Geometric Group Theory
Davis 60

An important source of Coxeter groups acting on hyperbolic space is the automorphism groups of Lorentzian lattices, i.e., integer quadratic forms of signature (n,1). Such a form is called reflective if isisometry group is generated up to finite index by reflections. It is known that there are only finitely many, and I have undertaken to find them all; I am guessing there are between 1,000 and 10,000. I will speak on parts of this work. It is motivated by the problem of classifying the "Lorentzian Lie algebras" defined by Gritsenko and Nikulin, which means the "best" kind of hyperbolic Kac-Moody algebra.

One of the main programs in Geometric Group Theory is the study of the QI equivalence relation on groups. In much the same way, Measurable Group Theory studies the ME (Measure Equivalence) relation between groups. In my talk I will define the notion of ME and explain how it provides a neat platform to express a variety of classical results. I will further present a collection of equivalence relations between groups. Finally, I will report on some new progress, focusing mainly on a recent result obtained with Furman and Sauer regarding hyperbolic lattices.

Let M be a finite volume complete negatively pinched manifold and S be a codimension two, totally geodesic, immersed submanifold with normal crossings. Allcock proved that the complement M-S of S is aspherical, yet little is known about its fundamental group. In a joint work with Chris Hruska, we show that the fundamental group of M-S is relatively hyperbolic if S is compact embedded, or if S has a sufficiently thick regular neighborhood. The proof builds on ideas of Bowditch, and the model space for proving relative hyperbolicity is the universal branched cover of M branched over S, which is a non-proper CAT(-1) space.

I will survey our knowledge of the topology and geometry of Out(F_n) and Outer space, particularly as they compare with mapping class groups and Teichmuller space. In the 20+ years since the definition of Outer space by Culler and Vogtmann we have succeeded (more or less) to understand the topology, and the serious work on the geometry is in its early stages.

The Dehn function of a finitely presented group expresses the area of loops (in a geometric model for the group) as a function of the length of the loops. satisfying higher finiteness conditions. These functions express the volumes of (k+1)-ball fillings of k-spheres (in suitable geometric models for the group) as functions of the volumes of the spheres.
We discuss some results about higher dimensional Dehn functions of groups.

I shall discuss several constructions, algebraic and geometric, that involve subdirect products of groups. In particular I'll outline a proof of the following theorem of Bridson-Howie-Miller-Short: a subdirect product of finitely presented groups is finitely presented if it virtually projects onto each pair of factors (the VSP criterion). I'll use this criterion to construct novel examples of finitely presented subdirect products of surface groups. An algorithmic version of the VSP criterion will be used to prove that the class of finitely presented residually free groups is recursively enumerable.

Résumé: Kac-Moody theory provides notably examples of locally compact (semi-)simples groups which possess discrete subgroups of finite covolume. While some of their properties are reminiscent of arithmetic groups in positive characteristic, they generally behave very differently than linear groups. A peculiar property of Kac-Moody lattices is that they admit a Weyl group, which is always a Coxeter group (usually non-amenable). The talk will illustrate this fact by presenting certain properties of Kac-Moody groups which are best understood when reduced to Coxeter group questions.

Outer space for a free group F_n may be described as the space of minimal actions of F_n on a tree. By a theorem of Culler and Morgan, such an action is uniquely determined by its length function, so an alternate description of outer space is the space of length functions on F_n. Motivated by the search for a good "outer space" for right-angled Artin groups, we prove an analogous theorem about length functions for actions of 2-dimensional right-angled Artin groups on CAT(0) cube complexes. (Joint work with Max Margolis)

We compare the volume of a hyperbolic 3 manifold and the complexity of its fundamental group.

Lowner-Gromov type inequalities are used to define a manifold invariant called the 'systolic category'. We will discuss an analogy between the systolic category and the Lusternik-Schnirelmann category. and the relation of the former to the cup-length.

