Filling invariants in systolic complexes and groups

Tadeusz Januszkiewicz Jacek Świa̧tkowski Department of Mathematics, The Ohio State University 231 W 18th Ave, Columbus, OH 43210, USA and Instytut Matematyczny, Uniwersytet Wrocławski pl. Grunwaldzki 2/4, 50-384 Wroc/l aw, Poland

     Email: tjan@math.ohio-state.edu and swiatkow@math.uni.wroc.pl

Abstract

Systolic complexes are simplicial analogues of nonpositively curved cubical complexes. We study their asymptotic filling properties and show that many groups acting on them geometrically are very different from classical ones. In particular, we prove existence in arbitrary cohomological dimension of hyperbolic groups that contain no subgroups isomorphic to fundamental groups of nonpositively curved riemannian manifolds of dimension greater than 2.

AMS Classification numbers Primary: 20F69, 20F67 Secondary: 20F65

Keywords: systolic complexes and groups, filling radius, word-hyperbolic groups, asymptotic invariants

Introduction

Systolic complexes were introduced in [JS2] and, independently, in [H]. These are simply connected simplicial complexes that satisfy certain curvature-like condition that we call simplicial nonpositive curvature (SNPC). The condition is local and purely combinatorial. It neither implies nor is implied by nonpositive curvature for geodesic metrics on complexes, but it has many similar consequences. It is a simplicial analogue of the combinatorial condition for cubical complexes equivalent to nonpositive curvature for the standard piecewise euclidean cubical metric, namely flagness of links at vertices.

A remarkable property of simplicial nonpositive curvature is that SNPC complexes of groups are developable. This allowed to construct in [JS2] numerous examples of interesting systolic complexes (pseudomanifolds, normal chamber complexes of arbitrary thickness) with large automorphism groups, in arbitrary dimension. The above automorphism groups led to the notion of systolic groups, defined in [JS2] as ones that act by simplicial atomorphisms properly discontinuously and cocompactly on a systolic complexes. It was shown that systolic groups are biautomatic and their existence was established in arbitrary cohomological dimension. A minor modification in construction yields also groups that are in addition word-hyperbolic.

Ideas related to systolic complexes allowed to answer various open questions. For example, there was isolated a simple combinatorial condition for simplicial complexes that implies word-hyperbolicity of their fundamental groups, and which works in dimensions greater than 2. Another simple combinatorial condition turned out to be sufficient for a simplicial complex to be CAT (0) with respect to the standard piecewise euclidean metric. Constructions of systolic complexes allow to obtain finite nonpositively curved (in metric sense) branched coverings of arbitrary finite simplicial piecewise euclidean pseudomanifolds. They also provide finite nonpositively curved developments of euclidean simplicial billiard tables in arbitrary dimension. The reader is referred to [JS2] for details concerning these results.

Some other properties of systolic groups were found by D. Wise [Wi]. For example, he showed that every finitely presented subgroup of a systolic group is systolic. He also revealed interesting connections between systolic concept and that of small cancellation.

In this paper, which is a continuation of [JS2], we study filling properties in systolic complexes and groups. One of our motivations was to show that in systolic world one meets objects very different from classical ones. The following result is a manifestation of this phenomenon.

Theorem A (see Corollary 5.4 in the text). Let G be a systolic group. Then G contains no subgroup isomorphic to the fundamental group of a closed nonpositively curved riemannian manifold of dimension greater than 2.

Theorem A together with the construction in [JS2] of word-hyperbolic sytolic groups of arbitrary cohomological dimension yields the following.

Corollary B. For each natural n > 2 there is a word-hyperbolic group of cohomological dimension n containing no subgroup isomorphic to the fundamental group of a closed nonpositively curved riemannian manifold of dimension greater than 2.

It is interesting to confront Corollary B with conjectural existence of a surface subgroup in every word-hyperbolic group of dimension greater than 1.

The above results are also related to a loose conjecture of M. Gromov saying that every construction of hyperbolic groups of large (rational) cohomological dimension uses arithmetic groups as building blocks. A more precise form of this conjecture, stated as Question Q 1.3 in [Be], asks wheather for every K > 0 there is N > 0 such that every wordhyperbolic group G of rational cohomological dimension ≥ N contains an arithmetic lattice of dimension ≥ K. Examples of word-hyperbolic systolic groups of large cohomological dimension, constructed in [JS2], show that this is not true. Cocompact lattices (of large cohomological dimension) are excluded as their subgroups by Theorem A, while cofinite volume ones by biautomaticity. The latter is due to the fact that infinite cyclic subgroups of biautomatic groups are undistorted and solvable subgroups are virtually abelian (see [Ep]).

The second main result of the paper deals with systolic groups that are products, showing that there are very few of them.

Theorem C (see Corollaries 7.5 and 7.7 in the text). If product of two infinite groups is systolic then both factor groups are virtually free. Product of more than two infinite groups is never systolic.

Theorems A and C are proved using the concept of the filling radius for spherical cycles. Roughly speaking, a k-spherical cycle in a space is a map from the sphere S^{k} to (an appropriate thickening of) the space, and its filling is a singular chain in the space bounded by this cycle. Filling radius of a spherical cycle C is the infimum over all fillings D of C of the maximal distance of a point in D from C. Typically, filling radius of C grows with size (e.g. diameter) of C. Strikingly, in systolic complexes filling radius of k-spherical cycles, for k ≥ 2, is universally bounded above. More precisely, we have the following.

Lemma (see Lemma 3.4 in the text). Let f : S → X be a simplicial map from a triangulation S of the sphere S^{k}, for some k ≥ 2, to a systolic complex X. Then f has a simplicial extension F : B → X, for some triangulation B of the ball B^{k+1} with ∂B = S, such that the image F (B) is contained in the full subcomplex of X spanned by the image f(S).

In Section 4 we introduce an asymptotic property of metric spaces inspired by the above lemma. We call this property S^{k}FRC, which is an abbreviation of the phrase ”kspherical cycles have filling radius constant (i.e. uniformly bounded)”. In fact, this gives a sequence of properties, one for each natural k. Properties S^{k}FRC are shown to be preserved by quasi-isometries and hence applicable to finitely generated groups equipped with word metrics. Theorem A above is a consequence of the following result concerning S^{k}FRC.

Theorem D.

1. (see Corolary 2.4 in combination with Lemma 4.3) If for some natural number k group G is S^{k}FRC then any subgroup H < G is S^{k}FRC.

2. (see Theorem 3.1 in combination with Lemma 4.3) Systolic groups are S^{k}FRC for any k ≥ 2.

3. (see Proposition 5.1) Let G be the fundamental group of a closed nonpositively curved riemannian manifold of dimension n. Then G is not S^{k}FRC for 1 ≤ k ≤ n − 1.

We think it is worth emphasizing that among numerous filling invariants it is the filling radius (and not more commonly used filling volume) that works particularly efficiently in the context of systolic complexes and groups. Next two facts concerning this invariant imply the second main result of the paper, Theorem C above.

Theorem E (see Proposition 6.1 in the text). Product of more than two infinite groups is never S^{2}FRC. If product of two infinite groups is S^{2}FRC then both factor groups are S^{1}FRC.

Theorem F (see Propositions 7.1 and 7.2 in the text). If a finitely presented group is S^{1}FRC then it is word-hyperbolic and its Gromov boundary has topological dimension 0. Consequently, G is virtually free.

As a byproduct of our arguments we obtain the following interesting corollary about groups with asymptotic dimension 1. For a background concerning asymptotic dimension the reader is referred to [R]. Corollary follows from Theorem F and Proposition 4.4.

Corollary G. A finitely presented group with asymptotic dimension 1 is virtually free.

One more idea contained in the paper and related to filling radius is the concept of asymptotic hereditary asphericity (AHA) introduced in Section 2. It implies S^{k}FRC for all k ≥ 2 and it is satisfied by systolic complexes and groups. As the name sugests, it is modelled on the property of hereditary asphericity for topological spaces (see [Da], [DD]). AHA (and in fact its minor strengthening, see Remark 3.6.1) is the strongest known to us geometric (i.e. quasi-isometry invariant) property of systolic complexes and groups. It is inherited by uniformly embedded subspaces and in particular by arbitrary finitely generated subgroups. In this paper we use AHA only to shorten expositon, but we suspect it may play a role in future developments.

Last section of the paper deals with filling volume in systolic complexes. The exposition is essentialy independent of the previous sections and the proofs use different methods. The main result, Theorem H below, was suggested to us by D. Wise. The reader is referred to Section 8 for the precise statement of this result.

Theorem H (see Theorem 8.2 in the text). Filling volume for 2-spherical cycles in systolic complexes is linearly bounded by volume of the cycles.

1. Systolic complexes and groups.

In this section we recall definitions, basic properties and some earlier obtained results concerninig systolic complexes and groups.

Simplicial complex X is flag if every finite subset of its vertices pairwise connected by edges spans a simplex of X. A cycle in a simplicial complex is a subcomplex homeomorphic to the circle S^{1}. Length |γ| of a cycle γ is the number of its edges. We say that a cycle in X has a diagonal if some two nonconsecutive vertices in this cycle are connected by an edge in X.

