Charles Cox

Generation properties for finite and infinite groups

In the study of groups - both finite and infinite - finding a ‘nice’ generating set is often helpful in developing our intuition and understanding of a group or family of groups. There are two notions, both from the world of finite groups, that I will discuss: invariable generation and spread. Invariable generation was considered by Jordan in 1872, but has been of recent importance in computational group theory, especially so for Galois groups. All finite groups are invariably generated, but a cornerstone of infinite group theory - the non-abelian free groups - are not. The recent work of Burness, Guralnick, and Harper beautifully resolved key questions for spread for finite groups, but we will see that this topic is wide open for infinite groups.

Our key example throughout the talk will be a group (generated by 2 elements) that can be thought of as an infinite generalisation of the finite symmetric groups. Any proofs will be short and non-technical (owing to the way these groups ‘feel’ a lot like symmetric groups). I will also aim to overview the current state of the literature for infinite groups for each of the above notions.