The degree of commutativity of wreath products with infinite cyclic top group

TL;DR

This paper investigates the probability that two elements commute in wreath products with an infinite cyclic top group, showing that this probability tends to zero regardless of the generating set, extending previous lamplighter group results.

Contribution

It proves that the degree of commutativity is zero for a broad class of wreath products with infinite cyclic top groups, generalizing prior lamplighter group findings.

Findings

Degree of commutativity is zero for these wreath products.

In large homomorphic images, the base group's image has density zero.

Extends Cox's results to more general wreath product structures.

Abstract

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups $G$: the degree of commutativity $\text{dc}_S(G)$, with respect to a given finite generating set $S$, results from considering the fractions of commuting pairs of elements in increasing balls around $1_G$ in the Cayley graph $\mathcal{C}(G,S)$. We focus on restricted wreath products the form $G = H \wr \langle t \rangle$, where $H \ne 1$ is finitely generated and the top group $\langle t \rangle$ is infinite cyclic. In accordance with a more general conjecture, we show that $\text{dc}_S(G) = 0$ for such groups $G$, regardless of the choice of $S$. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox's main auxiliary result: in `reasonably large' homomorphic images of wreath products $G$ as above, the image of the base group has density zero, with respect to certain types of generating sets.