Twisted conjugacy classes of Automorphisms of Baumslag–Solitar groups

Let $\phi:G\to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(\phi)$ the number of twisted $\phi$-conjugacy classes. Fel'shtyn and Hill [7] conjectured that if $\phi$ is injective, then $R(\phi)$ is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see [17] and [6]. It was showed in [12] that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for injective homomorphisms for the family of the Baumslag–Solitar groups $B(m,n)$ where $m\ne n$ and either $m$ or $n$ is greater than 1, and for automorphisms for the case $m=n>1$. family of the Baumslag–Solitar groups $B(m,n)$ where $m\ne n$.