Median spaces are non-discrete generalisations of CAT(0) cube complexes. Both property (T) of Kazhdan and Haagerup property (a-T-menability) can be reformulated in terms of actions on median spaces. This and the fact that the asymptotic geometry of a mapping class group is median imply results on homomorphisms into mapping class groups. The talk is on joint work with I. Chatterji and F. Haglund (first part), and on joint work with J. Behrstock and M. Sapir (second part).

This talk is a report on joint work with Pedro Ontaneda. We have been studying smooth bundles whose closed manifold fibers are equipped with negatively curved Riemannian metrics. Two such bundles with the same base B are equivalent if there is such a bundle with base Bx[0,1] whose restrictions to Bx0 and Bx1 are fiberwise isometric to the given bundles. We obtain results about the forget extra structure map from this theory to ordinary smooth bundle theory.

We characterize elements of Out(F__n) in terms of dynamic invariants and describe some applications.
This is joint work with Michael Handel.

In his seminal article on hyperbolic groups, M. Gromov defined a combinatorial-geometric structure called CAT(0) cube complexes. A fundamental example of CAT(0) cube complex was associated by M. Davis with any right-angled Coxeter system, so that the Coxeter group acts geometrically on the complex.
When a compact space M has (immersed) hyperplanes, its fundamental group acts on a CAT(0) cube complex (by the work of M. Sageev). We show the following : when the hyperplanes of M are embedded and the space M is negatively curved then the fundamental group π₁(M) acting on the CAT(0) cube complex (virtually) extends to a right-angled Coxeter group acting on its Davis' complex
This is joint work with D. Wise.

We explain how to construct a quasi-convex bicombing for the mapping class group and derive some applications to the structure of subgroups.

In his work classifying spherical and affine buildings, J.Tits proved that every (irreducible) thick affine building of rank at least 3 is associated with an algebraic group over a field. In particular, it follows that the finite Weyl groups of such buildings have to be crystallographic. We complete this picture by constructing rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.
This is a joint work with Arkady Berenstein.

We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.
This is a joint work with Indira Chatterji, Christophe Pittet, and Laurent Saloff-Coste.

L²-Betti numbers are invariants of compact manifolds obtained from the square integrable differential forms on the universal covering via the use of functional analysis (von Neumann traces), introduced by Atiyah and studied by Gromov, Lück and many others. By definition, they are non-negative real numbers.
However, for torsion-free fundamental groups the L²-Betti numbers are conjectured always to be integers (know for many groups). By work of Linnell this is equivalent to the fact that a certain overring DG of the group ring QG is a skew field.
Also for groups with torsion there is a precise conjecture on the possible values of L²-Betti numbers. The overring DG in this case can not be a skew field (as QG contains non-trivial zero divisors). We present work about a refinement of the Atiyah conjecture, and we explain how this is equivalent to a precise structure result for the ring DG.
We prove this refined Atiyah conjecture for certain classes of groups, including new cases of the original Atiyah conjecture.
This is joint work with Peter Linnell and Anselm Knebusch.

Right-angled Artin groups (RAAGs) are finitely-generated groups which are completely described by the fact that some of the generators commute; thus they interpolate between free groups at one extreme and free abelian groups at the other. RAAGs and their subgroups are the source of many important examples and counterexamples in geometric group theory. In this talk I will describe joint work with Ruth Charney on the outer automorphism groups of RAAGs, which can be though of analogously as interpolating between the outer automorphism groups of free groups and the general linear groups GL(n,Z). We are particularly interested in determining which properties shared by Out(F_n) and GL(n,Z) are in fact true for all automorphism groups of RAAGs.

I'll talk about partial results in determining the Dehn functions of arithmetic subgroups of semisimple Lie groups. (Joint with Mladen Bestvina and Alex Eskin.)

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n≥4.
This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). In this talk, I will discuss some of the background of the problem and sketch a proof that the Dehn function of SL(n,Z) is at most quartic when n≥5.