Abstract: Let G be a finitely generated group and let G>G_1>G_2 ... be a sequence of finite index normal subgroups of G with trivial intersection.

We expect that the asymptotic behaviour of various group theoretic invariants of the groups G_i should relate to algebraic, topological or measure theoretic properties of G.

A classic example of this is the Luck approximation theorem which says that the growth of the ordinary Betti numbers of sequence G_i is given by the L^2-Betti number of (the classifying space) of G.

Another example is a theorem of Abert and myself that the growth of the ranks (=minimal number of generators) of $G_i$ is given by the cost of the action of G in the profinite completion of G with respect to the chain (G_i). Work of Lackenby relates growth of ranks of the sequence (G_i) with virtual splittings of G as an amalgam or an HNN extension. Other very interesting invariants to study are the growth of deficiency, and growth of torsion of the integral homology of the groups $G_i$. In this survey talk I will focus on the highlights of whats is known so far and on the many exciting open questions in this subject.

We expect that the asymptotic behaviour of various group theoretic invariants of the groups G_i should relate to algebraic, topological or measure theoretic properties of G.

A classic example of this is the Luck approximation theorem which says that the growth of the ordinary Betti numbers of sequence G_i is given by the L^2-Betti number of (the classifying space) of G.

Another example is a theorem of Abert and myself that the growth of the ranks (=minimal number of generators) of $G_i$ is given by the cost of the action of G in the profinite completion of G with respect to the chain (G_i). Work of Lackenby relates growth of ranks of the sequence (G_i) with virtual splittings of G as an amalgam or an HNN extension. Other very interesting invariants to study are the growth of deficiency, and growth of torsion of the integral homology of the groups $G_i$. In this survey talk I will focus on the highlights of whats is known so far and on the many exciting open questions in this subject.

## Date:

Mon, 11/09/2017 - 11:00 to 12:00

## Location:

Feldman building, Room 128