Abstract
Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].
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