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 $(q,6) \neq 1$.Comment: 28 page

Groups generated by 3-state automata over a 2-letter alphabet, I

An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite, abelian, and free groups are classified. In addition, we provide detailed information and complete proofs for several groups from the class, with the intention of showing the main methods and techniques used in the classification.Comment: 37 pages, 52 figure

The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $(\mathbb Z_2^2)\wr\mathbb Z$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring $\mathbb Z_2[[t]]$ of formal power series over $\mathbb Z_2$.Comment: 18 pages, 2 figure