Malwina Zietek | Searcharxiv

arXiv2022

Towards a dichotomy for the Reidemeister zeta function

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$ where $\mathbb Z_p$ the group of p-adic integers. As a consequence, we obtain a dichotomy for the Reidemeister zeta function of a continuous map of a topological space with fundamental group that is non-finitely generated torsion abelian group. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitely generated torsion-free nilpotent groups, based on a weak commutativity condition.

math.GR↗

arXiv2019

Dynamical zeta functions of Reidemeister type and representations spaces

In this paper we continue to study the Reidemeister zeta function. We prove Pólya -- Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta function for a large class of automorphisms of Abelian groups. We also study dynamical representation theory zeta functions counting numbers of fixed irreducible representations for iterations of an endomorphism. The rationality and functional equation for these zeta functions are proven for several classes of groups. We find a connection between these zeta functions and the Reidemeister torsions of the corresponding mapping tori. We also establish the connection between the Reidemeister zeta function and dynamical representation theory zeta functions under restriction of endomorphism to a subgroup and to a quotient group.

math.GR↗