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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