Abstract

Burnside groups are groups where every element has bounded order. A major theme in group theory over the last hundred years is the challenge of determining when/which finitely generated Burnside groups can be infinite. In another direction, “random groups” are usually defined by taking quotients of a free group by a normal subgroup generated by suitably chosen random elements. Depending on exactly how one chooses the number and length of relations, one typically gets hyperbolic groups. These groups are infinite as long as not too many relations are chosen, and exhibit other interesting behaviour.

One could equally well consider what happens if one takes random quotients of other free objects, such as free Burnside groups. I will discuss recent joint work with Dominik Gruber where we find a reasonable model for random (infinite) Burnside groups, building on earlier tools developed by Coulon and Coulon–Gruber.