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 \({{\,\textrm{dc}\,}}_S(G)\) ${\text{dc}}_S(G)$ , with respect to a given finite generating set S, results from considering the fractions of commuting pairs of elements in increasing balls around \(1_G\) $1_G$ in the Cayley graph . We focus on restricted wreath products of the form \(G = H \hspace{1.111pt}{\wr }\hspace{1.111pt}\langle \hspace{1.111pt}t \rangle \) $G=H\wr ⟨t⟩$ , where \(H \ne 1\) $H\ne 1$ is finitely generated and the top group \(\langle \hspace{1.111pt}t \rangle \) $⟨t⟩$ is infinite cyclic. In accordance with a more general conjecture, we show that \({{\,\textrm{dc}\,}}_S(G) = 0\) ${\text{dc}}_S(G)=0$ 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.