The free group on n generators modulo n+u random relations as n goes to infinity

TL;DR

This paper investigates the asymptotic behavior of free groups with n generators subjected to n+u random relations, revealing convergence to a specific non-abelian random group related to Cohen-Lenstra heuristics.

Contribution

It explicitly characterizes the limiting random group obtained from free groups with increasing generators and random relations, extending understanding of random group models.

Findings

Convergence of free groups to a specific non-abelian random group as n approaches infinity.

Connection established between these random groups and Cohen-Lenstra heuristics.

Identification of the model as part of the few relator Gromov random group class.

Abstract

We show that, as n goes to infinity, the free group on n generators, modulo n+u random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen-Lenstra heuristics. For each n, these random groups belong to the few relator model in the Gromov model of random groups.