## Topology & Geometric Group Theory Seminar

## Fall 2007

### 2:45 - 3:45, Malott Bache Auditorium

Tuesday, October 2

**Laurent
Bartholdi**, Lausanne Swiss Polytechnic Institute of
Mathematics

*Amenable groups and algebras*

I will define a natural notion of amenability for associative algebras
(first considered by Gromov), and explain its relation to amenability
of groups.

Let G be a finitely generated group. Call the *algebraic
growth* of G the function computing the rank of successive
quotients along the lower central series of G. This function is a
lower bound for the usual (word) growth function of G.

I will prove the following result: if G is amenable, then its
algebraic growth is subexponential. This answers a question by
Vershik, and is a partial converse to the classical statement that
groups of subexponential (word) growth are amenable.

As a corollary, all the groups constructed by Golod and Shafarevich
are non-amenable. This gives the first examples of non-amenable,
residually-finite torsion groups.

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