} → T_{y}Δ_{L,j}^{n} is the restriction of the differential dP_{x} : T_{x}E^{n+1} → T_{y}Δ^{n} of the radial projection in E^{n+1} (with respect to the center of S^{n}) from an open neighbourhood U of Σ_{L}^{n} to the hyperplane containing Δ_{L}^{n}. Let V = V_{r} + V_{p}, where V_{r} is the radial component of V in E^{n+1} (parallel to the radius of S^{n} through x) and V_{p} is its component parallel to Δ_{L}^{n}. Since clearly dP_{x}(V_{r}) = 0 and dP_{x}(V_{p}) = a · V_{p}, where a ≤ 1 is the ratio of the distances from the center of S^{n} of the points y and x respectively, we get (dP_{L}^{n})_{x}(V) = a·V_{p}. To estimate the length of the component V_{p}, denote by α_{x} the angle between the radii in S^{n} through the barycenter of σ_{L}^{n} and through x. Since α_{x} is also the dihedral angle between the hyperplane tangent to Σ_{L}^{n} at x and the hyperplane containing Δ_{L}^{n}, we get ||V_{p}|| ≤ ||V ||/ cos α_{x}. But in our case we have α_{x} ≤ β_{L}^{n} and we obtain an estimate

||(dP_{L}^{n})_{x}(V)|| = ||a · V_{p}|| ≤ \frac{a}{cos β_{L}^{n}} ||V || ≤ \frac{1}{cos β_{L}^{n}} ||V ||.

This shows that

||(dP_{L}^{n})_{x}|| ≤ \frac{1}{cos β_{L}^{n}} for each x ∈ Σ_{L}^{n}.

By combining this with (16.7.1) and (16.7.2) the lemma follows.

20. Corollary. — If π/3 ≤ L ≤ π/2 then

max\{||∇φ_{L,j}^{n}(x)|| : x ∈ Σ_{L}^{n}\} ≤ \frac{(n + 1)\sqrt{2}}{2}

for j = 1, …, n.

Proof. — Note that the size of the simplex Δ_{L}^{n} increases with the increase of L and hence for L ≥ π/3 we have H_{L,j}^{n} ≥ H_{π/3,j}^{n}. A direct computation in the simplex Δ_{π/3}^{n} (which has side lengths 1) shows that

H_{π/3,j}^{n} ≥ \begin{matrix}√ & \overline{2} \\ √ & \overline{n + 1} \end{matrix}

for any 1 ≤ j ≤ n. On the other hand, if L ≤ π/2, we clearly have β_{L}^{n} ≤ β_{π/2}^{n}. By a direct computation in the right-angled spherical simplex Σ_{π/2}^{n} we get that cos β_{π/2}^{n} = 1/\sqrt{n + 1} which implies that

(16.8.2)  cos β_{L}^{n} ≥ \frac{1}{\sqrt{n + 1}}.

Combining the inequalities (16.8.1) and (16.8.2) with the inequality from Lemma 13.7 finishes the proof.

Proof (of Theorem 19). — Note that, due to the definition of the functions φ_{L,j}^{n} in terms of linear functions and radial projections, the restriction of any such function to a face Σ_{L}^{n′} in Σ_{L}^{n} is either constant equal to 1 or coincides with the appropriate function φ_{L,j′}^{n′}. Thus, the functions obtained from the functions φ_{L,j}^{n} (for all n and j) by adding natural constants are sufficient to construct distance-like functions as in the proof of Lemma 13.4 for metric complexes with all simplices spherical regular of side length L. Denoting by S_{L}^{n} the set of (isometry classes of) simplices Σ_{L}^{i} with 0 ≤ i ≤ n, and assuming that π/3 ≤ L ≤ π/2, we get from Corollary 20 that M_{S_{L}^{n}} = (n + 1)\sqrt{2}/2 works in Lemma 13.4 for S = S_{L}^{n}.

Let T be the set of isometry classes of the standard regular euclidean simplices of dimensions ≤ n. Then the set S, as in Corollary 20, of isometry classes of links at all faces for all simplices in T can be expressed as the union

S = ∪_{i=0}^{n−2}S_{L_{i}}^{n−1−i},

where each of the sets S_{L_{i}}^{n−1−i} consists of links at i-dimensional faces and the numbers L_{i} = arccos(1/(k + 2)) are the side lengths in such links, as a direct calculation shows. Since π/3 ≤ L_{i} < π/2, the argument above shows that we can take M_{S} = n\sqrt{2}/2 in the conclusion of Lemma 13.4 for S as above. Consequently, by referring to the end of proof of Proposition 12, we can take D_{S} = 1/M_{S} = \sqrt{2}/n in the conclusion of Corollary 19, for T and S as above, hence the theorem.

18. Locally 6-large simplices of groups

In this section we recall and adapt to our needs some notions and facts related to simplices of groups and simple complexes of groups. We will use them in the construction described in Section 19. Since both simplices of groups and simple complexes of groups are special cases of complexes of groups, some parts of this section repeat the exposition of Section 7 in these special cases. However, the exposition here is more detailed, more self-contained, and free from several technicalities. For example, we do not mention twisting elements g_{στρ}, since they are all assumed to be trivial, i.e. equal to the units in the corresponding groups. On the other hand, we discuss explicitly the notions related to developability. We also change slightly the notation to make it more convenient for our purposes. The reader is advised to consult Section 12 in Part II of [4] as a standard reference.

For a simplex Δ, denote by P_{Δ} the poset of all nonempty faces of Δ, including Δ itself, and denote by < the relation of being a proper (sub)face. A simplex of groups G over Δ is a family G_{σ} : σ ∈ P_{Δ} of groups, together with a family of homomorphisms ψ_{στ} : G_{τ} → G_{σ} for any pair σ < τ, such that ψ_{στ} ∘ ψ_{τρ} = ψ_{σρ} whenever σ < τ < ρ. We will call groups G_{σ} local groups of G and homomorphisms ψ_{στ} structure homomorphisms of G.

A morphism m : G → F from a simplex of groups G over Δ to a group F is a family m_{σ} : σ ∈ P_{Δ} of homomorphisms m_{σ} : G_{σ} → F which agree with the structure homomorphisms of G in the sense that m_{τ} = m_{σ} ∘ψ_{στ} whenever σ < τ. Given a simplex of groups G, denote by Ĝ the direct limit of G, i.e. the quotient group of the free product of the groups G_{σ} : σ ∈ P_{Δ} by the normal subgroup generated by relations of form g = ψ_{στ}(g) for all structure homomorphisms ψ_{στ} and all g ∈ G_{τ}. Denote by i_{G} : G → Ĝ the canonical morphism to the direct limit. This morphism has (or can be characterized by) the universal property saying that any morphism m : G → F factors through i_{G}, i.e. there is the homomorphism m̂ : Ĝ → F such that m = m̂ ∘ i_{G}. The homomorphism m̂ is unique and we call it the homomorphism induced by m.

