Automorphisms of Higher Rank Lamplighter Groups

Let ${\mathrm{\Gamma }}_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $D{L}_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut({\mathrm{\Gamma }}_d(q))$ and $Out({\mathrm{\Gamma }}_d(q))$ for $d\ge 2$, and apply our results to count twisted conjugacy classes in these groups when $d\ge 3$. Specifically, we show that when $d\ge 3$, the groups ${\mathrm{\Gamma }}_d(q)$ have property $R_{\mathrm{\infty }}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $\Gamma_2(q)=L_q = {\mathbb Z}q \wr {\mathbb Z}$haveproperty$R{\infty}$ifandonlyif$(q,6) \neq 1$.