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

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.