3rd Young Geometric Group Theory Meeting

Luminy, 20-24 January 2014

Titles and abstracts

**Pierre-Emmanuel Caprace:** *Geometric aspects of simple locally compact groups*

*Abstract:* The goal of this minicourse is to give an account on recent tools, developed in a joint work with Colin Reid and George Willis, designed to study the class S of compactly generated locally compact groups that are topologically simple and non-discrete. A motivation to study that class S comes from the fact that the isometry groups of many natural examples of proper metric spaces of non-positive curvature turn out to belong to S. In the first part of the course, I will describe some of those examples, with a special emphasis on the case where the metric space is a tree, or a more general CAT(0) cube complex. In the second part, I will present general properties of any group G in the class S, and explain how they can be used to produce in some cases a compact G-space whose properties have a geometric flavour which is reminiscent of the properties of the isometry groups of non-positively curved spaces.

**Rémi Coulon:** *Groups with a tree-graded asymptotic cone*

*Abstract:* Given a finitely generated group G, its asymptotic cones are a class of objects that capture the large scale properties of the group. Roughly speaking, they are obtained by looking at G from infinitely far away. The asymptotic cones of a hyperbolic group are trees. Conversely, if G is a finitely presented group such that some asymptotic cone of G is a tree, then G is hyperbolic. The goal of this talk is to present a generalization of this statement for the class of relatively hyperbolic groups. This is joint work with M. Hull and C. Kent.

**Julie Déserti:** *Some properties of the Cremona groups*

*Abstract:* We will give some properties of the Cremona group in dimension 2 (its generators, its finite subgroups, its finitely generated subgroups, its automorphism group...), and explain why the Cremona group in dimension 3 is quite different.

**Slavyana Geninska:** *The limit sets of finitely generated subgroups of irreducible lattices in PSL(2,R)^n*

*Abstract:* While lattices in higher rank Lie groups are very well studied, only little is known about discrete subgroups of infinite covolume of semi-simple Lie groups. The main class of examples are Schottky groups. In this talk we give a description of the limit sets of all finitely generated subgroups of irreducible lattices in PSL(2,R)^n.

**Étienne Ghys:** *My favorite groups*

*Abstract:* The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures.

**Lars Louder:** *Strong accessibility for finitely presented groups*

*Abstract:* A hierarchy for a group G is a tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions of subgroups of G, à la the Haken hierarchy. I will explain why certain kinds of hierarchies of finitely presented groups must be finite.

**Hee Oh:** *Dynamics on geometrically finite hyperbolic manifolds*

*Abstract:* We plan to discuss equidistribution of flows on 3-dimensional hyperbolic manifolds which are not necessarily of finite volume, with the goal of applying the results in counting problems for orbits of discrete hyperbolic groups. When the Kleinian group is geometrically finite and of critical exponent bigger than 1, these equidistribution results as well as the corresponding counting results can be made effective. We will highlight the recent counting results for Apollonian circle packings as well as for more general circle packings invariant under a geometrically finite group.

**John Pardon:** *Totally disconnected groups (not) acting on three-manifolds*

*Abstract:* Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three.

**Alessandro Sisto:** *Quasi-cocycles detect hyperbolically embedded subgroups*

*Abstract:* Hyperbolically embedded subgroups have been defined by Dahmani-Guirardel-Osin and they provide a common perspective on (relatively) hyperbolic groups, mapping class groups, Out(F_n), CAT(0) groups and many others. I will sketch how to extend a quasi-cocycle on a hyperbolically embedded subgroup H to a quasi-cocycle on the ambient group G. Also, I will discuss how some of those extended quasi-cocycles (of dimension 2 and higher) "contain" the information that H is hyperbolically embedded in G. This is joint work with Roberto Frigerio and Maria Beatrice Pozzetti.

**Juan Souto:** *Topology of hyperbolic 3-manifolds and first eigenvalue of the Laplacian*

*Abstract:* I will describe several results linking the topology and geometry of hyperbolic 3-manifolds M to the first eigenvalue lambda_1(M) of the Laplacian on M. The basic problem is whether there are "expander sequences" of hyperbolic 3-manifolds (M_i)_{i>0} satisfying some given topological conditions. Here, (M_i)_{i>0} is an expander sequence if lambda_1(M_i) remains bounded away from 0. The answer to this question depends on the nature of the imposed topological condition and in any case leads to interesting phenomena. For instance, the construction of expander sequences involves often perhaps unusual constructions of hyperbolic 3-manifolds. On the other hand, proving that there are no expander sequences satisfying some topological condition yields passes often through a much deeper understanding on the class of manifolds under consideration.