A morphism m : G → F is locally injective if all its homomorphisms m_{σ} are injective. It is surjective if the target group F is generated by the union ⋃_{σ∈P_{Δ}} m_{σ}(G_{σ}). A simplex of groups is developable if it admits a locally injective morphism (equivalently, if its canonical morphism to the direct limit is locally injective). Locally injective and surjective morphisms can be characterized in terms of the direct limit as being identical to the compositions q ∘ i_{G}, where q : Ĝ → Ĝ/N is the quotient homomorphism and N ⊂ Ĝ is a normal subgroup such that N ∩ (i_{G})_{σ}(G_{σ}) = {1} for any σ ∈ P_{Δ}.

Given a locally injective morphism m : G → F of a simplex of groups G over Δ, we define the development D(G, m) of G with respect to m as follows. First, identify the local groups G_{σ} with their images m_{σ}(G_{σ}) ⊂ F, and the structure homomorphisms ψ_{στ} with the inclusions of the corresponding subgroups of F. Define an equivalence relation ∼ on the set Δ × F by

(x, g) ∼ (y, h) iff x = y ∈ σ and g^{−1}h ∈ G_{σ} for some face σ of Δ.

Let [x, g] be the equivalence class of (x, g), [σ, g] := {[x, g] : x ∈ σ}, and put

D(G, m) = Δ × F/ ∼ .

We obtain then a multi-simplicial complex with the faces [σ, g]. These are injective images of σ × {g} under the quotient map of ∼. This complex is multi-simplicial and not just simplicial since the intersection of its faces is in general a union of faces and not just a single face. This construction is called the Basic Construction in [4, II.12]. We insist on using a coarser simplicial structure than [4] (who use the barycentric subdivisions of our faces).

Most of the simplices of groups in this paper will satisfy the property that the local group G_{Δ} (where Δ is the underlying simplex of G) is trivial, i.e. G_{Δ} = {1}. We will call such simplices of groups ∂-supported. The next proposition gathers general and well known properties of developments. We present these properties in the restricted context of ∂-supported simplices of groups, which simplifies formulations and is sufficient for the purposes of this paper. These results (including their proofs) can be found in [4, II.12] (compare also [JS, Proposition 3.2]).

14. Proposition. — Let G be a ∂-supported polytope of groups over a simplex Δ, and let m : G → F be a locally injective morphism.

1. The formula h[x, g] = [x, hg] defines an action of the group F on D(G, m) by automorphisms. The quotient map of this action is equal to the map induced by the projection Δ × F → Δ. The action is without inversions, i.e. a face preserved by an automorphisms is fixed pointwise. The stabilizer of a face [σ, g] is a conjugation G_{σ}^{g} := gG_{σ}g^{−1} (we still view the local groups of G as subgroups of F, via m).

2. D(G, m) is finite (as a complex) if and only if F is a finite group.

3. D(G, m) is locally finite if and only if the groups G_{σ} for all faces σ of Δ are finite. In fact, for local finiteness it is sufficient to require that the groups G_{v} for all vertices v of Δ are finite.

4. D(G, m) is connected if and only if the morphism m is surjective.

5. D(G, m) is a pure complex, i.e. it is the union of its top dimensional faces.

6. D(G, m) is gallery connected if and only if the subgroups G_{s} for all codimension-1 faces s of Δ generate F. Recall that gallery connected means that any two top dimensional faces are connected by a finite sequence of top dimensional faces such that consecutive faces have common codimension-1 subface.

7. D(G, m) is a pseudomanifold if and only if in addition to (3) and (6) the local groups G_{s} of G are isomorphic to Z_{2} for all codimension-1 faces s of Δ.

8. D(G, m) is an orientable pseudomanifold if and only if in addition to (7) there is a homomorphism ρ : F → Z_{2} whose restriction ρ_{s} : G_{s} → Z_{2} is an isomorphism for all codimension-1 faces s of Δ (equivalently, ρ∘m_{s} : G_{s} → Z_{2} is an isomorphism for any such s).

The next proposition describes the fundamental group of the development of a surjective morphism, in terms of the direct limit. Recall that we denote by m̂ : Ĝ → F the homomorphism induced by a morphism m : G → F. Here we do not need to assume that G is ∂-supported.

15. Proposition. — Let G be a developable simplex of groups and let m : G → F be a locally injective and surjective morphism. Then π_{1}(D(G, m)) = ker(m̂ : Ĝ → F). In particular, D(G, m) is simply connected F = Ĝ and m = i_{G}.

We will call development D(G, i_{G}) the universal development of a developable simplex of groups G (or the universal covering of G), and denote it shortly by G̃.

We now turn to discussion of links in developments. Given a simplex Δ and its face σ, the link Δ_{σ} of Δ at σ is the spherical simplex composed of the unit vectors tangent to Δ and orthogonal to σ at a fixed interior point of σ. The face poset P_{Δ_{σ}} of Δ_{σ} canonically identifies with the subposet (P_{Δ})_{σ} in P_{Δ} consisting of all faces τ such that τ properly contains σ.

If K is a multi-simplicial complex, and σ is its face, then the link K_{σ} is a union of the links τ_{σ} for all faces τ of K that properly contain σ, glued together into a multi-simplicial complex according to the equivalence relation on the disjoint union induced by the natural inclusions τ_{σ} ⊂ τ′_{σ} for all pairs τ ⊂ τ′.

Given a simplex of groups G over Δ and a face σ of Δ, consider its restriction G_{σ} := G|_{(P_{Δ})_{σ}} and view it as a simplex of groups over the link simplex Δ_{σ}. Put also i_{σ} := {ψ_{στ} : τ ∈ (P_{Δ})_{σ}} and note that i_{σ} : G_{σ} → G_{σ} is a morphism. Observe that since the morphism m is locally injective, all the homomorphisms ψ_{στ} are injective, and thus i_{σ} is a locally injective morphism.

16. Proposition. — Let G be a simplex of groups over Δ and let m : G → F be a locally injective morphism. Then, given a face [σ, g] in the development D(G, m), the link D(G, m)_{[σ,g]} is isomorphic to the development D(G_{σ}, i_{σ}). Moreover, this isomorphism is equivariant with respect to the action of the stabilizing subgroup Stab(F, [σ, g]) on D(G, m)_{[σ,g]} and the action of G_{σ} on D(G_{σ}, i_{σ}).