Given a natural number k ≥ 5, simplicial complex X is k-large if it is flag and every cycle in X of length 4 ≤ |γ| < k has a diagonal.

A simplicial complex X is locally k-large if its links at all (nonempty) simplices are k-large. X is k-systolic if it is locally k-large, connected and simply connected. Since the case of k = 6 is particularly important, we abbreviate the term ”6-systolic” to an easier to pronounce term systolic.

We view k-largeness as local ”curvature” condition for links of a complex, and local k-largeness as a kind of curvature bound from below. We sometimes call local 6-largeness simplicial nonpositive curvature (shortly SNPC), as it yields similar consequences as metric nonpositive curvature. Systolic complex is then the simplicial analogue of what is called CAT (0) or Hadamard space.

We now list several straightforward properties of the just introduced notions.

1.1 Fact.

(1) If k > m and X is k-large then X is also m-large.

(2) Any full subcomplex of a k-large simplicial complex is k-large.

(3) Links of a k-large complex are k-large, i.e. any k-large complex is locally k-large.

(4) If γ is a cycle of length |γ| = m < k in a k-large simplicial complex X then γ bounds in X a subcomplex Δ homeomerphic to the 2-disc consisting of m−2 2-simplices (and thus containing no other vertices than the vertices of γ).

Given a vertex v in a simplicial complex X, let balls in X centered at v be the subcomplexes of X defined inductively as follows. Let B_{0}(v, X) = v and put B_{n+1}(v, X) to be the union of all those simplices of X that intersect B_{n}(v, X). Implicit in the statement of Lemma 4.3 of [JS2] is the following.

1.2 Lemma. If X is a systolic simplicial complex and v is any vertex of X then for every integer n ≥ 0 the ball B_{n}(v, X) is a deformation retract of the ball B_{n+1}(v, X).

This lemma has several important consequences gathered in the following.

1.3 Corollary.

(1) (see Corollary 7.5 and Lemma 7.2(1) in [JS2]) Balls in systolic complexes are contractible.

(2) (see Theorem 4.1(1) in [JS2]) Systolic simplicial complexes are contractible.

(3) Locally 6-large simplicial complexes are aspherical (i.e. their universal covers are contractible). In particular, 6-large simplicial complexes are aspherical.

(4) Every full subcomplex in a 6-large simplicial complex is aspherical.

Corollary 1.3(2) above is a simplicial analogue of Cartan-Hadamard Theorem.

It is interesting to mention that there is no obvious relationship between local 6largeness (SNPC) and nonpositive curvature in metric sense. However, k-systolicity for k sufficiently large implies metric nonpositive curvature (see Section 15 in [JS2] for details). Also, 7-systolicity of a simplicial complex implies its hyperbolicity in the sense of Gromov. More precisely, we have the following.

1.4 Theorem (see Theorem 2.1 in [JS2]). Let X be a 7-systolic simplicial complex. Then the 1-skeleton of X with its standard geodesic metric is δ-hyperbolic with δ = 2\frac{1}{2}.

A group is k-systolic if it acts by simplicial automorphisms, properly discontinuously and cocomplactly, on a k-systolic simplicial complex. It is systolic, if it is k-systolic with k = 6. In Section 14 of [JS2] we have established the following.

1.5 Theorem. Every systolic group is biautomatic.

The reader is referred to [Ep] for definition and properties of biautomatic groups. Biautomaticity has several algorithmic and geometric consequences. For example, every biautomatic group satisfies quadratic isoperimetric inequality and hence has solvable word problem. Moreover, every solvable subgroup of a biautomatic group is virtually abelian and every abelian subgroup is undistorted.

As a consequence of Theorem 1.4 above we also have the following property of groups.

1.6 Corollary. Every 7-systolic group is word-hyperbolic.

Interestingly, systolic complexes and groups exist in any dimension. Constructions from Sections 19–21 in [JS2] imply, among others, the following.

1.7 Theorem.

(1) (see Corollary 21.2 in [JS2]) For each natural n and each k ≥ 6 there exists an ndimensional orientable compact simplicial pseudomanifold that is k-large.

(2) (see Corollary 21.3(1) in [JS2]) For each natural n and every k ≥ 6 there exists a developable simplex of groups whose fundamental group is k-systolic and has virtual cohomological dimension n.

The fundamental group of a pseudomanifold as in Theorem 1.7(1) is easily seen to be torsion-free and have cohomological dimension n. In both parts (1) and (2), if k ≥ 7 then the corresponding fundamental group is in addition word-hyperbolic.

To formulate results showing even more flexibility in our constructions we need several definitions. An n-dimensional simplicial complex X is a chamber complex if it is the union of its n-simplices (which are the chambers of X). Clearly, links of a chamber complex are chamber complexes (of lower dimension). A chamber complex is gallery connected if any two of its chambers can be connected by a finite sequence of chambers such that any two consecutive chambers in this sequence share a face of codimension 1. A chamber complex is normal if it is gallery connected and all of its links are gallery connected.

Thickness of a chamber complex X at a face σ of codimension 1 is the number of chambers of X containing σ. For example, X is a pseudomanifold if its thickness is uniformly 2. Chamber complex X is thick if its thickness at every face of codimension 1 is greater than 2. Thickness of X is its maximal thickness at some codimension 1 face. The next result follows directly from Proposition 21.1 in [JS2].

1.8 Theorem. For each natural n and every k ≥ 6 there is a compact n-dimensional thick chamber complex X, with arbitrarily large thickness (uniform or variable as a function on the set of codimension 1 faces) which is k-large.

A simplicial map f : X → Y between pseudomanifolds is a branched covering if it is nondegenerate and its restriction to the complements of codimension 2 skeleta in X and Y is a covering.

1.9 Theorem (see Proposition 22.2 in [JS2]). Given any k ≥ 6, every finite family of normal compact pseudomanifolds of the same dimension has a common compact branched covering which is k-large.

2. Asymptotic hereditary asphericity.

In this section we introduce the concept of asymptotic hereditary asphericity. We also show that it is invariant under quasi-isometries and even uniform embeddings. In particular, when applied to finitely generated groups with word metrics, it passes to arbitrary finitely generated subgroups. We finish the section with some comments and questions.

Recall that, given a metric space X and a real number r > 0, the Rips’ complex P_{r}(X) is the simplicial complex with X as the vertex set, in which a finite subset of X spans a simplex iff all distances between the points in this subset are not greater than r. Clearly, if 0 < r < R then P_{r}(X) is a subcomplex in P_{R}(X).

We view any subset A of a metric space X as a metric space equipped with the restricted metric.

2.1 Definition.

1. Given real numbers 0 < r ≤ R, a metric space X is (r, R)-aspherical if every simplicial map f : S → P_{r}(X), where S is any triangulation of sphere S^{k} with k ≥ 2, has a simplicial extension F : B → P_{R}(X) for some triangulation B of the ball B^{k+1} such that ∂B = S.

2. A metric space X is asymptotically hereditarily aspherical (shortly AHA) if for every r > 0 there is R ≥ r such that each subset A ⊂ X is (r, R)-aspherical.

Recall that a (not necessarily continuous) map h : (X_{1}, d_{1}) → (X_{2}, d_{2}) between metric spaces is a uniform embedding if there are real functions g_{1}, g_{2} with g_{i}(t) → +∞ as t → +∞, such that

g_{1}(d_{1}(x, y)) ≤ d_{2}(h(x), h(y)) ≤ g_{2}(d_{1}(x, y))

for all x, y ∈ X_{1}.

2.2 Proposition. Suppose that a metric space X_{1} uniformly embeds in a metric space X_{2} which is AHA. Then X_{1} is AHA.

Before giving a proof, we derive some consequennces from Proposition 2.2. First, note that since quasi-isometry is a special case of a uniform embedding, we get the following.

2.3 Corollary. Asymptotic hereditary asphericity is a quasi-isometry invariant.

As a consequence of Corollary 2.1, it makes sense to speak of AHA property for finitely generated groups, by requiring it to hold for the word metric associated to any finite generating set. Then, since a finitely generated subgroup of a finitely generated group uniformly embeds in this group (via inclusion), we get the following.

2.4 Corollary. If a finitely generated group Γ is AHA then every finitely generated subgroup of Γ is AHA.

Proof of Proposition 2.2:

Without loss of generality, we can assume that the functions g_{1}, g_{2} in the estimates for the uniform embedding h : X_{1} → X_{2} are monotone and that g_{1} ≤ g_{2} (otherwise we replace them with ĝ_{1}(t) := inf_{s≥t} g_{1}(s) and ĝ_{2}(t) := sup_{s≤t} g_{2}(s)).

