Group-theoretic applications of non-commutative toric geometry

Summary

Introduction

These lectures consider the way some geometric techniques can be used to approach questions about ‘small’ groups, in particular, their finite present ability and the structure of their automorphism groups. By a ‘small’ group I mean one containing no non-Abelian free subgroups. Most of the time this will be in the context of discrete groups but towards the end I shall observe that one can glean related information using counting methods in the world of pro-p groups. The common theme is that a small group is in some sense at most half the size of a free group. The geometric and pro-p approaches give different ways of making this latter statement more precise. Both techniques involve looking only at the residually finite images of the group; there is no information to be had in this way about infinite simple groups for example. At the other extreme, finite groups also do not register on the scale.

In fact we shall be looking at various aspects of the representation theory of the group and consequently group rings come into play. However, rather than concentrating on group rings, I shall try and convince you that some non-commutative relatives of familiar commutative Noetherian rings are worth studying and that their representation theory has applications in group theory.