We will call D(G_{σ}, i_{σ}) the local development (or the link) of G at σ. This coincides with the notion of the link L(G, σ) as defined in Section 7.

Following Definition 2, we say that a simplex of groups is locally k-large if all of its local developments are k-large (in particular truly simplicial, not just multi-simplicial). Theorem 15 implies then the following.

21. Corollary. — For k ≥ 6, any locally k-large simplex of groups is developable.

15. Remark. — Note that if the homotopical systole of a development of a locally k-large simplex of groups is ≥ 3 (which is a nontrivial condition for a multi-simplicial complex), then it is simplicial. To see this, observe first that a multi-simplicial complex X with simplicial links, which is not simplicial, must have a double edge (i.e. two edges with both endpoints coinciding). Second, note that the cycle consisting of these two edges is homotopically nontrivial in X. This follows from the fact that the universal covering of X is simplicial since, being locally 6-large, it can be obtained as the union of a sequence of small extensions (see Section 5), starting from a single simplex, and all these extensions together with their union are simplicial. This implies that the homotopical systole of X is 2, justifying the initial statement.

Our last goal in this section is to recall terminology related to the so called simple complexes of groups (examples of which are simplices of groups), and to formulate some results which extend the already mentioned results for simplices of groups. We will need these concepts and facts in the next section, in the proof of Proposition 17.

Let X be a simplicial complex and let F be a group acting on X by automorphisms. A subcomplex K ⊂ X is a strict fundamental domain of this action if the restricted quotient map K → F \X is an isomorphism of simplicial complexes. Given an action of F that admits a strict fundamental domain K, we associate to any face σ of K a group G_{σ} := Stab(F, σ), the stabilizer of σ in F. In fact, due to the existence of a strict fundamental domain, the stabilizer G_{σ} fixes the simplex σ pointwise. We have obtained a system {G_{σ}} of groups with inclusions G_{τ} ⊂ G_{σ} whenever σ ⊂ τ. We call this system the simple complex of groups associated to the action of F.

A simple complex of groups G over a simplicial complex Q is a system of groups G_{σ} associated to the faces of Q, equipped with a system of injective homomorphisms ψ_{στ} : G_{τ} → G_{σ} for all pairs σ ⊂ τ, such that ψ_{στ} ∘ ψ_{τρ} = ψ_{σρ} whenever σ ⊂ τ ⊂ ρ.

The notions of a morphism to a group, local injectivity and surjectivity of a morphism, developability of G and development D(G, m) associated to a locally injective morphism m : G → F have straightforward extensions from the case of G being a simplex of groups to that of a simple complex of groups. It is then clear that if m : G → F is a locally injective morphism then G is equivalent (isomorphic as a simple complex of groups) to the simple complex of groups associated to the action of F on the development D(G, m). Thus, developability of G can be characterized geometrically by saying that G is isomorphic to a simple complex of groups associated to an action. Moreover, the obvious analogue of Proposition 15 holds if the underlying complex Q of a simple complex of groups G is connected.

Now we extend the notion of the local development, as defined above for simplices of groups, to arbitrary simple complexes of groups. Let G be a simple complex of groups over Q and let σ be a face of Q. Consider the link Q_{σ} of Q at σ, and for any face τ in Q_{σ} denote by τ̄ the corresponding face of Q properly containing σ. Define then a simple complex of groups G_{σ} = ({G′_{τ}}, {ψ′_{τρ}}) over Q_{σ} by putting G′_{τ} := Gτ̄ and ψ′_{τρ} := ψ_{τ̄ρ̄}. Define also a locally injective morphism i_{σ} : G_{σ} → G_{σ} consisting of homomorphisms (i_{σ})_{τ} : G′_{τ} → G_{σ} given by (i_{σ})_{τ} := ψ_{στ̄}. The development D(G_{σ}, i_{σ}), equipped with the action of G_{σ}, is then called the local development of G at σ (or the link of G at σ). If G is developable then the local developments of G occur as links in the developments of G for all injective morphisms. More precisely, if m : G → F is an injective morphism, and [σ, g] a face in the corresponding development D(G, m), then the link D(G, m)_{[σ,g]}, with the induced action of the stabilizing subgroup of F, is equivariantly isomorphic to the local development D(G_{σ}, i_{σ}).

A simple complex of groups over Q is 6-large if all of its local developments are 6-large. Clearly, by Theorem 15, every locally 6-large simple complex of groups is developable.

19. Extra-tilability.

In this section we introduce a condition called extra-tilability which allows one to construct, inductively with respect to the dimension, simplices of groups admitting finite k-large developments (for arbitrary k ≥ 6). A construction of such developments is presented in Section 20. In this section we indicate various useful consequences of the introduced condition.

7. Definition. — A simplicial complex X equipped with an action of a group G by simplicial automorphisms is extra-tilable if the following conditions are satisfied:

1. the action is simply transitive on top-dimensional simplices of X and its quotient is a simplex (equivalently, any top-dimensional simplex is a strict fundamental domain for this action);

2. X is 6-large;

3. for any face σ of X the ball B_{1}(σ, X) is a strict fundamental domain for the restricted action of a subgroup of G on X.

A simplex of groups G is locally extra-tilable if local developments of G equipped with actions of the corresponding local groups are all extra-tilable.

3. Example. —

1. The Coxeter (or dihedral) group D_{n} = ⟨s_{1}, s_{2}|s_{1}^{2}, s_{2}^{2}, (s_{1}s_{2})^{n}⟩ with n = 6k or n = ∞, with its canonical action on the corresponding Coxeter complex (i.e. a division of S^{1} into 2n segments), is obviously extra-tilable.

2. Let X be the Coxeter complex of the triangle Coxeter group (6, 6, 6), which may be viewed as a triangulation of the hyperbolic plane by regular triangles with angles π/6. It follows from Poincare’s Theorem that the action of this group on X is extra-tilable.

3. The quotient simplex of groups associated to the action in (2) is locally extra-tilable.

Note that condition (1) in Definition 7 implies that the complex of groups associated to the action of G on X is a ∂-supported simplex of groups. Consequently, X is equivariantly isomorphic to a development of this simplex of groups. For this reason, we will often speak of extra-tilable developments of ∂-supported simplices of groups (rather than of extra-tilable complexes).

The reader can easily verify that if the pair X, G is extra-tilable then links of X equipped with the actions of the corresponding stabilizers in G are extra-tilable. Consequently, a simplex of groups that admits an extratilable development is locally extra-tilable. The next proposition provides the converse of this statement, together with a much stronger property that will be crucial in our later arguments.

