I am planning to briefly describe these two classes of groups, their most important properties, and spaces on which they act. I will try to explain how these different spaces fit together in a form of a dictionary relating the two theories. I will finish by listing some of the outstanding problems in the two subjects

I will discuss properties, techniques and examples related to hyperbolic-like groups.
For example, contracting geodesics, weakly proper discontinuous/acylindrical group actions.
Then I explain the construction of projections complexes and mention some of its applications

A theorem of Bestvina-Bromberg-Fujiwara asserts that the mapping class group of a hyperbolic surface of finite type has finite asymptotic dimension; its proof relies on an earlier result of Bell-Fujiwara stating that the curve complex has finite asymptotic dimension. The analogous statements are still open for Out(Fn). In joint work with Mladen Bestvina and Ric Wade, we give a first hint towards this, by obtaining a bound (linear in the rank n) on the topological dimension of the Gromov boundary of the graph of free factors of Fn (as well as some other hyperbolic Out(Fn)-graphs).

William Thurston's seminal construction of a hyperbolic 3-manifold fibering over the circle gave the first example of a Gromov hyperbolic surface-by-cyclic group. This breakthrough sparked a flurry of activity, and there has subsequently been much progress towards developing a general theory of hyperbolic group extensions. In this talk I will review some of this basic theory -- including combination theorems for ensuring a group extension is hyperbolic and structural theorems about general hyperbolic extensions -- and then discuss my work with Sam Taylor studying hyperbolicity in the specific context of free group extensions. For instance, we use the geometry of Outer space to show that every purely atoroidal subgroup of Out(F_n) that quasi-isometrically embeds into the free factor complex gives rise to a hyperbolic extension of F_n

This talk will be focused on the problem of: to what extent can the fundamental groups of compact 3-manifolds be distinguished by the finite quotients of their fundamental groups.

The talk will highlight examples (e.g. the figure eight knot complement) and introduce ideas and techniques used in attacking the problem

A surface group is the fundamental group of a closed surface of non-positive Euler characteristic.    A great deal of recent energy in geometric group theory has focussed on finding surface subgroups in various classes of groups of geometrical interest, especially word-hyperbolic groups. I will survey recent developments, highlights of which include Kahn—Markovic’s solution of the Surface Subgroup conjecture for Kleinian groups and Calegari—Walker’s discovery of surface subgroups in random groups

Homological stability is the classical phenomenon that for many natural families of moduli spaces the homology groups stabilize. Often the limit is the homology of another interesting space; for example, the homology of the braid groups converges to the homology of the space of self-maps of the Riemann sphere. Representation stability makes it possible to extend this to situations where classical homological stability simply does not hold, using ideas inspired by asymptotic representation theory. I will give a broad survey of homological stability and a gentle introduction to the tools and results of representation stability, focusing on its applications in topology.

Since Baumslag and Roseblade discovered that finitely presented subgroups of a product of two free groups must be virtually a product of free groups there has been an intensive research on properties of subdirect products of groups mainly focusing on the homological and homotopical properties, including the celebrated 1-2-3 Theorem by Bridson, Howier, Miller and Short. In this talk we will discuss how to extend some of these results to Lie algebras.

This is a joint work with Dessislava Kochloukova

Since the beginning of the 20th century, infinite torsion groups have been the source of numerous developments in group theory: Burnside groups Tarski monsters, Grigorchuck groups, etc. From a geometric point of view, one would like to understand on which metric spaces such groups may act in a non degenerated way (e.g. without a global fixed point).

In this talk we will focus on CAT(0) spaces and present two examples with rather curious properties. The first one is a non-amenable finitely generated torsion group acting properly on a CAT(0) cube complex. The second one is a non-abelian finitely generated Tarski-like monster : every finitely generated subgroup is either finite or has finite index. In addition this group is residually finite and does not have Kazdhan property (T).

(Joint work with Vincent Guirardel)