Fix any subset A ⊂ X_{1}, any r > 0, and any simplicial map f : S → P_{r}(A) from a triangulated sphere S^{k}, for some k ≥ 2. The map h induces the simplicial map h_{r} : P_{r}(X_{1}) → P_{g_{2}(r)}(X_{2}), and we consider the subset h(A) ⊂ X_{2} and the composed map h_{r} ∘ f : S → P_{g_{2}(r)}[h(A)]. Since X_{2} is AHA, there is R′ ≥ g_{2}(r) (depending only on h and r, and not on A and f) and a simplicial extension F′ : B → P_{R′}(h(A)) of h_{r} ∘ f, for some triangulation B of the ball B^{k+1} such that ∂B = S. Put R := sup{t : g_{1}(t) ≤ R′} and note that, since g_{1}(r) ≤ g_{2}(r) ≤ R′, we have R ≥ r. Moreover, if h is injective, the inverse map h^{−1} : h(A) → A induces the simplicial map (h^{−1})_{R′} : P_{R′}[h(A)] → P_{R}(A). The composed map F = (h^{−1})_{R′} ∘ F′ : B → P_{R}(A) is an extension of f showing that A is (r, R)-aspherical. Since the constant R above does not depend on A, the proposition follows.

In the case when h is not injective, we need to modify slightly the last part of the argument. Namely, we define a map F on the vertex set of B by requiring that F (v) ∈ h^{−1}(F′(v)) and F (v) = f(v) for v ∈ ∂B = S. These conditions are compatible, since h ∘ f(v) = F′(v) for v ∈ ∂B = S. Moreover, since d_{2}(F′(v), F′(w)) ≤ R′ for any two adjacent vertices in B, by definitions of R and F we have d_{1}(F (v), F (w)) ≤ R for any two such vertices. Therefore, F induces the simplicial map B → P_{R}(A) which is an extension of f, and the rest of the argument goes as at the end of the previous paragraph.

2.5 Comments and questions.

1. The property of asymptotic hereditary asphericity introduced in this section is modelled on the property of hereditary asphericity for topological (metric) spaces, see [Da]. The latter enabled to construct examples of spaces for which the cell-like maps do not raise dimension. Hereditarily aspherical spaces can have arbitrarily large dimension [DD].

2. Recall that according to the (unresolved) Whitehead’s conjecture every subcomplex of an aspherical 2-complex is aspherical. This provides the context for the following observations. 2.1. Let X be a complex of dimension ≤ 2 satisfying Whitehead’s conjecture. Using methods similar to those in Section 3 it is not hard to show that G = π_{1}X is AHA.

2.2. CAT (0) complexes of dimension ≤ 2 satisfy Whitehead’s conjecture and hence are AHA. As a consequence, groups acting properly discontinuously and cocompactly on such complexes are AHA. Thus, product of two (virtually) free groups is AHA. 2.3. It follows from 2.2 that virtually free groups are AHA. It is also not hard to see

that, more generally, groups Γ with asdimΓ ≤ 1 are AHA. 2.4. Whitehead’s conjecture is satisfied by various classes of small cancellation complexes. For example, every subcomplex of a C′(\frac{1}{6}) complex is C′(\frac{1}{6}), and hence aspherical. Consequently, C′(\frac{1}{6}) small cancellation groups are AHA. Note that this class contains many groups that are not finitely presented.

3. Are there any 2-dimensional groups that are not AHA? This question may be viewed as group theoretic asymptotic variant of Whitehead’s conjecture. Here dimension can have various meanings (virtual cohomological dimension, asymptotic dimension, etc.). Special cases of this question are the following. 3.1. Are word-hyperbolic groups with 1-dimensional boundary AHA? 3.2. Are random groups of Gromov [G] or generic groups of Champetier [Ch] AHA

(compare comment 4.5.5)?

4. There are AHA groups with both virtual cohomological and asymptotic dimensions arbitrarily large. These are for example hyperbolic systolic groups of arbitrarily large vcd constructed in [JS2] (compare Theorem 1.7 (2)). Their asymptotic dimension is large since for word-hyperbolic groups we have asdimΓ ≥ dim ∂Γ + 1 = vcdΓ (see [BM] and [Sw]).

5. Is generic (in any reasonable sense) group of dimension ≥ 3 AHA?

6. Damian Osajda [O2] has proved that Gromov boundary of a 7-systolic group is strongly hereditary aspherical. We have reasons to believe that the same is true for all hyperbolic AHA groups. Is the converse true, i.e. is a word-hyperbolic group with hereditarily aspherical Gromov boundary necessarily AHA?

7. For what classes of subgroups H is the free product with amalgamation along H of AHA groups again AHA? Is this true for H finite, virtually cyclic, of asymptotic dimension 1, AHA? What about HNN-extensions?

3. Systolic complexes and groups are AHA.

This section is entirely devoted to the proof of the following theorem and to the statement of its corollaries.

3.1 Theorem. Each systolic simplicial complex is asymptotically hereditarily aspherical. More precisely, for each r > 0 every subset A ⊂ X^{(0)} in the vertex set X^{(0)} of a systolic simplicial complex X, with the induced from X polygonal distance, is (r, 8r+17)-aspherical.

Before getting to the proof, we list interesting corollaries. Recall that a group is systolic if it acts simplicially properly discontinuously and cocompactly on a systolic simplicial complex. Since such a group, equiped with a word metric, is clearly quasi-isometric to the corresponding systolic complex, in view of Corollary 2.3 we get the following.

3.2 Corollary. Every systolic group is asymptotically hereditarily aspherical.

Due to Corollary 2.4 we get also the following stronger result.

3.3 Corollary. Every finitely generated subgroup of a systolic group is AHA.

We now turn to the proof of Theorem 3.1. The first step is the following observation concerning systolic simplicial complexes.

3.4 Lemma. Let f : S → X be a simplicial map from a triangulation S of the sphere S^{k}, for some k ≥ 2, to a systolic complex X. Then f has a simplicial extension F : B → X, for some triangulation B of the ball B^{k+1} with ∂B = S, such that the image F (B) is contained in the full subcomplex of X spanned by the image f(S).

Proof: Denote by K the full subcomplex of X spanned by f(S). Since X is systolic, it is 6-large, and since any full subcomplex of a 6-large complex is 6-large, K is 6-large. In particular, K is aspherical. It follows that the map f, viewed as a continuous map, can be contracted in K to a point. Modifying this contraction to a simplicial map we get an extension F as required.

We proceed with the proof of Theorem 3.1. Let X be a systolic complex. Due to Corollary 2.3, it is sufficient to show that the set X^{(0)} of vertices in X, equipped with the metric of polygonal distance in the 1-skeleton of X, is AHA. Fix r > 0, a subset A ⊂ X^{(0)} and consider arbitrary k-spherical cycle f : S → P_{r}(A), for some k ≥ 2.

3.5 Claim. There is a subdivision S_{∗} of S and a simplicial map f_{∗} : S_{∗} → X such that (1) f_{∗}(v) = f(v) for every vertex v in the initial triangulation S;

(2) for every simplex σ of S, viewed as a subcomplex of S_{∗}, we have diam(f_{∗}(σ)) ≤ 4r +8.

We skip the proof of the claim until the end of this section (before Remark 3.6), and continue the proof of Theorem 3.1, using the claim. Take a simplicial map f_{∗} : S_{∗} → X as in the claim and put C = 4r + 8. For each vertex v in S_{∗} consider its image f_{∗}(v) and denote by f_{0}(v) a vertex in A at distance not greater than C from f_{∗}(v) in the polygonal distance in X^{(0)}. If v is a vertex of the initial triangulation S (before subdividing into S_{∗}), put f_{0}(v) := f(v) = f_{∗}(v). Choices as above are possible due to conditions (1) and (2) in Claim 3.5. The map f_{0} induces the simplicial map S_{∗} → P_{2C+1}(A) which we also denote f_{0}. Moreover, maps f_{0} and f are clearly simplicially homotopic as maps S^{k} → P_{2C+1}(A), in the following sense. There is a triangulation T of the product S^{k} ×[0, 1] and a simplicial map h : T → P_{2C+1}(A) such that

(h1) triangulation T restricted to S^{k} × {0} and S^{k} × {1} is isomorphic to S and S_{∗}

respectively; (h2) restrictions of h to S^{k} × {0} and S^{k} × {1} coincide with f and f_{0} respectively.

To get the assertion, it is sufficient to get an extension F_{0} : B_{0} → P_{2C+1}(A) of f_{0}, for some triangulation B_{0} of the ball B_{k+1} such that ∂B_{0} = S_{∗}. Indeed, an extension F of f as asserted can be then combined out of F_{0} and h.

To get F_{0} as above, apply Lemma 3.4 to the map f_{∗} : S_{∗} → X. It gives an extension F_{∗} : B_{∗} → X with image F_{∗}(B_{∗}) contained in the full subcomplex K in X spanned by the image f_{∗}(S_{∗}). Define F_{0} in a similar way as f_{0}. For every vertex v of B_{∗} choose F_{0}(v) to be a vertex in A at polygonal distance in X^{(0)} not greater than C. However, for boundary vertices v ∈ ∂B_{∗} put F_{0}(v) := f_{0}(v), which makes sense since ∂B_{∗} = S_{∗}. Choices as above are possible due to condition (2) in Claim 3.5 and due to inclusion F_{∗}(B_{∗}) ⊂ K. It is clear from the description that F_{0} induces the simplicial map B_{∗} → P_{2C+1}(A) which is an extension of f_{0} as required. Since 2C + 1 = 8r + 17, the theorem follows.