17. Proposition. — Let G be a locally extra-tilable simplex of groups. Then the action of the direct limit Ĝ on the universal development G̃ = D(G, i_{G}) has the following property: each n-ball B = B_{n}(σ, G̃) in G̃, for any natural number n, is a strict fundamental domain for the action of a unique subgroup H_{B} of Ĝ. In particular, G̃ equipped with the action of Ĝ is extratilable.

To prove Proposition 17 we need the following.

39. Lemma. — Let G be a ∂-supported locally 6-large simplex of groups over a simplex Δ, and let m : G → G be a locally injective and surjective morphism. Suppose that for some simplex σ ⊂ D(G, m) the ball B := B_{1}(σ, D(G, m)) is a strict fundamental domain for the action of a subgroup H < G. Denote by H the simple complex of groups over B associated to the action of H on D(G, m), and by ν : H → H the associated morphism. Then

1. ν is surjective, i.e. H is generated by the union of the images ν_{σ}(H_{σ}) of the local groups H_{σ} of H;

2. B determines the subgroup H uniquely.

Proof. — To prove part (1), note first that the development D(G, m) is, by surjectivity of m, connected. Since any simple complex of groups with connected development is surjective, we get surjectivity of H by the fact that D(H, ν) = D(G, m).

The proof of (2) goes by induction on n = dim Δ. Let H′ < G be another subgroup for which B is a strict fundamental domain. Denote by H′ the simple complex of groups over B associated to the action of H′ on D(G, m), and by H′_{σ} its local groups at simplices σ of B.

Suppose first that dim Δ = 1. Since G is ∂-supported, the local groups of both H and H′ at edges are all trivial. We will show that for every vertex v of B the local groups H_{v} and H′_{v} coincide. By applying (1), this property implies that H = H′, hence (2).

The equality H_{v} = H′_{v} is obvious for vertices v from the interior of B (i.e. vertices of the central simplex σ), since then both groups are trivial. For the remaining vertices v both these groups coincide with the stabilizer of G at v, which one easily deduces from the fact that there is exactly one edge in B adjacent to v (and from simple transitivity of G on the edges of D(G, m)).

In the general case, note that for any simplex σ of B both groups H_{σ}, H′_{σ} act on the link [D(G, m)]_{σ} = D(G_{σ}, m_{σ}) with the strict fundamental domain B_{σ}. The inductive assumption implies that H_{σ} = H′_{σ}, and again the proof is concluded by applying (1).

Proof (of Proposition 18.3). — Note that since G is locally extra-tilable, it is in particular locally 6-large. Thus, by Corollary 21, G is developable and hence it makes sense to speak of the universal development G̃ = D(G, i_{G}). By Proposition 15, G̃ is simply connected, and hence it is a systolic complex. For any ball B in G̃ consider a simple complex of groups H = ({H_{σ}}, {ϕ_{τσ}}) over B defined as follows. For any face σ of B consider the link (G̃)_{σ} and the action of the stabilizer Stab(Ĝ, σ) on it. Let σ_{0} be the image of σ under the quotient map G̃ → G̃/Ĝ = Δ. Then the action of Stab(Ĝ, σ) on (G̃)_{σ} is equivariantly isomorphic to the action of the local group G_{σ_{0}} of G on the local development D(G_{σ_{0}}, i_{σ_{0}}) and hence it is extra-tilable. By strict convexity of balls (Corollary 7), the link B_{σ} either coincides with (G̃)_{σ} or has a form B_{1}(τ, (G̃)_{σ}) for some simplex τ ⊂ (G̃)_{σ}. In any case, by local extra-tilability of G, B_{σ} is a strict fundamental domain for the action of a subgroup of Stab(Ĝ, σ) on (G̃)_{σ}. Moreover, due to Lemma 39, this subgroup is unique, and we take it as the local group H_{σ} in H. Note that if σ ⊂ τ then H_{τ} ⊂ H_{σ}. In fact, H_{τ} can be identified as a subgroup of H_{σ} more precisely as follows. Denote by τ′ the face in the link (G̃)_{σ} corresponding to τ. Then, viewing H_{σ} as acting on (G̃)_{σ} , H_{τ} is equal to the stabilizer of τ′ in this action. We take as the structure homomorphism ϕ_{στ} for H the inclusion homomorphism from H_{τ} to H_{σ}, for any relevant pair σ, τ of simplices in B.

Consider the morphism j : H → Ĝ given by the inclusions of the local groups H_{σ} in Ĝ, and denote by ĵ : Ĥ → Ĝ the corresponding homomorphism between the direct limits. Since j is locally injective, H is developable and we denote by H̃ the universal development of H. The ball B, identified with the subcomplex [B, 1] in H̃, is clearly a strict fundamental domain for the action of Ĥ on H̃. To prove the proposition, we will show that there is a ĵ-equivariant isomorphism between G̃ and H̃ that is identical on B. If this is the case, B is a strict fundamental domain for the subgroup H_{B} := ĵ(Ĥ) < Ĝ.

Let J : H̃ → G̃ be a simplicial map given by J([x, g]) := ĵ(g) · x for any x ∈ B (where x on the right lies in B ⊂ G̃). From what was said above about local groups of H, it follows that the local development of H at a face σ of B is equivariantly isomorphic to the link (G̃)_{σ} acted upon by the group H_{σ}. This implies that the map J induces isomorphisms at links of all simplices in H̃, and hence it is a covering. Since both complexes H̃ and G̃ are connected and simply connected, it follows that J is an isomorphism as required.

It remains to prove uniqueness of H_{B}. In view of the strong convexity of B (see Corollary 9), this follows by the argument as in the proof of Lemma 39.2. Thus the proposition follows.

The above arguments yield in fact the following more general result.

18. Proposition. — Let G be a locally extra-tilable simplex of groups and let Q be any strongly convex subcomplex in the universal development G̃ = D(G, i_{G}). Then Q is a strict fundamental domain for the action of a unique subgroup H_{Q} of the direct limit Ĝ.

In the next proposition only part (2) is important for further applications. We include part (1) to indicate the relationship of the phenomena that we obtain with residual finiteness of involved groups.

19. Proposition. — Let G be a locally extra-tilable simplex of finite groups. Then

1. the direct limit group Ĝ is residually finite;

2. for any natural k there is an injective morphism m : G → F into a finite group F such that we have sys_{h}[D(G, m)] ≥ k.

