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 $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(\Gamma_d(q))$ and $Out(\Gamma_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $\Gamma_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $\Gamma_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ have property $R_{\infty}$ if and only if $(q,6) \neq 1$.