It remains to prove Claim 3.5. For every 1-simplex (uw) in S choose a path p_{uw} in the 1-skeleton of X, of length

1 ≤ L_{uw} ≤ r + 2, connecting the vertices f(u) and f(w). Subdivide (uw) into L_{uw} 1simplices and define f_{∗} on the subdivided (uw) as its obvious simplicial map on the path p_{uw}.

For every 2-simplex σ = (uwz) in S, its boundary ∂σ is already subdivided and the map f_{∗} is already defined on ∂σ. The image f_{∗}(∂σ) is easily seen to be contained in the combinatorial ball B_{2k}(f(u), X), where k = [r + 2] is the largest integer not greater than r + 2. Since balls in systolic complexes are contractible, there is a subdivision σ_{∗} of σ, compatible with the subdivision on ∂σ, and a simplicial map f_{∗}^{σ} : σ_{∗} → B_{2n}(f(u), X) which extends the already defined map f_{∗} on ∂σ. Moreover, the diameter of the image f_{∗}^{σ}(σ_{∗}) in X is clearly not greater than 4k ≤ 4r + 8. Note that in the same way we can define (by induction with respect to dimension) subdivisions σ_{∗} and extensions f_{∗}^{σ} for all simplices σ in S, with the same estimate for diameters of images.

Composing a subdivision S_{∗} out of the subdivisions σ_{∗} above, and taking f_{∗} to be the union of the maps f_{∗}^{σ}, we get a map f_{∗} : S_{∗} → X as required in the claim.

3.6 Comments and questions.

1. The arguments in this section give the following more general filling property in systolic complexes: every simplicial map from a simply connected simlicial complex to a systolic complex can be contracted to a point in the simplicial span of its image. In particular, every simplicial cycle modelled on a simply connected space can be filled in the simplicial span of its image by a chain modelled on the cone over this space. The latter can be converted into an asymptotic property slightly stronger than AHA, namely that filling radius for cone fillings of simply connected cycles is uniformly bounded. This property is the strongest known to us coarse property of systolic spaces and groups.

2. Systolic groups are, by definition, finitely presented. Are there any finitely presented AHA groups that are not systolic? Dani Wise [Wi] conjectures that product of two nonabelian free groups is not systolic. Since this product is AHA (see 2.5.2.2), it is a candidate in dimension 2.

3. Are there any AHA groups in dimension ≥ 3 that are not systolic?

4. Are simply connected cubical complexes with 6-large links AHA? Note that for 5-large links this is not true, for example due to the cubical tesselation with all links icosahedral of the hyperbolic 3-space. Similar example can be also constructed in dimension 4. However, there are serious restrictions on the topology of 5-large simplicial complexes above dimension 3. For example, no sphere of dimension above 3 admits a 5-large triangulation, see [JS1, Section 2.4]. Are simply connected cubical complexes with 5-large links S^{k}FRC for k ≥ 4?

4. Filling radius of spherical cycles.

To prove the main results of this paper concerning systolic groups, we make use only of a rather limited part of the fact that these groups are AHA. We need only the S^{2}FRC property, as introduced below in this section. This property, and the related properties S^{k}FRC for k ≥ 2, are easily implied by asymptotic hereditary asphericity. Since these properties seem to be interesting independantly, we include some side comments and questions.

To define properties S^{k}FRC we need some preparations. Our exposition uses simplicial chain complexes with arbitrary coefficients.

4.1 Definition. A k-spherical cycle in a simplicial complex X is a simplicial map f : S → X (possibly degenerate) from an oriented simplicial k-sphere S. Denote by C_{f} the simplicial cycle induced by f (the image through f of the fundamental cycle in S). A filling of a k-spherical cycle f is a simplicial (k + 1)-chain D in X such that ∂D = C_{f}. Given a simplicial chain C = ∑_{i} t_{i}σ_{i} of dimension n in X, its support is the set supp(C) consisting of all vertices in all underlying simplices |σ_{i}| for which the coefficients t_{i} are non-zero (here we assume that if σ′ is obtained from σ by taking opposite orientation then σ′ = −σ, and that the underlying simplices |σ_{i}| of the oriented simplices σ_{i} occuring in C = ∑_{i} t_{i}σ_{i} are distinct for distinct i). The support of a spherical cycle f : S → X, denoted supp(f), is the image through f of the vertex set of S.

The filling radius of a cycle C in a simplicial complex X is the minimum over all fillings D of the maximal distance of a vertex in supp(D) from the support of C. Usually, this filling radius grows with the diameter of cycles C. However, it easily follows from Lemma 3.4 that in systolic complexes X the filling radius for k-spherical cycles with k ≥ 2 is 0 (i.e. each such cycle f has a filling with support contained in supp(f)). This is the model behaviour that motivates the next definition, which describes an asymptotic version of this phenomenon.

4.2 Definition. A metric space X has filling radius for k-spherical cycles constant (shortly, X is S^{k}FRC), if for every r > 0 there is R ≥ r such that any k-spherical cycle f : S → P_{r}(X) which is null-homologous in P_{r}(X) has a filling D in P_{R}(X) with supp(D) ⊂ supp(f).

We skip an easy argument, similar to that in the proof of Proposition 2.2, showing that for each k the S^{k}FRC property is inherited by spaces which uniformly embed into S^{k}FRC spaces, and hence it is a quasi-isometry invariant. For completeness, we include a proof of the following.

4.3 Lemma. Let X be a metric space which is asymptotically hereditarily aspherical. Then X is S^{k}FRC for all k ≥ 2.

Proof: For a fixed r > 0 let R ≥ r be the corresponding constant for X as in the definition of AHA. We will show that each k-spherical cycle in P_{r}(X) (no matter if null-homologous in P_{r}(X) or not) has a desired filling in P_{R}(X), for the same constant R.

Let f : S → P_{r}(X) be any k-spherical cycle. If we denote by A ⊂ X the set of images through f of all vertices in S, we can view f as a map to the Rips’ complex P_{r}(A). By the choice of R, there is a simplicial extension F : B → P_{R}(A) of f as in the definition of AHA (Definition 2.1). We can then take as a filling D of f the chain C_{F} induced by the map F, i.e. the image in P_{R}(A) of the fundamental chain of B. This finishes the proof.

One more straightforward observation concerning S^{k}FRC property, Proposition 4.4 below, describes its relationship to the asymptotic dimension of a metric space. We refer the reader to [G] or [R] for the definition and some basic facts concerning the latter notion (in [G] this notion is denoted asdim_{+} and called large scale dimension). For the proof (which is rather straightforward) we refer the reader to [Sw], where this result occurs as unnumbered proposition in the appendix (p. 220).

4.4 Proposition. If X is a metric space and asdimX = p then X is S^{k}FRC for each k ≥ p.

4.5 Comments and questions.

1. There are many ways in which one can strengthen S^{k}FRC. One is to replace homological fillings by (simplicial) homotopical ones. Another is to require thin fillings for all spherical cycles, and not only for fillable ones (call this S^{k}-thinness). Also, we can require S^{k}FRC (or its stronger variants) to hold uniformly in k (at least for k ≥ 2 or k sufficiently large), i.e. with R not depending on k for any given r. All these strengthenings still follow, for all k ≥ 2, from asymptotic hereditary asphericity.

2. Another way to strengthen S^{k}FRC is to require constant filling radius for all nullhomologous k-dimensional cycles, and not only for spherical ones (call this k-FRC). The same requirement for all cycles, not only for null-homologous ones, is still stronger (call it k-thinness). Of course, one can require both conditions above to hold uniformly for appropriate dimensions k. However, k-FRC and k-thinness are not implied by AHA. These conditions are closely related to asymptotic dimension, see 4.5.6.

3. Proofs of Lemma 3.4 and Theorem 3.1, with no changes, show that systolic complexes and groups are k-thin for all cycles that are simply connected (call this SC-k-thin), uniformly in k ≥ 2 (compare comment 3.6.1 where even stronger, homotopical variant of this property is mentioned). SC-k-thinnes is inherited by uniformly embedded subspaces, and in particular by arbitrary finitely generated subgroups. See Comment 5.7.2.1 for an interesting application of SC-k-thinness.

4. Note that if X is k-thin for k ≥ 2 uniformly, then for each r > 0 there is R ≥ r such that for any A ⊂ X and any k ≥ 2 the map H_{k}[P_{r}(A)] → H_{k}[P_{R}(A)] has trivial image. This may be vieved as homological analogue of AHA.

5. It follows from the results and arguments in Section 5 of [Sw] that if Γ is a random group as described by M. Gromov in [G, Chapter 9] (and called strongly isoperimetric group in [Sw]), then Γ is k-thin uniformly for k ≥ 2. The same is true for groups satisfying various types of small cancellation conditions.

6. By Proposition in the appendix of [Sw] (p. 220), bounds on the asymptotic dimension of a metric space X imply strengthened variants of S^{k}FRC. For example, if asdimX = p then X is p-FRC and it is k-thin for k ≥ p + 1 uniformly. Uniform k-FRC for k ≥ p defines what is called in [Sw] (Definition 1.2) the asymptotic homological dimension, denoted asdim_{h}.