Proof. — Let Δ be the underlying simplex of G. To prove (1), recall that a group G is residually finite if for any g ∈ G with g ≠ 1 there is a normal subgroup N < G of finite index such that g ∉ N. Let g ∈ Ĝ, g ≠ 1. Consider a ball B in the universal development G̃ centered at [Δ, 1] and containing [Δ, g]. By Proposition 17, there is a subgroup H_{B} < Ĝ for which B is a strict fundamental domain. Note that g ∉ H_{B} because each orbit of H_{B} intersects B only once. Moreover, since G̃ is locally finite, B is finite (as a complex), and since Ĝ acts simply transitively on top-dimensional faces of G̃, it follows that H_{B} is a finite index subgroup of Ĝ. Thus the normalization N = ⋂_{h∈Ĝ} hH_{B}h^{−1} has also finite index in Ĝ, and clearly g ∉ N. This finishes the proof of part (1).

To prove (2), consider the face [Δ, 1] in G̃ and the ball B = B_{k}([Δ, 1], G̃) centered at this face. Observe that a polygonal path in G̃ connecting a vertex of [Δ, 1] with a vertex outside B has length greater than k (i.e. consists of more than k edges). Let H_{B} be the subgroup of Ĝ for which B is a strict fundamental domain, and let N = ⋂_{h∈Ĝ} hH_{B}h^{−1}. As before, N is a finite index subgroup in Ĝ.

Recall that for each vertex v of Δ the local group G_{v} is identified with the stabilizer of Ĝ at [v, 1] (in its action on G̃). Thus, since G is ∂-supported, we have G_{v} ∩ H_{B} = {1}, and hence also G_{v} ∩ N = {1}. It follows that the composition G → Ĝ → Ĝ/N is a locally injective morphism to a finite group Ĝ/N. We take this morphism as m and the quotient Ĝ/N as F.

We now estimate from below the homotopical systole of the development D(G, m). Since N is a subgroup of H_{B}, the orbit of a vertex v of [Δ, 1] under the action of N on G̃ intersects B only at v. Thus the polygonal distance between v and any other vertex from this orbit is greater than k (in fact, this distance is even ≥ 2k, but we don’t need this sharper estimate). It follows that any homotopically nontrivial closed polygonal path in D(G, m) passing through a vertex of [Δ, 1] has length > k. On the other hand, D(G, m) is acted upon by the quotient group Ĝ/N and this action is transitive on top-dimensional faces. Thus any homotopically nontrivial path in D(G, m) can be mapped by an automorphism of D(G, m) to a path that intersects [Δ, 1]. Thus, the homotopical systole of D(G, m) is greater than k, which finishes the proof of part (2).

We say that a locally injective morphism m : G → F from a simplex of groups G is extra-tilable if the development D(G, m) acted upon by the group F is extra-tilable. Obviously, to have a extra-tilable morphism, a simplex of groups has to be locally extra-tilable. The next proposition, a culmination of the results in this section, will be the key technical tool in the arguments involved in the main construction presented in the next section.

20. Proposition. — Let G be a locally k-large simplex of finite groups, for some k ≥ 6, and suppose G is locally extra-tilable. Then G admits an extratilable morphism μ : G → E to a finite group E such that the development D(G, μ) is k-large.

Proof. — Since it follows from our assumptions that G is locally 6large, let m : G → F be a locally injective morphism to a finite group F as prescribed by Proposition 19.2, i.e. such that sys_{h}[D(G, m)] ≥ k. Then D(G, m) is clearly k-large (see Corollary 1). Denote by K = ker m̂ the kernel of the homomorphism m̂ : Ĝ → F induced by m, and note that K has finite index in Ĝ.

For any face σ ⊂ G̃ consider the ball B^{σ} := B_{1}(σ, G̃) and the subgroup H^{σ} < Ĝ for which B^{σ} is a strict fundamental domain. Clearly, H^{σ} is a finite index subgroup for each σ. Consider the intersection K ∩ ⋂_{σ⊂[Δ,1]} H^{σ}, which is still of finite index in Ĝ, and normalize it to get a finite index normal subgroup N of Ĝ. Put E := Ĝ/N and denote by μ the natural morphism from G to E. Since N ⊂ K, the development D(G, μ) is a covering of the development D(G, m) and, since the latter is k-large, the former is k-large too. It remains to prove that μ is extra-tilable.

By the fact that [Δ, 1] is a fundamental domain for the action of Ĝ on G̃, for any face σ ⊂ G̃ there is a face σ_{0} ⊂ [Δ, 1] and an element g ∈ Ĝ such that H_{σ} = gH_{σ_{0}}g^{−1}. In particular, since the subgroup N is contained in H_{σ_{0}} and normal, it is also contained in H_{σ}. Denote by p : G̃ → D(G, μ) the covering map induced by the quotient homomorphism Ĝ → E = Ĝ/N. It follows that the image p(B^{σ}) is a strict fundamental domain for the action of the subgroup H^{σ}/N ⊂ E on D(G, μ). Since the images p(B^{σ}) for all simplices σ in G̃ exhaust the balls of radius 1 centered at faces in D(G, μ), the action of E on D(G, μ) is extra-tilable, and the proposition follows.

20. Existence of k-large developments.

In this section we give a rather general construction of finite k-large developments of simplices of groups in arbitrary dimension. This construction allows to get examples of complexes with various interesting properties. Our main result is the following.

21. Proposition. — Let Δ be a simplex and suppose that for any codimension 1 face s of Δ we are given a finite group A_{s}. Then for any k ≥ 6 there exists a ∂-supported simplex of finite groups G = ({G_{σ}}, {ψ_{στ}}) and a locally injective and surjective morphism m : G → F to a finite group F such that G_{s} = A_{s} for any codimension 1 face s of Δ and

1. G is locally extra-tilable;

2. the development D(G, m) is (finite and) k-large.

Proof. — We will construct appropriate groups G_{σ} inductively with respect to the codimension of σ in Δ. Here we will view F as G_{∅}, the group associated to the “empty face” ∅ of Δ of codimension dim(Δ) + 1.

By the requirements of the proposition, we have to put G_{Δ} = {1} and G_{s} = A_{s} for all faces s of codimension 1. This gives the starting point for our induction. Suppose that finite groups G_{σ} are already defined for all faces σ of codimension ≤ i, together with injective homomorphisms ψ_{στ} as required. Suppose also that for all such σ the following condition (which will be an additional part of the inductive hypothesis) is satisfied: The groups G_{τ} : σ ⊂ τ form a simplex of groups G^{σ} over the link Δ_{σ} and the homomorphisms ψ_{στ} form a locally injective and surjective morphism m^{σ} : G^{σ} → G_{σ} such that the development D(G^{σ}, m^{σ}) is k-large and locally extra-tilable. Note that for i = 1 these inductive assumptions are fulfilled. For any face ρ of codimension i + 1 in Δ consider the simplex of groups G^{ρ} over the link Δ_{ρ} formed of the groups G_{σ} : ρ ⊂ σ. By the inductive assumptions, this gives a locally k-large ∂-supported and locally extra-tilable simplex of groups. By Proposition 20, there is a surjective extra-tilable morphism μ : G^{ρ} → E to a finite group E such that the development D(G^{ρ}, μ) is k-large. By putting G_{ρ} := E and ψ_{ρσ} := μ_{σ} we get the inductive hypothesis for i + 1. This finishes the proof.

