Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups

Abstract

Let $ϕ:G \to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(ϕ)$ the number of twisted $ϕ$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $ϕ$ is injective, then $R(ϕ)$ is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see \cite{ll} and \cite {fel:1}. It was showed in \cite {gw:2} that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for the Baumslag-Solitar groups $B(m,n)$, where either $|m|$ or $|n|$ is greater than 1 and $|m|\ne |n|$. We also show that in the cases where $|m|=|n|>1$ or $mn=-1$ the conjecture is true for automorphisms. In addition, we derive few results about the coincidence Reidemeister number.