7. Define asymptotic spherical homological dimension asdim_{h}^{s} as follows: asdim_{h}^{s}X ≤ p iff X is S^{k}FRC for all k ≥ p. We have inequalities asdim_{h}^{s} ≤ asdim_{h} ≤ asdim (see [Sw] for the latter inequality). It is also clear that asdim_{h}^{s}(X) ≤ 2 for each metric space X which is AHA, in particular for systolic complexes and groups. On the other hand, the asymptotic dimensions asdim and asdim_{h} of a systolic complex or group can be arbitrarily large (see Comment 2.5.4).

8. Find a group Γ with asdim_{h}^{s}Γ ≤ 2 which is not AHA. Find a group which is S^{k}FRC for k ≥ 2 but not uniformly, or which is not S^{k}-thin for some k ≥ 2. What other reasons can make a group with asdim_{sh}Γ ≤ 2 not AHA? The same problems concern groups Γ with asdim_{h}Γ ≤ 2, or even asdimΓ ≤ 2, compare Question 2.5.3.

9. Systolic groups are (in general) not k-thin for any natural k. However, they may have filling radius uniformly bounded for all cycles modelled on a fixed k-dimensional manifold or pseudomanifold M, whatever k and M are. If this is true, the following interesting problem arises. What is the dependence of the uniform filling radius bound on the topology of M? Note that this potentially yields some topological complexity hierarchy in which simply connected manifolds are the least complex ones (compare 4.5.3).

5. Nonpositively curved manifolds.

In this section we look at the S^{k}FRC condition in the context of nonpositively curved manifolds. Our main objective is to show that systolic groups contain no fundamental groups of nonpositively curved manifolds other than surface groups. A crucial observation is the following.

5.1 Proposition. Let X be a simply connected nonpositively curved complete riemannian manifold with dim(X) = n. Then X is not S^{k}FRC for 1 ≤ k ≤ n − 1.

Before proving the proposition we derive its consequences. By applying Lemma 4.3 we get the following.

5.2 Corollary. If X is a simply connected nonpositively curved complete riemannian manifold with dim(X) ≥ 3 then X is not S^{2}FRC. In particular, X is not AHA.

To formulate next corollaries we introduce some terminology. A group is called a nonpositively curved manifold group of dimension n if it acts isometrically, properly discontinuously and cocompactly on a simply connected nonpositively curved complete riemannian manifold of dimension n (examples are the fundamental groups of closed nonpositively curved manifolds). Since such groups are clearly quasi-isometric to the corresponding manifolds, Corollary 5.2 implies the following.

5.3 Corollary. Nonpositively curved manifold groups of dimension greater than 2 are not S^{2}FRC. In particular, they are not AHA.

The above corollary together with Corollary 3.3 imply the following results concerning systolic groups.

5.4 Corollary. A systolic group contains no nonpositively curved manifold group of dimension greater than 2. In particular, it contains no fundamental group of a closed nonpositively curved manifold other than the infinite cyclic group or a surface group (including the torus group Z ⊕ Z).

5.5 Corollary. An abelian subgroup of a systolic group has rank at most 2.

5.6 Corollary. A systolic group contains no product of fundamental groups of closed nonpositively curved manifolds other than Z ⊕ Z.

Proof of Proposition 5.1:

Fix some k as in the statement and some r > 0. For every R ≥ r we will construct a k-spherical cycle f : S → P_{r}(X) which is null-homologous in P_{r}(X) but has no filling in P_{R}(X) with support contained in the the support of f.

Fix a point p ∈ X and consider the exponential map E : T_{p}X → X. By Cartan-Hadamard Theorem, this map is a distance non-decreasing homeomorphism. Consider the sphere of radius 1000R in the tangent space T_{p}X, centered at 0, and an equatorial sub-sphere S_{k} of dimension k in it. Denote by N_{R} the R-neighbourhood of S_{k} in T_{p}X, i.e. the set consisting of all points in T_{p}X at distance not greater than R from S_{k}. Note that the inclusion of S_{k} in N_{R} is a homotopy equivalence, and in particular the element [S_{k}] induced by S_{k} in the singular homology H_{k}(N_{R}) is nontrivial. The same is clearly true for the element [E(S_{k})] in H_{k}(E(N_{R})).

Fix a triangulation S of the image sphere E(S_{k}) so that the adjacent vertices are at distance at most r in X. The inclusion of vertices of S in X induces a k-spherical cycle f : S → P_{r}(X). Since X is contractible, this cycle is clearly null-homologous in P_{r}(X). Suppose D is a filling of f in P_{R}(X) with support contained in the support of f. Denote by U_{R} the R-neighbourhood of the image sphere E(S_{k}) in X. Since U_{R} ⊂ E(N_{R}) (because the exponential map E is distance non-decreasing), and since supp(f) ⊂ E(S_{k}), it is not hard to make out of D a singular chain in E(N_{R}) which fills some singular cycle representing the homology class [E(S_{k})]. This contradicts the fact that 0 ≠ [E(S_{k})] ∈ H_{k}(E(N_{R})), hence the proposition.

5.7 Comments and questions.

1. Is the fundamental group of every closed aspherical manifold of dimension n ≥ 3 not S^{k}FRC for some 2 ≤ k < n? 1.1. Note that, due to Theorem 3b.2 (iii) in [DJ] (and its proof), if X is a closed PL n-manifold with large links then metric spheres in X are homeomorphic to the sphere S^{n−1}. In particular, the proof of Proposition 5.1 above applies to the universal cover of such manifold and, consequently, its fundamental group is not S^{k}FRC for 1 ≤ k ≤ n−1. There are such manifolds that (after smoothing) admit no riemannian metric of nonpositive curvature.

2. Is it true that systolic (or more generally AHA) groups contain no fundamental groups of closed aspherical manifolds of dimension greater than 2? In view of Corollary 2.4 and Lemma 4.3, positive answer to Question 1 above would imply positive answer to the present question. 2.1. One can answer Question 2 in positive, referring to SC-k-thinness (see Comment

4.5.3) rather than to AHA, for a class of aspherical manifolds described in Section 5c of [DJ]. This is some class of negatively curved topological manifolds N with no negatively curved PL metric, existing in any dimension n ≥ 5. Crucial property of these manifolds is that metric spheres in their universal covers, though not homeomorhic to S^{n−1}, are all simply connected. In particular, fundamental groups of these manifolds N are not SC-(n − 1)-thin by the argument as in the proof of Proposition 5.1. In view of Comment 4.5.3, fundamental groups of these manifolds are not systolic, and cannot occur as subgroups in systolic groups. These groups are different from fundamental groups of riemanian or PL negatively curved manifolds since the ideal boundary is not a manifold.

2.2. Damian Osajda [O1] has proved that higher homotopy groups at infinity for systolic complexes are trivial. Since, according to a result of D. Wise [Wi], finitely presented subgroups of systolic groups are systolic, this means that systolic groups contain no subgroups isomorphic to the fundamental groups of closed aspherical manifolds covered by R^{n}, for n ≥ 3.

3. Is it true that systolic (or more generally AHA) groups contain no Poincare duality subgroups of dimension greater than 2? This would answer in positive the question of L. Mosher and M. Sageev [MS] about the existence of hyperbolic groups of large dimension that have no Poincare duality subgroup of dimension ≥ 3.

6. Products.

In this section we characterize products of groups that are S^{2}FRC. Together with the results in the next section, this provides restrictions on product groups that are asymptotically hereditarily aspherical, or that are systolic. The main result is the following.

6.1 Proposition. Product of more than two infinite groups is never S^{2}FRC. If product of two infinite groups is S^{2}FRC then both factor groups are S^{1}FRC.

In view of the obvious fact that an infinite group is not S^{0}FRC, the above proposition is a direct consequence of the following.

6.2 Proposition. If G_{1} is a group which is not S^{k}FRC for some k ∈ {0, 1}, and G_{2} is infinite, then the product G_{1} × G_{2} is not S^{k+1}FRC.

Proof: We will work with fixed word metrics d_{1}, d_{2} on the groups G_{1}, G_{2} respectively, and with the word metric d on G_{1} × G_{2} corresponding to the union of generating sets in the factor groups. This metric d can be also described by

d((g_{1}, g_{2}), (g′_{1}, g′_{2})) = d_{1}(g_{1}, g′_{1}) + d_{2}(g_{2}, g′_{2}).

Since G_{1} is not S^{k}FRC, there is r > 0 such that for every R ≥ r there is a k-spherical cycle f : S → P_{r}(G_{1}) which is null-homologous in P_{r}(G_{1}) but has no filling in P_{R}(G_{1}) with support contained in the support of f. We will construct out of f a (k + 1)-spherical cycle f_{c} : S_{c} → P_{r+1}(G_{1} × G_{2}) null-homologous in P_{r+1}(G_{1} × G_{2}) but having no filling in P_{R}(G_{1} × G_{2}) with support contained in the support of f_{c}. This will show that G_{1} × G_{2} is not S^{k+1}FRC.

