We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. We will discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity, as well as theorems about the distribution of non-abelian analogs of class groups of function fields that motivate this work.
Speaker(s): Melanie Matchett Wood (University of Wisconsin-Madison)