In [A. L. Fel’shtyn and R. Hill, The Reidemeister zeta function with applications to Nielsen theory and connection with Reidemeister torsion, K-Theory 8 (1994), 367–393] we have conjectured that the Reidemeister number is infinite as long as an endomorphism of a discrete group is injective and the group has exponential growth. In the paper we prove this conjecture for any automorphism of a non-elementary, Gromov hyperbolic group. We also prove some generalisations of this result. The main results of the paper have topological counterparts.
"Reidemeister numbers." Topol. Methods Nonlinear Anal. 21 (1) 147 - 154, 2003.