| Résumé: | We consider compact metric spaces equipped with a countable family of predicates and relations, so that the sets determined by these relations are closed in the product topology. Two such structures are equivalent when there is a map between them that is simultaneously a homeomorphism and an isomorphism for this additional structure. We show that one may use this notion to obtain upper bounds for the complexity of equivalence relations in the framework of Borel reducibility. As an example we show that, when considered up to topological group isomorphism, the families of locally-compact and Roelcke-precompact Polish groups have strictly simpler classification problems than the class of all Polish groups. This is joint work with Christian Rosendal. |
| Résumé: | In this talk we will present three seemingly different results about randomness in a finitely generated nilpotent group: an asymptotic shape theorem for First Passage Percolation (FPP); a generalization to random metrics of Pansu's theorem that the unique asymptotic cone of a nilpotent group is a particularly nice nilpotent Lie group; a Subadditive Ergodic Theorem for nilpotent groups. The results are all related, and the proof involves sub-Riemannian geometry and Ergodic Theory. Joint with Alex Furman. |
| Résumé: | Let X be a CAT(0) cube complex, and G an essential and non-elementary group of automorphisms of X. We choose a probability measure m whose support generates G and consider the associated random walk: we draw independently elements of G according to the law m and form their products. We are interested in the asymptotic behaviour this walk and its action on X. We prove in particular that almost surely the random walk converges to a point in the boundary of X. Furthermore, only a small part on the boundary, with strong geometric properties, can be obtained this way. This is a work in progress with Talia Fernós and Frédéric Mathéus. |
| Résumé: | Extensive amenability is a strong form of amenability of a group action, that has recently turned out to be a useful tool to prove amenability of > groups. I will discuss this property in general and show some stability results for extensively amenable actions. Then I will use these to establish amenability of some subgroups of the group IET of interval exchange transformations. Joint work with K. Juschenko, N. Monod and M. de la Salle. |
| Résumé: | I will explain the proof of a criterion that applies to certain groups that are known to have finitely generated homology up to degree d-1. In a certain setup if the group is residually p-finite the criterion shows that mod-p-homology in degree d is infinite. Kevin Wortman has proven infiniteness of mod-p-homology in degree d for S-arithmetic subgroups of almost simple groups over function fields. The criterion is an adaptation of his proof and also applies to cell stabilizers in affine and hyperbolic Kac-Moody groups over finite fields. The talk is based on joint work with Abramenko, Bux, Thomas, Wortman. |
| Résumé: | Branch groups act faithfully on infinite rooted trees in a particular way, in some sense imitating the full automorphism group of the tree. The class of branch groups contains many examples with remarkable properties such as infinite finitely generated torsion groups, groups of intermediate word growth, amenable but not elementary amenable groups, etc. In the talk I shall explain how the subgroup structure of a branch group is determined by its action on a tree and how this structure in turn determines all possible branch actions. I will also show some applications. |
| Résumé: | In this talk, we show every compactly generated t.d.l.c. group is exhausted by a finite series of closed normal subgroups so that the subquotients are either compact, discrete, or topologically characteristically simple. The notion of associated factors is then developed. Using the relation of association, we prove a Jordan-Hölder theorem for compactly generated t.d.l.c. groups and derive a number of new group invariants. (Joint with Colin Reid.) |