From the fact that a k-spherical cycle f : S → P_{r}(G_{1}) as above is null-homologous in P_{r}(G_{1}) it follows that there is a triangulation D of the (k + 1)-disc, extending the triangulation S viewed as the boundary triangulation of D, and a simplicial map F : D → P_{r}(G_{1}) extending f. Note that here we use the assumption that k = 0, 1, since in higher dimensions spheres can bound topologically more complicated chains. For the second factor group G_{2}, consider the triangulation L of the line segment consisting of 3l + 1 edges, for some l ≥ R, with vertices a_{0}, a_{1}, …, a_{3l+1} placed in this order on L, and an isometric simplicial embedding λ : L → P_{1}(G_{2}), i.e. a simplicial map such that d_{2}(λ(a_{0}), λ(a_{3l+1})) = 3l + 1. Such λ clearly exists since the group G_{2} is infinite. Take the product D × L, with the product structure of a polyhedral cell complex, and choose a triangulation (D × L)^{t} subdividing the cell structure of D × L without introducing new vertices. The map of the vertex set of D × L to G_{1} × G_{2} determined by F and λ induces the simplicial map F × λ : (D × L)^{t} → P_{r+1}(G_{1} × G_{2}). Since (D × L)^{t} is topologically a (k + 2)-disc, its boundary ∂(D × L)^{t} is a triangulation of the (k + 1)-sphere, and we define the (k + 1)-spherical cycle f_{c} to be the restriction

f_{c} = (F × λ)|_{∂(D×L)^{t}} : ∂(D × L)^{t} → P_{r+1}(G_{1} × G_{2}).

Clearly, the map F × λ shows that f_{c} is null-homologous in P_{r+1}(G_{1} × G_{2}).

It remains to show that there is no filling of f_{c} in P_{R}(G_{1} ×G_{2}) with support contained in the support of f_{c}.

To any simplicial n-dimensional chain C in P_{R}(G_{1} × G_{2}) we associate a simplicial (n − 1)-dimensional chain Δ(C) in P_{R}(G_{1}), in the following three steps. First, we take C′ to be the chain in P_{R}(G_{1} × G_{2}) consisting of those n-simplices in C which have at least one vertex not contained in the subcomplex P_{R}(G_{1} × B), where B = {g ∈ G_{2} : d_{2}(g, λ(a_{0})) ≤ 2l}. Next, we take as ∂_{B}C′ the subchain in the boundary ∂C′ consisting of those (n − 1)-simplices that have all vertices in G_{1} × B. Finally, we project (the vertices of) ∂_{B}C′ on G_{1}, getting the chain Δ(C) in P_{R}(G_{1}).

Note that the above description of the operator Δ corresponds, when applied to cycles, to the standard way of defining the homomorphisms

H_{n}(P_{R}(G_{1} × G_{2})) → H_{n}(P_{R}(G_{1} × G_{2}), P_{R}(G_{1} × B)) →
→ H_{n−1}(P_{R}(G_{1} × B)) → H_{n−1}(P_{R}(G_{1}))

on the level of simplicial cycles. We mention without including further arguments the following easy to verify properties of Δ:

(1) if C is a cycle then Δ(C) is a cycle,

(2) Δ(∂C) = −∂(Δ(C)) for any chain C, (3) Δ(f_{c}) = f.

Now, suppose that D is a filling of f_{c} in P_{R}(G_{1} × G_{2}) with support contained in supp(f_{c}). Then, by the above properties of Δ, the chain Δ(D) is (up to sign) a filling of f in P_{R}(G_{1}). We check the properties of the support of Δ(D). Note first that all vertices in the simplices of D′ lie at distance greater than R from G_{1} × {λ(a_{0})}. Consequently, the vertices of ∂_{B}D′ lie at distance greater than R from G_{1} × {λ(a_{0}), λ(a_{3l+1})}. Since, denoting by L^{(0)} the set of vertices in L, we heave

supp(f_{c}) ⊂ G_{1} × {λ(a_{0}), λ(a_{3l+1})} ∪ supp(f) × λ(L^{(0))}

and

supp(∂_{B}D′) ⊂ supp(D) ⊂ supp(f_{c}),

it follows that supp(∂_{B}D′) ⊂ supp(f) × λ(L^{(0)}). After projecting to G_{1}, we clearly get supp(Δ(D)) ⊂ supp(f), a contradiction. This completes the proof.

6.3 Comments and questions.

1. A method used in the proof of Proposition 6.2 allows to show that, for any k, a product of an infinite group with a group which is not k-FRC is not (k + 1)-FRC. Is the statement of Proposition 6.2 still true for k > 1?

2. For k, l ≥ 2, is product of a group which is not k-FRC with a group which is not l-FRC always not (k + l)-FRC? For the same k and l, is product of a group which is not S^{k}FRC with a group which is not S^{l}FRC always not S^{k+l}FRC?

3. Is product of any two S^{1}FRC groups S^{2}FRC (compare 7.9.3)? The same question can be asked for other products of S^{k}FRC or k-FRC groups.

7. S^{1}FRC groups.

In this section we establish several properties of S^{1}FRC groups. We use them, together with results from the previous section, to get restrictions on product groups that are AHA or systolic.

Our main results are the following.

7.1 Proposition. If a finitely presented group is S^{1}FRC then it is hyperbolic.

7.2 Proposition. Topological dimension of the Gromov boundary of an S^{1}FRC hyperbolic group is zero.

In view of the well known fact that a hyperbolic group with zero-dimensional boundary is virtually free, the above propositions imply the following.

7.3 Corollary. Any finitely presented S^{1}FRC group is virtually free.

Before proving Propositions 7.1 and 7.2 we derive some of their consequences for S^{2}FRC, systolic and AHA groups.

7.4 Corollary. A product of two infinite finitely presented groups is S^{2}FRC iff both factors are virtually free.

Proof: One implication in Theorem 6.1 follows from the fact that asymptotic dimension of a finitely generated free group is 1, while that of a product of two such groups is 2, and from Proposition 4.4. The other implication follows from Proposition 6.1 and Corollary 7.3.

7.5 Corollary. If product of two infinite groups is systolic then both factor groups are virtually free.

Proof: Since it acts simplicially properly discontinuosly and cocompactly on a simplicial complex, any systolic group is finitely presented. If it is a product then all factors are finitely presented. The rest of the argument is then as in the proof of Theorem 7.4.

Since AHA groups are S^{2}FRC (Lemma 4.3), we also get the following.

7.6 Corollary. If product of two infinite finitely presented groups is AHA then both factors are virtually free.

Note that Proposition 6.1 alone, without the help of results from this section, implies the following result that complements Corollaries 7.5 and 7.6.

7.7 Corollary. Product of more than two infinite groups is never AHA and, in particular, never systolic.

Proof of Proposition 7.1:

Let Γ be any finitely presented group which is S^{1}FRC. Fix a finite generating set Σ and the corresponding word metric d_{Σ} in Γ. Since Γ is finitely presented, there is r ≥ 1 such that each 1-spherical cycle in the Rips’ complex P_{1}(Γ) is null-homologous in P_{r}(Γ). For example, we can take as r the maximal length of a relation from some finite presentation for Γ with Σ as the generating set. Let R ≥ r be a constant occuring in the definition of S^{1}FRC property, i.e. such that each 1-spherical cycle f : S → P_{r}(Γ) which is null-homologous in P_{r}(Γ), has a filling D in P_{R}(Γ) with support contained in the R-neighbourhood of the support of f. We will show that Γ is δ-hyperbolic for δ = 100R. The constant δ we obtain is obviously far from optimal.

Let T be a geodesic triangle in the Cayley graph C(Γ, Σ), with vertices in Γ. View T as a 1-spherical cycle in P_{1}(Γ), and denote the corresponding simplicial map from a 1-sphere by f_{T} : S_{T} → P_{1}(Γ). Since f_{T} has a filling in P_{r}(Γ) (due to finite presentability, as discussed above), there is a filling D of f_{T} in P_{R}(Γ) such that supp(D) ⊂ supp(f_{T}). This implies the following

Claim 1. The map f_{T} (viewed as a continuous map) is a loop which is contractible (i.e. homotopic to a constant map) in the Rips’ complex P_{R}[supp(f_{T})].

To prove δ-hyperbolicity of Γ, for δ = 100R, we need to show that, for any side A of the triangle T and for every vertex x on A, there is a vertex y in the union B ∪ C of two other sides such that d_{Σ}(x, y) ≤ 100R. Note that if length of A is not greater than 200R then this condition is trivially satisfied. If length of A is greater than 200R, chooose vertices p_{0}, p_{1}, …, p_{m} on A, occuring in this order, with p_{0}, p_{m} the endpoints of A, such that for i = 0, 1, …, m − 1 we have

50R < d_{Σ}(p_{i}, p_{i+1}) ≤ 98R.

