Commuting probabilities of infinite groups

TL;DR

This paper investigates the probability that two elements commute in infinite groups under certain measures, linking high commuting probabilities to structural properties like the existence of abelian quotients.

Contribution

It extends known results by quantifying how high commuting probabilities imply specific algebraic structures in infinite groups.

Findings

High commuting probability implies existence of a normal subgroup with abelian quotient.

Results generalize finite group theorems to infinite groups with specific measures.

Provides conditions under which algebraic consequences follow from commuting probabilities.

Abstract

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.