22. Corollary. — For each natural n and each k ≥ 6 there exists an ndimensional compact simplicial pseudomanifold that is k-large. Moreover, this pseudomanifold can be obtained to be orientable.

Proof. — In view of Proposition 14.7, the first statement in the corollary follows from Proposition 21 by putting A_{s} = Z_{2} for all codimension 1 faces s.

To ensure orientability, we need to modify slightly constructions in the proofs of Propositions 21 and 20. Recall from Proposition 14.8 that a necessary condition for the development D(G, m) associated to a morphism m : G → F to be an orientable pseudomanifold is the existence of a homomorphism r : F → Z_{2} such that the composed morphism r ∘ m maps the local groups G_{s} at codimension 1 faces s isomorphically to Z_{2}. Thus, when constructing local groups G_{σ}, we need to have additional homomorphisms r_{σ} : G_{σ} → Z_{2}, forming together a morphism from G to Z_{2}, such that the compositions r_{σ} ∘ ψ_{σs} : G_{s} → Z_{2} are isomorphisms. By the inductive assumption concerning this property, there is always a homomorphism r̂_{σ} : Ĝ^{σ} → Z_{2} from the direct limit of the simplex of groups G^{σ}, with the desired property. Thus, to have the appropriate r_{σ}, it is necessary that the normal subgroup N giving G_{σ} as the quotient Ĝ^{σ}/N is contained in the kernel of r̂_{σ}. Since this can be obtained by passing to a finite index subgroup in the previously chosen N, the corollary follows.

We mention further consequences of Corollary 22.

23. Corollary. —

1. For each natural n there exists a developable simplex of groups whose fundamental group is Gromov-hyperbolic, virtually torsion-free, and has cohomological dimension n.

2. For each natural n there exists an n-dimensional compact simplicial orientable pseudomanifold whose universal cover is CAT (0) with respect to the standard piecewise euclidean metric.

3. For each natural n and each real number d > 0 there exists an ndimensional compact simplicial orientable pseudomanifold whose universal cover is CAT (−1) with respect to the piecewise hyperbolic metric for which the simplices are regular hyperbolic with edge lengths d.

Proof. — By Corollary 22, for every natural n there exists an n-dimensional compact simplicial orientable pseudomanifold X which is 7-large. It is obtained as a development of a certain simplex of finite groups G. The fundamental group Γ of X is a subgroup of finite index in the fundamental group of G, and it is torsion-free. To see this, note that X is aspherical (Theorem 14.1), and hence it is a classifying space for Γ. Since X is finite dimensional, Γ cannot contain a finite subgroup. Moreover, since (being a compact pseudomanifold of dimension n) X has nontrivial cohomology in dimension n, the group Γ has cohomological dimension equal to n. Finally, by Corollary 2, the group Γ is Gromov hyperbolic. This proves (1).

Parts (2) and (3) follow from Corollary 22 in view of Theorem 17. Parts (2) and (3) of the above corollary give an affirmative answer, in

arbitrary dimension n, to a question raised by D. Burago [5, p. 292]. The answer for n = 3 has been given in [6].

As one more application we note that Corollary 22 allows an alternative approach to the main result of our paper [16] stating that for each natural n there exists a Gromov hyperbolic Coxeter group with virtual cohomological dimension n. As we have shown in [16], to construct such a group it is sufficient to construct a compact orientable n-dimensional pseudomanifold which satisfies “flag-no-square” condition (which is equivalent to 5-largeness). Since k-largeness for k ≥ 6 implies 5-largeness, we get such pseudomanifolds by the construction of Proposition 21 (improved as in the proof of orientability in Corollary 22), which is different from the construction in [16].

21. Non-positively curved branched covers

In this section we use the idea of extra-tilability to show the existence of nonpositively curved finite branched covers for a class of compact piecewise euclidean pseudomanifolds that contains all manifolds. This answers a question of M. Gromov. Using the same method, we show that any finite complex K is homotopy equivalent to the classifying space for proper G-bundles of a CAT (−1) (hence Gromov hyperbolic) group G. This answers a question of I. Leary.

We start by recalling some terminology. A chamber in a simplicial pseudomanifold is any of its top-dimensional faces. A simplicial pseudomanifold is normal if all of its links are gallery-connected (See Proposition 14.6). We borrow the term “normal” from M. Goresky and R. MacPherson [12]). The property of being normal does not depend on a triangulation of a pseudomanifold (see Section 4.1 in [12]). Manifolds are obviously normal.

A branched covering of a simplicial pseudomanifold X is a simplicial pseudomanifold Y equipped with a nondegenerate simplicial map p : Y → X which is a covering map outside codimension 2 skeleta.

The main results in this section are the following two theorems.

20. Theorem. — Let X be a compact connected normal simplicial pseudomanifold with a piecewise euclidean (respectively, piecewise hyperbolic) metric. Then X has a compact branched covering Y which is nonpositively curved (respectively, has curvature κ ≤ −1) with respect to the induced piecewise constant curvature metric.

21. Theorem. — For any finite complex K there is a CAT (−1) space X and a group G acting properly discontinuously and cocompactly by isometries on X, so that the quotient G\X is homotopy equivalent to K.

Both theorems above are corollaries to a stronger technical result contained in Proposition 22. To formulate this proposition we need more definitions. We say that a simplex of groups G over a simplex Δ, with local groups G_{s} at all codimension 1 faces s isomorphic to Z_{2}, is symmetric if it satisfies the following conditions:

1. the local groups G_{σ} are generated by their local subgroups at faces of codimension 1 (i.e. at those codimension 1 faces s which contain σ);

2. any automorphism f of the underlying simplex Δ extends to an automorphism φ of G.

Note that, due to condition (1), automorphisms φ from condition (2) are uniquely determined by automorphisms f.

