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 {\, 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 1G 1 G in the Cayley graph "Equation missing". We focus on restricted wreath products of the form G = H 1. 111pt 1. 111pt 1. 111ptt G = H ≀ ⟨ t ⟩, where H 1 H ≠ 1 is finitely generated and the top group 1. 111ptt ⟨ t ⟩ is infinite cyclic. In accordance with a more general conjecture, we show that {\, 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.