This clearly can be done, for example by taking all distances minimal (i.e. as close from above to 50R as possible), except the last two possibly bigger. For each i = 0, 1, …, m put U_{i} := {g ∈ Γ : d_{Σ}(g, p_{i}) ≤ 50R}. We then have:

(a1) U_{i} ∩ U_{i+1} ≠ ∅ for i = 0, 1, …, m − 1,

(a2) U_{i} ∩ U_{j} = ∅ for |i − j| > 1,
(a3) A^{(0)} ⊂ ∪_{i=0}^{m}U_{i}, where A^{(0)} is the set of vertices on the side A.

We skip the straightforward checking of these properties.

For every vertex v ∈ B^{(0)} ∪ C^{(0)} put U_{v} := {g ∈ Γ : d_{Σ}(g, v) ≤ R}. Denote by U the family consisting of the sets U_{i} for i = 0, 1, …, m and the sets U_{v} for all v ∈ B^{(0)} ∪ C^{(0)}. It follows from property (a3) that

(a4) supp(f_{T}) ⊂ ∪U. We will show the following

Claim 2. For each i = 0, 1, …, m−1 there is v ∈ B^{(0)} ∪C^{(0)} such that U_{i} ∩U_{i+1} ∩U_{v} ≠ ∅.

Before proving the claim we will show how to use it to conclude the proof of Proposition 7.1. For each x ∈ A^{(0)} there is i such that d_{Σ}(x, p_{i}) ≤ 49R. By Claim 2, there is v ∈ B^{(0)} ∪ C^{(0)} with d_{Σ}(p_{i}, v) ≤ 50R + R = 51R. Consequently, d_{Σ}(x, v) ≤ 100R, and hence Γ is δ-hyperbolic with δ = 100R.

It remains to prove Claim 2. To do this, denote by N(U) the nerve of the family U, i.e a simplicial complex whose vertices are the sets from U and whose simplices correspond to subfamilies of U that have nonempty intersection. For each vertex w ∈ supp(f_{T}) choose a set from U containing w, denoting it by h(w), as follows. If w ∈ A^{(0)} choose as h(w) one of the sets U_{i} and if w ∈ B^{(0)} ∪ C^{(0)} \ {p_{0}, p_{m}} put h(w) = U_{w}. Note that if vertices w_{1}, w_{2} ∈ supp(f_{T}) are adjacent in the Rips’ complex P_{R}Γ then the corresponding sets h(w_{1}), h(w_{2}) have nonempty intersection. Consequently, h induces a simplicial map H : P_{R}[supp(f_{T})] → N(U).

Suppose, contrary to Claim 2, that there is i such that U_{i} ∩ U_{i+1} ∩ U_{v} = ∅ for each v ∈ B^{(0)} ∪ C^{(0)}. Together with property (a2) above this implies that the edge (U_{i}, U_{i+1}) in the nerve N(U) is isolated, i.e. not contained in any 2-simplex. We claim that then the composed map H ∘ f_{T} : S_{T} → N(U) is not contractible. To see this, note that h(p_{i}) = U_{i}, h(p_{i+1}) = U_{i+1} and h(w) ∈ {U_{i}, U_{i+1}} for all vertices w contained in A lying between p_{i} and p_{i+1}. This means that part of the 1-sphere S_{T} corresponding to the segment [p_{i}, p_{i+1}] ⊂ A is mapped through H on the edge (U_{i}, U_{i+1}). Moreover, by definition of h, no edge in S_{T} not contained in the above segment [p_{i}, p_{i+1}] is mapped on the edge (U_{i}, U_{i+1}) through H. As a consequence, the loop H ∘ f_{T} : S_{T} → N(U) passes through the edge (U_{i}, U_{i+1}) odd number of times, and since the edge is isolated, the loop is not contractible. Since this contradicts the assertion of Claim 1, the proof of Proposition 7.1 is completed.

Proof of Proposition 7.2:

We will prove that if G is a hyperbolic group such that dim ∂_{∞}G ≥ 1 then G is not S^{1}FRC.

Given ϵ > 0, an ϵ-path in a metric space (X, d) connecting points a, b ∈ X is a finite sequence x_{0}, x_{1}, …, x_{n} of points in X such that x_{0} = a, x_{n} = b, and d(x_{i−1}, x_{i}) < ϵ for i = 1, …, n. We start with the following auxilliary fact, certainly well known, including its proof for completeness.

7.8 Fact. Let (X, d) be a compact metric space with dim X ≥ 1. Then there are distinct points a, b ∈ X which can be connected by an ϵ-path in X for every ϵ > 0.

Proof: We will show that if points a, b as in the assertion do not exist then dim X = 0. More precisely, we will show that every point a ∈ X has arbitrarily small neighbourhoods that are open and closed.

Let U be any open neighbourhood of a in X. Consider a function f : X \ {a} → R defined by

f(x) := inf{ϵ > 0 : x is connected with a by an ϵ-path in X}.

By our assumption in the proof, function f is positive at every point. Moreover, it is easily seen to be locally constant, hence continuous. Since the complement X \ U is compact, f has minimum m > 0 at this complement. Take any 0 < δ < m and put

V := {x ∈ X : x can be connected with a by a δ-path in X}.

It is straightforward that V is open and closed, and that a ∈ V ⊂ U. This completes the proof of Fact 7.8.

Coming bact to the proof of Proposition 7.1, note that Gromov boundary ∂_{∞}G is a compact metric space. Existence of ϵ-paths as in Fact 7.8 between some points a, b ∈ ∂_{∞}G allows to construct S^{1}-cycles f : S → P_{r}G, for some fixed r > 0, with arbitrarily large filling radii, using a method from [Sw]. More precisely, if dim ∂_{∞}G ≥ 1, there is a proper compact subset K ⊂ ∂_{∞}G with dim K ≥ 1, and distinct points a, b ∈ K which can be connected by an ϵ-path c_{ϵ} in K for every ϵ > 0. Fixing point s ∈ ∂_{∞}G \ K, it is possible to project paths c_{ϵ} to various horospheres in G centered at s, by the projections described in Section 2 of [Sw]. If ϵ is small enough, we can construct an S^{1}-cycle by taking projections of c_{ϵ} on two sufficiently distant horospheres, and connecting them with two segments contained in projection rays corresponding to points a, b (the endpoints of c_{ϵ}). The fact that S^{1}-cycles in some P_{r}G constructed in this way have arbitrarily large filling radii follows from the argument as in the proof of Theorem 3.1 in [Sw]. This whole argument is the same as in the proof of Theorem 4.1 in [Sw]. We skip further details.

7.9 Comments and questions.

1. Results of this section motivate the following general question. Which products of infinite groups are ”2-dimensional”? ”Dimension 2” in this question can be meant in any of the following senses: S^{2}FRC, AHA, systolic, asymptotically or (virtually) cohomologically 2-dimensional, S^{2}-thin, 2-FRC, 2-thin, etc. Here are some observations and more detailed questions concerning this subject. 1.1. Note that, since product of two virtually free groups is asymptotically 2-dimen-

sional (as product of two asymptotically 1-dimensional factors), AHA (Comment 2.5.2.2), and S^{2}-thin, and since S^{2}-FRC is implied by any of these three conditions (Lemma 4.4), Corollary 7.4 can be strengthened as follows: a product of two infinite finitely presented groups is asymptotically 2-dimensional (or AHA or S^{2}- thin) iff both factor groups are virtually free.

1.2. Dani Wise [Wi] conjectures that product of two non-abelian free groups is not

systolic. 1.3. Note that product of two groups of asymptotic dimension 1 has asymptotic di-

mension at most 2 (see [R, Proposition 9.11]) and hence, due to Proposition 4.4, is S^{2}FRC. Is it AHA? Are there S^{1}FRC groups that are not asymptotically 1dimensional? If the answer to the last question is negative then product of any two S^{1}FRC groups is S^{2}FRC.

2. Which extensions of infinite groups by infinite groups are systolic, S^{2}FRC or AHA? Note that in systolic case this class probably coincides with the class of systolic products, because systolic groups are biautomatic [JS2]. (Biautomatic groups satisfy quadratic isoperimetric inequality, their solvable subgroups are virtually abelian and their infinite cyclic subgroups are undistorted. This probably excludes most of non-product extensions.) On the other hand, Baumslag-Solitar groups are AHA as they act on the product of a tree and the line (compare 2.5.2.1).

3. What are the relationships between the following classes of ”1-dimensional” groups: virtually free, of asymptotic dimension 1, S^{1}-thin, S^{1}-FRC? In finitely presented case all these classes coincide, and in infinitely presented one there are obvious inclusions. A similar question can be asked for various classes of ”2-dimensional” and ”higher dimensional” groups.

8. Second isoperimetric inequality.

In this section we show that the isoperimetric function for 2-spherical cycles (the so called second isoperimetric function) is linear in systolic complexes. Our exposition uses simplicial chain complexes with integer coefficients.

To formulate the main result of this section we need the following.