A morphism m : G → F is a symmetric morphism if m is surjective, G is a symmetric simplex of groups, and for any automorphism φ of G as in (2) there is an automorphism a_{φ} of F such that m ∘ φ = a_{φ} ∘ m. Note that, due to surjectivity of m, automorphisms a_{φ} are uniquely determined by automorphisms φ.

A symmetric development is the development associated to a symmetric morphism.

22. Proposition. — Given any k ≥ 6, every finite family of compact connected normal simplicial pseudomanifolds {X_{i}} of the same dimension has a common compact branched covering Y which is an extra-tilable symmetric development of a simplex of finite groups, and which is k-large. Moreover, for each X_{i} there is a group Γ_{i} of simplicial automorphisms of the universal cover Ỹ of Y such that X_{i} is isomorphic to the quotient Γ_{i}\Ỹ.

Before giving a proof of the proposition, we show how it implies Theorem 20. The proof of Theorem 21, together with discussion of its consequences, occupies the last part of the section (after the proof of Proposition 22).

Proof (of Theorem 20 (using Proposition 22)). — Given a metric pseudomanifold as in the theorem, denote by Π the set of all shapes of simplices of X and note that Π is finite. Clearly, any branched covering Y of X equipped with the lifted metric satisfies the condition Shapes(Y) ⊂ Π. Let k ≥ 6 be a natural number associated to Π as in the assertion of Theorem 17. By Proposition 22, X has a compact branched covering Y which is k-large, and hence also locally k-large. By Theorem 17, Y is then nonpositively curved (respectively, has curvature κ ≤ −1), as required.

Proof (of Proposition 22). — We use induction with respect to the dimension n of pseudomanifolds X_{i}.

For n = 1, each X_{i} is a triangulation of the circle, and we denote by l_{i} the number of edges in X_{i}. Let L be a common multiple of all numbers l_{i} and 12. Put Y to be the triangulation of the circle consisting of L edges. Then Y is as asserted in the proposition. To see this, note that due to divisibility of L by 12, Y is an extra-tilable development of the ∂-supported edge of groups with groups Z_{2} at vertices. The other assertions of the proposition are in this case obvious.

We now pass to the case of arbitrary dimension n. Consider the family X of all links at vertices in all pseudomanifolds X_{i}. Due to compactness of X_{i}’s, this family is finite. Moreover, since links of normal pseudomanifolds are normal, X consists of compact connected normal pseudomanifolds of the same dimension n − 1. By applying inductive hypothesis to the family X, we obtain an extra-tilable symmetric morphism m : G → F from an (n − 1)- dimensional simplex of finite groups G to a finite group F such that the development D(G, m) satisfies all assertions of the proposition relative to X. Let H be an n-dimensional simplex of groups described as follows. For local groups at faces of codimension < n take the local groups of G at faces of the same codimension (which are all isomorphic due to symmetry of G). For local groups at vertices take the group F. Symmetry of G and m allows to take as structure homomorphisms for H the homomorphisms occurring in G and in the morphism m. The so obtained simplex of finite groups H is clearly symmetric, locally k-large and locally extra-tilable. Since, being locally k-large, H is developable, consider its universal development H̃. Our next aim is to show that H̃ is a common branched covering of pseudomanifolds X_{i}. However, since H̃ is not compact, this will not yet finish the proof.

Fix one of the pseudomanifolds X_{i}, a chamber C in it, and any isomorphism p_{0} : D_{0} → C of some chamber D_{0} of H̃ with C. We will show that p_{0} can be extended to a branched covering p : H̃ → X_{i}. For this, note that any gallery γ in H̃ starting at the chamber D_{0} determines uniquely the map p_{γ} : D → X_{i} from the final chamber D in γ, by means of unfolding γ on X_{i} starting with p_{0}. Then, since (by Proposition 14.6) H̃ is gallery connected, we define p separately on each chamber D in H̃ by putting p|_{D} = p_{γ} for some choice of a gallery γ connecting D_{0} to D. To see that p is well defined we need to show that p_{γ} : D → X_{i} does not depend on the choice of γ. Equivalently, we need to show that for any gallery γ starting and terminating at D_{0} we have p_{γ} = p_{0}.

Since H̃ is simply connected, γ can be expressed, up to cancellation of back and forth subpaths, as the concatenation of elementary closed galleries started at D_{0}, i.e. galleries of form

D_{0}, D_{1}, …, D_{l}, D_{l}^{1}, D_{l}^{2}, …, D_{l}^{m}, D_{l}, D_{l−1}, …, D_{0}

with chambers D_{l}, D_{l}^{1}, …, D_{l}^{m} contained in the residue of a single vertex of H̃. Clearly, it is then sufficient to show that p_{γ} = p_{0} for any elementary closed gallery γ started at D_{0}. This however follows directly from the fact that links of H̃ at vertices, which are all isomorphic to the development D(G, m), are symmetric branched coverings of the links of X_{i} at vertices. Thus, the map p is well defined, and the fact that it is a branched covering follows easily from its definition.

Denote by Sym(H̃) the full group of simplicial automorphisms of H̃. Due to symmetry of H̃, and rigidity implied by the fact that H̃ is a pseudomanifold, this group is a semidirect extension of the direct limit Ĥ by the group of automorphisms of the underlying simplex of H. We will now show that for each X_{i} there exists a subgroup Γ_{i} < Sym(H̃) such that the quotient Γ_{i}\H̃ is isomorphic to X_{i}.

Consider the set p^{−1}(C) of all chambers in H̃ which are mapped through p on C. Clearly, this set contains our distinguished chamber D_{0}. For any chamber D ∈ p^{−1}(C) consider the isomorphism u_{D} : D_{0} → D such that p ∘ u_{D} = p_{0}. Clearly, u_{D} can be extended uniquely to an automorphism of H̃, and we denote this automorphism by g_{D}. Moreover, each automorphism g_{D} obviously commutes with p. Consequently, the set {g_{D} : D ∈ p^{−1}(C)} coincides with the group of all automorphisms of H̃ that commute with p. We denote this group by Γ_{i} and note that it acts simply transitively on the set p^{−1}(C). Furthermore, for any chamber C′ adjacent to C along a codimension 1 face, and for any chamber D ∈ p^{−1}(C), there is exactly one chamber D′ ∈ p^{−1}(C′) adjacent to C′. Moreover, the assignment D → D′ establishes 1-1 correspondence between the sets of chambers p^{−1}(C) and p^{−1}(C′). It follows that the group Γ_{i} acts simply transitively on the set p^{−1}(C′). Since X_{i} is gallery connected, the same argument gives the same conclusion for the set p^{−1}(C′′), for any chamber C′′ of X_{i}. This implies that the map P : Γ_{i}\H̃ → X_{i} induced by p is an isomorphism, as required. It is also important to note that, since each X_{i} is compact, each of the groups Γ_{i} has finite index in Sym(H̃).

We want now to find a compact development of H which will be k-large and which still be a branched covering of all X_{i}’s. Since H is locally klarge and locally extra-tilable, by Proposition 20 there exists an extra-tilable morphism μ : H → E to a finite group E such that the development D(H, μ) is k-large. Denote by K the kernel of the induced homomorphism i_{μ} : Ĥ → E. Take the intersection K ∩ ⋂_{i} Γ_{i} and normalize it in Sym(H̃) to get a normal subgroup N in Ĥ for which the induced morphism μ : H → Ĥ/N is symmetric (due to normalization in Sym(H̃)), still extra-tilable, and whose development D(H, μ) is still k-large (last two properties due to the inclusion N < K). Since, due to the inclusions N < Γ_{i}, D(H, μ) is still a common branched covering of X_{i}’s, the proposition follows.

Proof (of Theorem 21). — Let Z be a compact simplicial manifold with boundary having the same homotopy type as the complex K. It can be obtained for example by embedding K in R^{N}, and taking its regular neighbourhood in a sufficiently fine triangulation. Denote by X the double of Z, i.e. the closed manifold obtained by gluing two copies of Z by the identity map of their boundaries. It follows from Proposition 22 that for any k ≥ 6 there is a k-systolic pseudomanifold Ỹ and a group Γ acting simplicially, properly discontinuously, and cocompactly on it, such that X is isomorphic to the quotient Γ \Ỹ. By Theorem 17, taking k sufficiently large, we can arrange that Ỹ is CAT (−1) with respect to some piecewise hyperbolic metric with regular simplices, and then Γ acts by isometries.

Denote by i : X → X the involution which exchanges the copies of Z, used in the construction X, fixing their common boundary. We claim that, if Ỹ is taken to be the universal development H̃ as in the proof of Proposition 22, then i can be lifted to an isomorphism ĩ of Ỹ. To see that, fix a chamber C in X and consider lifts D and D′ of C and i(C) respectively, to H̃. Now, take as ĩ the isomorphism from the group Sym(H̃) induced by the map i_{0} : D → D′ such that i_{0} commutes with i through the covering H̃ → X. Due to rigidity implied by the fact that we deal with galleryconnected pseudomanifolds, ĩ is a lift of i as required. Using ĩ we get the extension G of Γ, of index 2, whose action on H̃ projects to the action of Z_{2} generated by i on X. Consequently, the quotient G\H̃ is isomorphic to Z = Z_{2}\X, and the theorem follows.

Theorem 21 has interesting corollaries. We refer to [21] for the background on Corollary 24.

24. Corollary. — Any finite complex K is homotopy equivalent to the classifying space for proper G-bundles of a CAT (−1) (hence Gromov hyperbolic) group G.

25. Corollary. — Any homotopy type of a finite complex occurs as the quotient G\R_{d}(G) of the Rips’ complex R_{d}(G) (with sufficiently large d) of some Gromov hyperbolic group G.

Corollary 25 follows from Proposition 22 in view of the following observation. Given a CAT (−1) space X and a group G acting on X properly discontinuously cocompactly by isometries, for sufficiently large d, the action of G on the Rips’ complex R_{d}(G) is equivariantly homotopy equivalent to the action on X. This follows from the fact that if G is Gromov hyperbolic then, for sufficiently large d, the quotient of the Rips’ complex G\R_{d}(G) is the classifying space for proper G-bundles, and the latter is uniquely determined up to homotopy equivalence (see [19]).

Corollaries 24 and 25 give answer to questions of Ian Leary (see [2, Question 1.24] and [20]).

REFERENCES

1. J. ALONSO, M. BRIDSON, Semihyperbolic groups, Proc. London Math. Soc. (3) 70, (1995), 56–114.

2. M.BESTVINA, Questions in Geometric Group Theory, http://www.math.utah.edu/∼bestvina

3. M. BRIDSON, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc. 127(1999), no. 7, 2143–2146.

4. M. BRIDSON, A. HAEFLIGER, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, 1999.

5. D. BURAGO, Hard Balls Gas and Alexandrov Spaces of Curvature Bounded Above, Documenta Mathematica, Extra Volume ICM II (1998), 289-298.

6. D. BURAGO, S. FERLEGER, B. KLEINER, A. KONONENKO, Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold, Proc. AMS 129 (2001), Number 5, 1493–1498.

7. R. CHARNEY, M. DAVIS, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math. 115 (1993), no. 5, 929–1009.

8. G. A. DIRAC, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76.

9. D. EPSTEIN, J. CANNON, D. HOLT, S. LEVY, M. PATERSON, W. THURSTON, Word Processing in Groups, Jones and Barlett, Boston MA, 1992.

10. E. GHYS, P. dE LA HARPE (eds.), Sur les Groupes Hyperboliques d’apres Mikhael Gromov, Progr. Math. vol. 83, Birkhäuser, Boston MA, 1990.

11. C. McA. GORDON, D. D. LONG, A. W. REID, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004), 135–148.

12. M. GORESKY, R. MACPHERSON, Intersection homology theory, Topology 19 (1980), no. 2, 135–162.

13. M. GROMOV, Asymptotic invariants of infinite groups, Geometric Group Theory, G. Niblo and M. Roller eds., LMS lecture Notes Series 182, Vol. 2, Cambridge Univ. Press, 1993.

14. M. GROMOV, Hyperbolic groups, Essays in group theory, S. Gersten, ed., Springer, MSRI Publ. 8 (1987), 75–263.

15. F. HAGLUND, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71, 2003.

16. T. JANUSZKIEWICZ, J. ´ SWIA̧TKOWSKI, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003), 555–583.

17. T. JANUSZKIEWICZ, J. ´ SWIA̧TKOWSKI, Filling invariants in systolic complexes and groups, submitted, 2005.

18. T. JANUSZKIEWICZ, J. ´ SWIA̧TKOWSKI, Nonpositively curved developments of billiards, preprint, 2006.

19. D. MEINTRUP, T. SCHICK, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1–7.

20. I. LEARY, A metric Kan–Thurston theorem, in preparation.

21. I. LEARY, B. NUCINKIS, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001), 539–550.

22. R. LYNDON, P. SCHUPP, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, 1977.

23. J. ´ SWIA̧TKOWSKI, Regular path systems and (bi)automatic groups, Geometriae Dedicata, to appear.

Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA, E-mail: tjan@math.ohio-state.edu

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, E-mail: swiatkow@math.uni.wroc.pl

Manuscrit reçu le date.