Automorphisms of higher rank lamplighter groups
Let τd(q) denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph DLd(q), as described by Bartholdi, Neuhauser and Woess. We compute both Aut(τd(q)) and Out(τd(q)) for d ≥ 2, and apply our results to count twisted conjugacy classes in these groups when d ≥ 3. Specifically, we show that when d ≥ 3, the groups τd(q) have property R∞, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when d = 2 the lamplighter groups τ2(q) = Lq = Zq Z have property R∞ if and only if (q, 6)≠1.
Stein, Melanie; Taback, Jennifer; and Wong, Peter, "Automorphisms of higher rank lamplighter groups" (2015). Mathematics Faculty Publications. 47.