8.1 Definition. Let C = ∑_{i} t_{i}σ_{i} be an n-chain in a simplicial complex X. Suppose (without loosing generality) that the underlying simplices |σ_{i}| for which t_{i} ≠ 0 are distinct for distinct i. The (n-dimensional) volume of C, V_{n}(C), is defined as V_{n}(C) := ∑_{i} |t_{i}|. The (2-dimensional) volume of a spherical 2-cycle f : S → X, V_{2}(f), is the number of 2-simplices in S that are mapped by f on 2-simplices (i.e. the number of 2-simplices on which f is injective).

8.2 Theorem. Given any spherical 2-cycle f : S → X in a systolic simplicial complex X, there exists a filling D of f such that

V_{3}(D) ≤ \frac{3}{2} · V_{2}(f).

Proof: The argument goes by induction with respect to the number of 2-simplices in the triangulation S of the 2-sphere.

The smallest triangulation S of the 2-sphere consists of four 2-simplices (the boundary triangulation of a 3-simplex). Then any nontrivial (i.e. such that [f] ≠ 0) spherical 2-cycle f : S → X has volume V_{2}(f) = 4. Moreover, by the fact that X is flag, there is a filling D of f consisting of a single simplex, i.e. such that V_{3}(D) = 1. Asseretion the theorem is clearly satisfied in this case.

Now, let S be any triangulation of the 2-sphere, and let f : S → X be any simplicial map. We consider subsequently three cases. In first two cases we deal with rather degenerate situations, making no use of the fact that X is systolic. We need these cases for completeness of the proof. The key systolic argument occurs in the third (most regular) case. Case 1: f is degenerate.

Consider a 1-simplex e = (v, w) in S for which f(v) = f(w), and let τ_{1} = (v, w, u_{1}), τ_{2} = (v, w, u_{2}) be the 2-simplices adjacent to e in S. Delete from S the interior of the union τ_{1} ∪τ_{2} and glue the obtained boundary edges in pairs: (u_{1}, v) with (u_{1}, w) and (u_{2}, v) with (u_{2}, w). As a result we obtain a multisimplicial triangulation S′ of the 2-sphere, which may contain potentially a double edge or a double triangle (i.e two 2-simplices that share boundaries), but never a loop. Suppose first that S′ is simplicial. The simplicial map f : S → X induces the simplicial map f′ : S′ → X in the obvious way. Moreover, we have the equality [f′] = [f] for the induced cycles, and thus every filling of f′ is also a filling of f. Consequently, since S′ consists of less 2-simplices than S, the assertion follows for f by induction.

It remains to deal with the case when S′ is not a simplicial comple8. This case splits into two subcases. First is when S′ consists of two 2-simplices that share their boundaries. But then [f′] = 0, hence [f] = 0, and the assertion holds trivially. Otherwise, there must be a vertex y in S such that (v, y) and (w, y) are both the 1-simplices in S. Since by passing from S to S′ the vertices v and w are identified, this leads to a double edge in S′. We still denote by (w, y) and (v, y) the corresponding edges in S′. Let y_{i} be all the vertices in S satisfying the property as y above. Denote by e_{i} = (v, y_{i}) and e′_{i} = (w, y_{i}) the corresponding edges in S′. The union of all these edges splits S′ into open discs D_{j}, and the boundary of each D_{j} is the union of edges from some pair e_{i}, e′_{i} or from two such pairs. In any case, we identify the eges from the pairs e_{i}, e′_{i} in the closure of each D_{j}, getting 2-spheres S_{j} of the following two kinds. Some of these spheres may consist of two 2-simplices identified along whole boundaries. It is not hard to see that all other spheres S_{j} are the honest simplicial spheres. We will denote them as S_{j_{k}}. Clearly, the map f : S → X induces simplicial maps f_{j_{k}} : S_{j_{k}} → X and we have ∑_{k}[f_{j_{k}}] = [f]. It follows that the sum ∑_{k} D_{j_{k}} of any fillings D_{j_{k}} of f_{j_{k}} is a filling of f. Since moreover ∑_{k} V_{2}(f_{j_{k}}) ≤ V_{2}(f) − 2, the assertion follows for f by induction.

Case 2: f is nondegenerate and the images of some two adjacent 2-simplices in S coincide.

Denote by τ_{1} = (v, w, u_{1}) and τ_{2} = (v, w, u_{2}) the two adjacent simplices in S whose images through f coincide. Remove from S the interior of the union τ_{1} ∪ τ_{2}, and glue the edges of the resulting boundary in pairs: (u_{1}, v) with (u_{2}, v) and (u_{1}, w) with (u_{2}, w). After glueing we obtain a cell decomposition S′ of the 2-sphere. We omit the easy case when S′ is a simplicial triangulation and pass directly to the opposite case.

The only reason for S′ not to be simplicial is that there is an edge e = (u_{1}, u_{2}) in S (which becomes a loop in S′, still denoted e) or that there are pairs of edges e_{i} = (u_{1}, z_{i}) and e′_{i} = (u_{2}, z_{i}) in S (which become the double edges in S′, still denoted e_{i}, e′_{i}). Split S′ into open discs D_{j} along the union of all edges e_{i}, e′_{i} and e as above. Consider those discs D_{j_{k}} in the splitting that contain more than two open 2-simplices of S′. Observe that the loop e (if exists) is not contained in the boundary of any D_{j_{k}}. Moreover, the boundary of each D_{j_{k}} is the union of edges from some pair e_{i}, e′_{i} or from two such pairs. We identify the eges from the pairs e_{i}, e′_{i} in the closure of each D_{j_{k}}, getting 2-spheres S_{j_{k}}. It is not hard to see that each S_{j_{k}} is simplicial. Note that the map f induces in the obvious way the simplicial maps f_{j_{k}} : S_{j_{k}} → X. The rest of the argument is as in Case 1.

Case 3: f is nondegenerate and no two adjacent 2-simplices from S are mapped to the same simple8.

By the Euler characteristic argument (or combinatorial Gauss-Bonnet theorem), there exists a vertex v in S for which the link S_{v} is a polygonal cycle consisting of m < 6 edges. Since the map f is nondegenerate, it induces the nondegenerate simplicial map f_{v} : S_{v} → X_{f(v)} on links. Since no two adjacent 2-simplices of S have the same image, the map f_{v} is locally injective, i.e. no two adjacent edges in S_{v} are mapped to the same edge. Since length m of S_{v} is at most 5, the map f_{v} is in fact injective. Its image in the link X_{f(v)} is a cycle of length m. Since X is systolic, the link X_{f(v)} is 6-large, and thus the cycle f_{v}(S_{v}) has a disc filling Δ_{1} in X_{f(v)} consisting of m − 2 simplices of dimension 2. Viewing the link X_{f(v)} as a subcomplex in X, the cycle f_{v}(S_{v}) has also the disc filling Δ_{2} in X equal to the join of the cycle with v. Moreover, there is a 3-chain C in X, consisting of m − 2 simplices of dimension 3, such that Δ_{2} − Δ_{1} = ∂C and the support of C consists of v and the vertices in f_{v}(S_{v}).

We now modify the spherical 2-cycle f : S → X by removing from S the open star of the vertex v, and by glueing to the obtained boundary S_{v} the disc Δ made of m − 2 simplices of dimension 2 in the way compatible with the filling Δ_{1} of f_{v}(S_{v}). If the obtained 2-sphere S′ is simplicial, we define the simplicial map f′ : S′ → X by taking the compatibility map Δ → Δ_{1} as the restriction of f′ to Δ, and by putting f′ = f outside Δ. Note that V_{2}(f′) = V_{2}(f) − 2 and the difference [f] − [f′] is equal to Δ_{2} − Δ_{1}. Let D be a filling of f′ satisfying the assertion of the theorem (which exists due to inductive assumption). By what was said above about the chain C, the sum D + C is a filling of f with V_{3}(D + C) ≤ V_{3}(D) + m − 2. Since m − 2 ≤ 3, the chain D + C is a filling of f as required.

It remains to consider the subcase in which the obtained 2-sphere S′ is not simplicial. The only reason for S′ not to be simplicial is as follows. For an edge δ of the inserted disc Δ not contained in the boundary ∂Δ, its endpoints (viewed as vertices in S_{v}) may be connected by an edge e in S. In S′ we then get two edges δ and e connecting the same pair of vertices. Let (δ_{i}, e_{i}) be all pairs of double edges in S′ as above. Note that for distinct i the corresponding edges δ_{i} are distinct, and hence there are at most m − 3 such pairs. The union of all edges δ_{i} and e_{i} from these pairs splits S′ into open discs D_{j}, and the boundary of each D_{j} is the union of edges from some pair (δ_{i}, e_{i}) or from two such pairs. Similarly as in Case 1, we identify the edges from the pairs (δ_{i}, e_{i}) in the closure of each D_{j}, getting 2-spheres S_{j}. The rest of the argument is as at the end of Case 1, combined with the estimates above in this case (for S′ simplicial). We omit the obvious details.

8.3 Comments and questions.

1. One is tempted to investigate higher dimensional isoperimetric inequalities for systolic complexes, similar to that in Theorem 8.2. It seems that proving them requires techniques significantly different from ours. On the other hand, tangible meaning and applications of such results remains unclear to us at this stage.

2. What are the possibilities for second isoperimetric function for AHA complexes and groups?

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