Identifying measures on non-abelian groups and modules by their moments via reduction to a local problem

TL;DR

This paper demonstrates that certain probability measures on profinite groups and modules are uniquely determined by their moments, using a local reduction approach to handle complex group structures.

Contribution

It introduces a method to identify measures on non-abelian groups and modules via local reductions, extending previous heuristics and including recent models.

Findings

Measures are determined by moments for a broad class of profinite groups.

Reduction to simple group powers simplifies the problem.

Results apply to random modules over algebras.

Abstract

Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to know that these measures are determined by their moments, which in this context are the expected number of surjections to a fixed finite group. We show a wide class of measures, including those appearing in a recent paper of Liu, Wood, and Zurieck-Brown, have this property. The method is to work "locally" with groups that are extensions of a fixed group by a product of finite simple groups. This eventually reduces the problem to the case of powers of a fixed finite simple group, which can be handled by a simple explicit calculation. We can also prove a similar theorem for random modules over an algebra.