Title: Spread and infinite groups

Abstract: The notion of the spread of a finite group was introduced by Brenner and Wiegold in 1975. A group G has spread zero if there exists a g in G\{1} such that <g, h> \ne G for every h in G. As examples, we might first think of the cyclic groups. We then might consider the direct product of two cyclic groups. To finite group theorists, the finite simple groups would then be a natural testing ground. The class of finite groups with spread zero has now been completed* (in 2021) by Burness-Guralnick-Harper. A brief overview of the results for finite groups will motivate questions involving spread for 2-generated infinite groups. We will then answer one of these. The examples will all be concrete (and can be considered as subgroups of the ‘limit’ of the finite symmetric groups).

*They are described by an easy to state algebraic condition, but I will motivate this in the talk.