|Résumé:||The divergence of a geodesic metric space measures the cost of going from a point to another avoiding a large ball. Drutu, Mozes and Sapir proved that the growth rate of the divergence function is closely related to the existence of cut-points in asymptotic cones. In this talk I will introduce these notions and discuss the case of the special linear group over a polynomial ring.|
|Résumé:||Je discuterai des propriétés des théories élémentaires des groupes, plus particulièrement de celles des groupes hyperboliques et des groupes qui agissent sur les arbres, ainsi que des liens avec la théorie géométrique des groupes.|
|Résumé:||In the sixties, starting with a polyhedron of the real projective space and projective reflection across each facet of the polyhedron, Vinberg (and also Tits) build a convex open set Ω on which the Coxeter group W associated to the polyhedron acts properly. Vinberg gives a necessary and sufficient condition for the action to be cocompact. I will give sufficient condition for the action to be of finite covolume or convex-cocompact or geometrically finite.|
|Résumé:||There is a well-known necessary and sufficient condition for a Coxeter group to be Gromov-hyperbolic, and it is equivalent to the existence of a negatively curved metric on the Davis complex. However, there exist non-hyperbolic Coxeter groups acting nicely on negatively curved spaces, such as groups corresponding to hyperbolic simplices with ideal vertices. In my talk I would like to discuss actions of non-affine Coxeter groups on CAT(-1) metric spaces and related curvature properties. I will also point out how such actions, together with appropriate hyperbolic filling constructions, might be possibly combined to produce CAT(-1) quotients of Coxeter groups. Some applications, for example the behavior of the reflection length, will be presented as well.|
|Résumé:||A Cartan subalgebra is a regular maximal abelian subalgebra A ⊂ M in a von Neumann algebra M. Regular means that the normalizer of A inside M generates the whole algebra M. Given a countable standard Borel equivalence relation R ⊂ X × X and a 2-cocycle ω, one can construct a Cartan subalgebra L∞(X) ⊂ Lω R. Feldman and Moore showed that any Cartan subalgebras is of this form. Recently, Popa and Vaes provided the first example of countable discrete group Γ such that for any free ergodic probability measure preserving action of Γ on (X, μ) we have a unique Cartan decomposition for the von Neumann algebra L∞(X) ⋊ Γ. In this talk, I will present some sufficient conditions on equivalence relations R such that the von Neumann algebras Lω R have unique Cartan decompositions. This is a joint work with Stefaan Vaes.|
|Résumé:||Dans cet exposé, je présenterai différents résultats et conjectures concernant les propriétés asymptotiques des actions d'un sous-groupe d'un groupe de Lie G sur un quotient de volume fini de G. J'évoquerai aussi leurs applications en théorie des nombres.|
|Wednesday 23 January 2013||Thursday 24 January 2013|
Property (T): motivation, definitions, examples
An application: expander graphs
Isometric affine actions and 1-cohomology
Random walks on linear groups
Transience of algebraic varieties in Zariski dense subgroups
1-cohomology (Shalom's theorem)
Random subgroups of linear groups are free
Colloquium (independent talk; see below)
|Résumé:||Considérons un graphe fini X, qu'on voit comme espace métrique discret. La distorsion-Lp de X est un invariant numérique qui mesure la difficulté à plonger X isométriquement dans Lp. Un résultat fondamental de Bourgain (1984) dit que la distorsion-Lp est bornée supérieurement par une fonction logarithmique du nombre de sommets du graphe. Nous expliquerons une application de ce résultat à un problème algorithmique issu de l'informatique théorique: MINCUT (trouver la coupure minimale dans un graphe). Pour terminer, nous présenterons un résultat obtenu récemment avec P.-N. Jolissaint, qui donne une borne inférieure sur la distorsion d'un graphe connexe fini, en terme du trou spectral, c'est-à-dire de la première valeur propre du laplacien combinatoire du graphe. On en déduit une preuve très simple du résultat de Linial-London-Rabinovich (1995) que les graphes expanseurs se plongent dans Lp avec distorsion logarithmique.|
|Résumé:||I will introduce some classical notions of geometric group theory (like growth and amenability) in the setting of associative algebras, and I will show how they interact with other classical invariants (like the Gelfand-Kirillov dimension and the lower transcendence degree). Hopefully the content of this talk will look like the embryonic stage of its title.|
|Résumé:||We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Drutu and Sapir. As a corollary we deduce that relatively hyperbolic groups have finite Assouad-Nagata dimension if and only if each of their parabolic subgroups does. We derive a similar conclusion about the existence of "good" embeddings of such groups into ℓp-spaces. We hope to also discuss corollaries for mapping class groups.|
An important feature of affine buildings (and Euclidean buildings in general) is the spherical building at infinity. For example if one wants to classify Euclidean buildings one applies the classification of spherical buildings to this building at infinity. If the Euclidean building is discrete and the building at infinity is Mo- ufang, this building at infinity determines the affine building because of al- gebraic reasons, most importantly the result of F. K. Schmidt that a field allows at most one complete discrete valuation.
In this talk I will present the following theorem.
Theorem. If X is a locally finite (possibly non-discrete) irreducible Euclidean building of rank at least two, then its building at infinity completely determines X.
The main ingredient of the proof is an analogous result for trees with a two-transitive action on the set of ends. In addition a construction of counterexamples in the locally infinite (discrete) case will be sketched.
|Résumé:||Relatively hyperbolic groups are a useful generalisation of hyperbolic groups, and they are used for example in the proof of the Virtual Haken Conjecture. I will discuss their boundaries, and especially how such boundaries change when modifying the peripheral structure and which maps between boundaries are induced by quasi-isometries. The common theme between these results is a new approach to Thurston's Dehn Filling Theorem.|
|Résumé:||Free Bogoliubov crossed products are a way to associate interesting von Neumann algebras with orthogonal representations of countable discrete groups. In this talk we will first explain basics on von Neumann algebras. Then we will speak about some recent results demonstrating the complexity of the classification of free Bogoliubov crossed products already in the case of representations of the integers. Finally, we also explain how certain structural properties of a free Bogoliubov crossed product are related to properties of the representation it was constructed from.|
|Résumé:||An action of a group on a set commensurates a subset A if the symmetric difference of A and gA is finite for all g. I'll indicate how the study commensurating actions are a useful tool to provide obstructions for groups to act on CAT(0) cube complexes.|
In my talk I would like to discuss the subtle interplay between the
various axioms of topological twin buildings. Although one is very far
away from a compact situation, one still very much tries to use
compactness arguments. Therefore, currently a key assumption for a
sufficiently deep theory of topological twin buildings is that panels be
compact; this might also explain why Kac-Moody groups over locally
compact fields currently are easier accessible from a topological point
of view than Kac-Moody groups over arbitrary topological fields.
Among the results whose proofs currently depend on the assumption that
panels be compact are:
*) A morphism between two topological twin buildings is continuous if and only if its restriction and corestriction to the union of all panels around a fixed chamber and its image are continuous.
*) The topology on the chamber set of a topological twin building is induced by the topologies on the vertex sets of that topological twin building and vice versa.