Degree of commutativity of infinite groups

TL;DR

This paper investigates the proportion of commuting pairs in various infinite groups, establishing conditions under which this proportion is positive or zero, with implications for group structure and growth types.

Contribution

It characterizes when finitely generated residually finite groups of subexponential growth have a positive proportion of commuting pairs, linking this to being virtually abelian.

Findings

Positive proportion of commuting pairs iff the group is virtually abelian

In hyperbolic groups, the proportion of commuting pairs is always zero

Results apply to groups with polynomial growth, including virtually nilpotent groups

Abstract

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.