Asymptotic Approximations of Finitely Generated Groups

The concept of approximation is ubiquitous in mathematics. A classical idea is to approximate objects of interest by ones simpler to investigate, and which have the required characteristics in order to reflect properties and behavior of the elusive objects one started with. Looking for approximation in geometric group theory, first we adapt this fundamental approach. We discuss both its well-established appearance in residual properties of groups and its recent manifestation via metric approximations of groups such as sofic and hyperlinear approximations. We focus on approximations of Gromov hyperbolic groups, comment open problems, and suggest a conjecture in this setting. Then we turn over this classical way and initiate the study of approximations by groups usually known as being not so elementary to investigate. This allows to see that many interesting groups (still unknown to have algebraic or metric approximations) admit this new type of approximations which we call asymptotic approximations. We give many examples of asymptotically sofic/hyperlinear groups, as well as of asymptotically non-residually finite groups. In particular, we provide the first examples of infinite simple asymptotically residually finite (resp. asymptotically amenable) groups with Kazhdan’s property (T). The present text is a transcript of a talk the author gave since 2008 on several occasions, namely at the universities of Neuchâtel, Copenhagen, Aix-Marseille, at the ENS Lyon, ETH Zurich, MF Oberwolfach, and CRM Barcelona. The author is grateful to these institutions for their support and hospitality.