Geodesic
The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 229-240

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It is shown that the number of twisted conjugancy classes is infinite for any automorphism of a nonelementary, Gromov hyperbolic group. An analog of the Selberg theory for twisted conjugacy classes is suggested. 
@article{ZNSL_2001_279_a14,
     author = {A. L. Fel'shtyn},
     title = {The {Reidemeister} number of any automorphism of a {Gromov} hyperbolic group is infinite},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {229--240},
     publisher = {mathdoc},
     volume = {279},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a14/}
}
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A. L. Fel'shtyn. The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 229-240. http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a14/