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

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 $${{\,\textrm{dc}\,}}_S(G)$$ 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 "Equation missing". 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 ≀ ⟨ t ⟩ , where $$H \ne 1$$ H ≠ 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 $${{\,\textrm{dc}\,}}_S(G) = 0$$ 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.