Automorphisms of Higher Rank Lamplighter Groups

Abstract

Let ${\Gamma }_d(q)$ 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 $Aut({\Gamma }_d(q))$ and $Out({\Gamma }_d(q))$ 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 ${\Gamma }_d(q)$ 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 $(q,6)\ne 1$.

28 pages