The reidemeister number of any automorphism of a Baumslag-Solitar group is infinite

Let phi : G --> G be a group endomorphism where G is a finitely generated group of exponential growth, and let R(phi) denote the number of phi-conjugacy classes. Fel'shtyn and Hill [10] conjectured that if phi is injective, then R(phi) is infinite. In this paper, we show that the conjecture holds for the Baumslag-Solitar groups B(m,n), where either vertical bar m vertical bar or vertical bar n vertical bar is greater than 1 and vertical bar m vertical bar not equal vertical bar n vertical bar. We also show that in the cases where vertical bar m vertical bar = vertical bar n vertical bar > 1 or mn = -1 the conjecture is true for automorphisms. In addition, we derive few results about the coincidence Reidemeister number.