Automorphisms Of Higher Rank Lamplighter Groups

Let Gamma(d)(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(Gamma(d)(q)) and Out(Gamma(d)(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 Gamma(d)(q) have property R-infinity, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when d = 2 the lamplighter groups Gamma(2)(q) = Lq = Z(q) Z have property R-infinity if and only if (q, 6) not equal 1.