The degree of commutativity and lamplighter groups

TL;DR

This paper calculates the probability that two elements commute in certain infinite groups, specifically wreath products involving the integers and finite groups, extending previous finite group results.

Contribution

It provides explicit computations of the degree of commutativity for wreath products like old old old and  old old, broadening understanding of this measure in infinite groups.

Findings

Degree of commutativity for old old old groups computed.

Degree of commutativity for wreath products with finite groups determined.

Results extend previous finite group analyses to infinite wreath products.

Abstract

The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this for finite groups. In [AMV17], this was generalised to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form $\mathbb{Z}\wr \mathbb{Z}$ and $F\wr \mathbb{Z}$ where $F$ is any finite group.