Aspect géométriques et dynamiques des groupes discrets

Lille, 4-6 septembre 2013

Titres et résumés

Titles and abstracts

**Uri Bader:** A perspective on Super-Rigidity

*Résumé :* In my talks I will prove in details a rigidity result which implies and extends Margulis and Zimmer Super-Rigidity. The proof is based on the the study of algebraic representations of actions, which extend representations of groups. The proof is fairly simple and I intend to explain everything in details. It is based on joint work with Alex Furman.

**Virginie Charette:** Variétés lorentziennes de dimension trois

*Résumé :* Le but de ce mini-cours est de décrire des 3-variétés lorentziennes complètes et plates; autrement dit, ce sont des quotients de R^3 par des groupes d'isométries lorentziennes affines agissant de façon libre et proprement discontinue. Nous nous intéresserons particulièrement au cas où les groupes sont librement engendrés, notamment lorsque leur partie linéaire est un groupe de Schottky. Une condition nécessaire pour qu'un tel groupe agisse correctement est ce qu'on appelle le "lemme des signes opposés" de Margulis. Quant à la condition suffisante, nous montrerons comment construire des "domaines fondamentaux croches", dûs à Drumm.
- Cours 1 : mise en contexte

a- petit historique

b- actions proprement discontinues sur R^3

c- l'espace de Minkowski et ses isométries

d- isométries hyperboliques affines et invariant de Margulis
- Cours 2 : actions propres

a- lemme des signes opposés : énoncé et preuve

b- plans croches et domaines fondamentaux croches

c- théorème de Drumm : énoncé et esquisse de preuve
- Cours 3 : applications du théorème de Drumm

a- espace des actions propres pour les surfaces de rang deux

b- généralisations de l'invariant de Margulis

**Spencer Dowdall:** Dynamics for splittings of free-by-cyclic groups

*Résumé :* The combined work of Thurston and Fried shows that the monodromies associated to the fibrations of a hyperbolic 3-manifold are intimately related in important topological, geometric, and dynamical ways. After reviewing this picture, I will describe recent results showing that similar relationships hold for the monodromies associated to splittings of a free-by-cyclic group. In particular, when the free-by-cyclic group is the mapping-torus group of a fully irreducible atoroidal automorphism of F_n, we will see that the monodromies of all nearby splittings are also fully irreducible and have similar stretch factors. If time permits I will also describe a certain polynomial invariant that simultaneously encodes the strech factors of all these monodromies. This is joint work with I. Kapovich and C. Leininger.

**Anna Erschler:** Limit order for finitely generated groups

*Résumé :* We consider a preorder on the space of finitely generated groups up to isomorphism which corresponds to taking limits in Cayley-Grigorchuk topology. The associated oriented graph can be described as follows: its vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in (G,S_i) and in (H,T) coincide. We study the structure on this graph and the interplay of this structure with algebraic and geometric properties of finitely generated groups. This is joint work with Laurent Bartholdi.

**Vincent Guirardel:** Wise's malnormal special quotient theorem

*Résumé :* Wise's malnormal special quotient theorem is a fundamental ingredient of Agol's theorem saying that every hyperbolic cubulable group is virtually special. I will explain how to give a proof this result using Gromov's geometric small cancellation theory.

**Thomas Haettel:** Braid groups, non-crossing partitions and curvature

*Résumé :* Tom Brady constructed a simplicial classifying space for the n-strand braid group, whose vertex link is isomorphic to complex of non-crossing partitions of n points. Using the metric locally induced by a spherical building, I will show that the braid group is CAT(0) up to 7 strands. This is a work in progress, with Dawid Kielak and Petra Schwer.

**Peter Haïssinsky:** Hyperbolic groups with planar boundaries

*Résumé :* The purpose of this series of talks is to provide necessary and sufficient conditions for a word hyperbolic group
with a planar boundary to be virtually isomorphic to a convex-cocompact Kleinian group.

The first talk should be devoted to establishing dynamical properties of convergence groups on the sphere. The second talk will concentrate on the analytical aspects of the problem. The third will deal with the more algebraic aspects, concluding with the proposed characterizations.

**Anders Karlsson:** Metric spectral theory

*Résumé :* I will discuss analogs of concepts and results from functional analysis in the setting of metric spaces. Applications include a generalization of Thurston's spectral theorem for surface homeomorphisms.

**Chloé Perin:** Forking independence in the free group

*Résumé :* Model theorists define, in structures whose first-order theory is "stable" (i.e. suitably nice), a notion of independence between elements. This notion coincides for example with linear independence when the structure considered is a vector space, and with algebraic independence when it is an algebraically closed field. Sela showed that the theory of the free group is stable. In joint work with Rizos Sklinos, we give some interpretation of this model theoretic notion of independence in the free group using cyclic JSJ decompositions.

**Wenyuan Yang:** Growth of quotients of a relatively hyperbolic group

*Résumé :* A finitely generated group G is called growth tight if for any finite generating set S, the growth rate of G relative to S is strictly bigger than that of any quotient of G by an infinite normal subgroup. It was known that all (non-elementary) relatively hyperbolic groups are growth tight. In this talk, I will construct, for any relatively hyperbolic group G, a sequence of relatively hyperbolic quotients G_n such that the growth rate of G_n tends to that of G.

Our approach is based on the study of Patterson-Sullivan measures on Bowditch boundary of a relatively hyperbolic group. We prove a variant of the Sullivan Shadow Lemma, called Partial Shadow Lemma. This tool allows to prove several results on growth functions of balls and cones. One central result is the existence of a sequence of geodesic trees with growth rates sufficiently close to the growth rate of G, and with transition points uniformly spaced in trees. By taking small cancellation over hyperbolic elements of high powers, we can embed these trees into properly constructed quotients G_n of G. This proves our results.