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 $(q,6) eq 1$.
In this paper, we compute the {Sigma}^n(G) and {Omega}^n(G) invariants when 1 rightarrow H rightarrow G rightarrow K rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have t
he R_{infty} property in terms of {Omega}^n(H) and {Omega}^n(K) when either K is finite or the sequence splits. As an application, we construct a group F rtimes? Z_2 where F is the R. Thompsons group F and show that F rtimes Z_2 has the R_{infty} property while F is not characteristic.
We compute the {Omega}^1(G) invariant when 1 {to} H {to} G {to} K {to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy classes and to
finite generation of commutator subgroups are also discussed.
In this note, we compute the {Sigma}^1(G) invariant when 1 {to} H {to} G {to} K {to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R. Thompsons group F
and show that F semidirect Z_2 has the R-infinity property while F is not characteristic. Furthermore, we construct a finite extension G with finitely generated commutator subgroup G but has a finite index normal subgroup H with infinitely generated H.
