### Reidemeister numbers

DOI: http://dx.doi.org/10.12775/TMNA.2003.009

#### Abstract

In [A. L. Fel’shtyn and R. Hill, < i> The Reidemeister zeta function with applications to

Nielsen theory and connection with Reidemeister torsion< /i> , K-Theory < b> 8< /b>

(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.

Nielsen theory and connection with Reidemeister torsion< /i> , K-Theory < b> 8< /b>

(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.

#### Keywords

Reidemeister number; twisted conjugacy classes; Gromov hyperbolic group; fundamental group

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