# Journées ANR de théorie géométrique des groupes

Du 8 au 9 juin 2009

IRMA

Les journées ANR "Théorie géométrique des groupes" se dérouleront les 8 et 9 juin 2009 à l'IRMA.

Organisateurs : Thomas Delzant, Sylvain Maillot

Financement : ANR Groupes (ANR BLAN07-2_183619)

## Programme

• #### Lundi 8 juin 2009

##### B. Rémy, Lyon

Rigidity and quasi-isometry classes of simple twin building lattices (joint with P.-E. Caprace)

The main subject matter of the talk will be twin building lattices that have been shown to be simple in a previous work. Simplicity holds "generically" when the buildings under consideration are not Euclidean; the most well-understood class of such groups is provided by Kac-Moody theory. These groups are often finitely presented and enjoy Kazhdan's property (T). The latter property suggests to investigate the rigidity properties of the natural action of the groups: we have a "higher-rank versus hyperbolic" rigidity result. One point is that "higher-rank" has to be given a suitable sense; the statement can probably be improved. In the same spirit, our most recent result is the fact that we can obtain infinitely many quasi-isometry classes of finitely presented simple groups.

##### E. Breuillard, Orsay

Version forte de l'alternative de Tits et applications

Je parlerai des liens entre l'alternative de Tits forte, une version adélique du lemme de Margulis, l'approximation diophantienne non-commutative et la conjecture du trou spectral en liaison avec mes travaux récents et ceux de Bourgain-Gamburd.

##### F. Dahmani, Toulouse

Commutations among interval exchange transformations.

An interval exchange transformation is a left-continuous piecewise isometric bijection of the interval $[0,1[$ into itself, with only finitely many discontinuity points. Individual such transformations have been extensively studied, but subgroups of the group of interval exchange transformations are rather poorly understood. In this talk, we report on a joint work with Guirardel and Fujiwara. We show that if such a subgroup is isomorphic to a connected Lie group, it is abelian (answering a question of Franks). An intriguing question is whether the group of interval exchange transformations contains a non-abelian free subgroup. We show that in the set of pairs of transformations $(S,T)$ such that $S$ has at most 2 discontinuity points, there is an open dense subset that cannot generate a non-abelian free subgroup, hence suggesting that free subgroups might be hard to find.

##### V. Guirardel, Toulouse

Small cancellation in the mapping class group

We prove that there exist normal free subgroups of the mappling class groups all of whose non-trivial elements are pseudo-Anosov. This is done using small cancellation techniques for the action on the complex of curves. As further applications, we get that any non-elementary finitely generated subgroup of the mapping class group has many quotients. In particular, this gives a new proof that lattices in higher rank simple Lie groups don't embed in the mapping class group.

##### C. Perin, Jerusalem

Induced definable structure on cyclic subgroups of the free group

Let E(x, a) be a first-order formula in the language of groups with one free variable, and constants a which lie in a group G. The definable set associated to E is the set of elements g of G for which E(g,a) holds in G. The collection of definable sets on G is called the definable structure of G. For example it can be shown that, up to finite sets, the definable sets of the group of integers Z are finite unions of arithmetic progressions (subsets of the form kZ+a). Consider now Z to be embedded in a finitely generated free group F. Is the definable structure induced by F on Z richer than this, or is it the same? In other words, if D is a definable set in F, is the intersection of Z with D definable in Z? We answer this question positively in the special case where D corresponds to a formula with one quantifier. That is, we show that the intersection of Z with D is, up to finitely many elements, a finite union of arithmetic progressions. To do this, we use techniques developped by Sela to find formal solutions to system of equations and inequations with parameters over free groups. These are based on analyzing real trees with Rips theory, and use of the shortening argument.
• #### Mardi 9 juin 2009

• ##### 09:45 - 10:45

Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

##### I. Chatterji, Orléans

Distorted centers and bounded cohomology.

Let G be a connected Lie group. Let R denote its radical (the largest connected solvable normal subgroup). We show that G has all its Borel cohomology with Z-coefficients represented by bounded cocycles if and only if R is linear. We will discuss a few consequences of this result. This is joint work with G. Mislin, Ch. Pittet and L. Saloff-Coste.

##### P.-E. Caprace, Louvain

Is the non-discrete side of locally compact groups more tractable ?

Simple groups play a prominent role in the structure theory of finite groups and of Lie groups. However the position and relevance of simple groups within the category of finitely generated discrete groups is much less obvious. This is best illustrated by residually finite groups. In this talk, we will explain that a compactly generated locally compact group which is residually finite (or more generally residually discrete) is necessarily compact-by-discrete. This may be used to establish decomposition results for non-discrete locally compact groups beyond the case of almost connected groups, thereby providing a motivation to the study of compactly generated simple groups.

##### A. Erschler, Orsay

Spaces of harmonic functions on groups

##### D. Vavrichek, Binghampton

Quasi-isometry invariant commensurizer subgroups.

I will present results about the quasi-isometry invariance of certain infinite cyclic subgroups and their commensurizers in one-ended finitely presented groups. I will discuss several applications of these results, including the quasi-isometry invariance of certain vertex groups of the Scott-Swarup JSJ decomposition for groups.