Geodesic
Twisted conjugacy classes of Automorphisms of Baumslag--Solitar groups
Algebra and discrete mathematics, no. 3 (2006), pp. 36-48.

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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 $m\ne n$.
Keywords: Reidemeister number, twisted conjugacy classes, Baumslag–Solitar groups. 
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     author = {Alexander Fel'shtyn and Daciberg L. Gon\c{c}alves},
     title = {Twisted conjugacy classes of {Automorphisms} of {Baumslag--Solitar} groups},
     journal = {Algebra and discrete mathematics},
     pages = {36--48},
     publisher = {mathdoc},
     number = {3},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2006_3_a3/}
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Alexander Fel'shtyn; Daciberg L. Gonçalves. Twisted conjugacy classes of Automorphisms of Baumslag--Solitar groups. Algebra and discrete mathematics, no. 3 (2006), pp. 36-48. http://geodesic.mathdoc.fr/item/ADM_2006_3_a3/