Homological stability is a remarkable phenomenon in the study of groups and spaces. For certain sequences G_n of groups, for example G_n=GL(n,Z), it states that the homology group H_i(G_n) does not depend on n for big enough n. There are many natural sequences G_n, from pure braid groups to congruence groups to Torelli groups, for which homological stability fails horribly. In these cases the rank of H_i(G_n) blows up to infinity, and in many (e.g. the latter two) cases almost nothing is known about H_i(G_n); indeed there may be no nice "closed form" for the answers. While doing some homology computations for the Torelli group, Tom Church and I found what looked to us like the shadow of a broad pattern. In order to explain it and formulate a specific conjecture, we came up with a notion of "stability of a sequence of representations of groups G_n". We began to realize that this notion can be used to make other predictions: from group representations to Malcev Lie algebras to the homology of congruence groups. Some of these predictions are known results, while others are not known. In this talk I will explain our broad conjectural picture via some of its many instances. No knowledge of either representation theory or group homology will be assumed.

We will discuss CAT(0) cube complexes and their connection to various topics in geometric group theory.

I will report on a recent joint work with Vincent Emery in which we determine the minimal covolume arithmetic subgroups of the group of isometries of the hyperbolic n-space for n odd and >3. Together with the previously known results it completes the solution of the minimal volume problem for arithmetic subgroups of hyperbolic isometries thus answering in part a question which was raised by C.L. Siegel in his 1945 paper.

It is proved that every automorphism of the Chevalley
group over local ring (with 1/3 for the root system G_{2}, with
1/2 for all other root systems and also without 1/2 for the systems
A_{l},D_{l},E_{l}, l>2) is standard, i.e. is a composition of ring,
inner, diagramm and central automorphisms.

In this talk I will discuss various classification problems for surface bundles, e.g. up to bundle isomorphism, homeomorphism, etc. Some of the main ingredients in the proofs will come from combinatorial/geometric group theory.

I am going to talk about a formal rewriting process which has different names: Makanin-Razborov process, Rips machine, Rauzy-Veech induction, Elimination process. Solution of different problems from group theory, logic, topology, ergodic theory and dynamical systems was independently reduced to the study of the properties of this process. These exciting connections can be further developed to obtain new results and formulate new problems.

Braid groups can be constructed from the symmetric groups in a number of different ways but there is only one construction (that I know of) that applies equally well to the real orthogonal groups. In this talk I will describe this elementary way of constructing new groups from old and discuss two sets of interrelated results. The first focuses on braid groups and some CAT(0) spaces on which they act (joint with Tom Brady). The second views the real orthogonal group as something like a continuous Coxeter group and constructs a "braided" version, i.e. something that can be viewed as the corresponding continuous Artin group. These usual groups have a word problem that is decidable (in an appropriate sense) and a free and vertex-transitive action on a finite-dimensional CAT(0) simplicial complex whose metric structure is a direct product of the real line and what appears to be a Euclidean building.

I will talk about a generalization of relative hyperbolicity based on the notion of a hyperbolically embedded subgroup. Examples of such subgroups include peripheral subgroups of relatively hyperbolic groups. They also appear in many other groups acting "nicely" on hyperbolic spaces (e.g. mapping class groups, outer automorphism groups of free groups, and groups acting on trees). A substantial part of the theory of relatively hyperbolic groups can be generalized in the new context. Applications to mapping class groups will be discussed.

I will discuss a theorem which states that every proper, cocompact, essential action on an irreducible CAT(0) cube complex contains a rank one isometry. This is joint work with Pierre-Emmanuel Caprace.

Z^n-free groups (or groups acting freely on Z^n-trees) form a big subclass of groups which are hyperbolic relative to their maximal abelian subgroups. The fact that all such groups possess free length functions with values in Z^n makes it possible to explore their properties using infinite words techniques. In my talk I am going to show how any Z^n-free group G can be embedded into a finite chain of HNN-extensions of special type so that the length function on G is preserved under the embedding.

A complete Kac-Moody group over a finite field is a totally disconnected, locally compact group, which may be thought of as an "infinite-dimensional Lie group". An example is G = SL(n,K) with K the field of formal Laurent series over a finite field. We study lattices in such G of rank 2, where the associated Bruhat-Tits building is a tree. We use the group action on the tree and finite group theory. This is joint work with Inna (Korchagina) Capdeboscq.

Let F be a finitely generated free group. We present an algorithm such that, given a subgroup H of F, decides whether H is the fixed subgroup of some family of automorphisms, or family of endomorphisms of F and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods. These results are in contrast with the fact that the same questions for a single automorphism, or endomorphism, remain open.