Mixed moments and the joint distribution of random groups

TL;DR

This paper investigates the joint distribution of random abelian and non-abelian groups, extending known results by analyzing mixed moments to determine their distributions.

Contribution

It generalizes the distribution results for random $p$-adic matrices and random groups using mixed moments, unifying abelian and non-abelian cases.

Findings

Universality results for joint distributions of cokernels

Explicit computation of distributions for quotients of free profinite groups

Extension of previous work on distributions of random groups

Abstract

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random $p$-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random $p$-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups.