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Trace Formulae, Zeta Functions, Congruences and Reidemeister Torsion in Nielsen Theory

Alexander Fel'shtyn·Richard Hill
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Abstract

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We continue to study analytical properties of the Nielsen zeta function. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced map on the unitary dual of the group. As a consequence we obtain a relation between a special value of the Reidemeister zeta function and a certain Reidemeister torsion. We also prove congruences for Reidemeister numbers of iterates of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group which are the same as those found by Dold for Lefschetz numbers.

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Alexander Fel'shtyn, Richard Hill. 1998-02-04. Trace Formulae, Zeta Functions, Congruences and Reidemeister Torsion in Nielsen Theory. https://arxiv.org/abs/math/9802017

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