Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups

Abstract

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 \(m\ne n\).