arXiv subjects

Alexander Fel'shtyn

Publications and source records attributed to Alexander Fel'shtyn.

At least 19 recordsLinked to original sources

arXiv2026

The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

In the present paper, taking a dynamical point on view, we study the growth rate and asymptotic behavior of the sequences of the Reidemeister numbers and the sequences of the Reidemeister and the Nielsen coincidence numbers. We also prove the Gauss congruences for the sequence $\{R(φ^n,ψ^n)\}$ of the Reidemeister coincidence numbers of the tame pair $(φ,ψ)$ of endomorphisms of a torsion-free nilpotent group~$G$ of finite Prüfer rank. Furthermore, we prove the rationality of the Nielsen coincidence zeta function, the Gauss congruences for the sequence $\{N(f^n, g^n)\}$ of the Nielsen coincidence numbers and show that the growth rate exists for the sequence \{$N(f^n, g^n)\}$ of tame pair of maps $(f,g)$ of a compact nilmanifold to itself.

math.DS↗

arXiv2026

Synchronization points: growth, asymptotics, congruences, and the synchronization zeta function

In this paper, we introduce the synchronization zeta function associated with a pair of self-maps of a topological space and investigate its properties. We also define the growth rate of synchronization points and derive an explicit formula in the setting of endomorphisms of compact, connected Abelian groups. In addition, we establish Gauss congruences and describe the asymptotic behavior for the sequence of numbers of synchronization points, under the assumption that the synchronization zeta function is rational. Further, we discuss connections with topological entropy and Reidemeister torsion.

math.DS↗

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↗

arXiv2022

Pólya-Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya-Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.

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↗

arXiv2018

New zeta functions of Reidemeister type and twisted Burnside-Frobenius theory

We introduce new zeta functions related to an endomorphism $ϕ$ of a discrete group $Γ$. They are of two types: counting numbers of fixed ($ρ\sim ρ\circϕ^n$) irreducible representations for iterations of $ϕ$ from an appropriate dual space of $Γ$ and counting Reidemeister numbers $R(ϕ^n)$ of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases it is proved that these zeta functions coincide. The Gauss congruences are proved. Useful asymptotic formulas for the zeta functions are found. Rationality is proved for some examples, which give also the first counterexamples simultaneously for TBFT ($R(ϕ)$=the number of fixed irreducible unitary representations) and TBFT$_f$ ($R(ϕ)$=the number of fixed irreducible unitary finite-dimensional representations) for an automorphism $ϕ$ with $R(ϕ)<\infty$.

math.GR↗

arXiv2017

Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ property.

math.GR↗

arXiv2016

Reidemeister theory of iterations of endomorphisms and poly-Bieberbach groups

We develop the Reidemeister theory of iterations of a group endomorphism $φ$ and study the asymptotic behavior of the sequence of the Reidemeister numbers of iterations $\{R(φ^k)\}$, the essential periodic $[φ]$-orbits and the heights of $φ$ on poly-Bieberbach groups.

math.GR↗

arXiv2015

The Nielsen numbers of iterations of maps on infra-solvmanifolds of type $R$ and periodic points

We study the asymptotic behavior of the sequence of the Nielsen numbers $\{N(f^k)\}$, the essential periodic orbits of $f$ and the homotopy minimal periods of $f$ by using the Nielsen theory of maps $f$ on infra-solvmanifolds of type $R$. We give a linear lower bound for the number of essential periodic orbits of such a map, which sharpens well-known results of Shub and Sullivan for periodic points and of Babenko and Bogatyi for periodic orbits. We also verify that a constant multiple of infinitely many prime numbers occur as homotopy minimal periods of such a map.

math.DS↗

arXiv2015

The $R_{\infty}$ and $S_{\infty}$ properties for linear algebraic groups

In the paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the $R_\infty$ and $S_\infty$ properties.

math.GR↗

arXiv2014

Growth rate for endomorphisms of finitely generated nilpotent groups and solvable groups

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.

math.GR↗

arXiv2014

The Nielsen and the Reidemeister numbers of maps on infra-solvmanifolds of type (R)

We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type $\R$. We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type $\R$, then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type $\R$ induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type $\R$ whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type $\R$.

math.GR↗

arXiv2012

Twisted conjugacy separable groups

We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this property and study its behavior under some extensions. An affirmative answer to the twisted Dehn conjugacy problem for polycyclic-by-finite group is obtained. Some problems for the further study are indicated.

math.GR↗

arXiv2012

Twisted conjugacy classes in residually finite groups

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of this automorphism. Also, we prove that any finitely generated residually finite non-amenable group has the R-infinity property (any automorphism has infinitely many twisted conjugacy classes). This gives a lot of new examples and covers many known classes of such groups.

math.GR↗

arXiv2012

The growth rate of Floer homology and symplectic zeta function

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence.This also establishes a connection between Floer homology and geometry of 3-manifolds.

math.SG↗

arXiv2011

Twisted conjugacy classes for polyfree groups

Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.

math.GR↗

arXiv2009

Reidemeister spectrum for metabelian groups of the form ${Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$, $p$ prime

In this note we study the Reidemeister spectrum for metabelian groups of the form ${\mathbb Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$. Particular attention is given to the $R_{\infty}$ property of a subfamily of these groups. We also define a Nielsen spectrum of a space and discuss some examples.

math.GR↗

arXiv2009

Geometry of Reidemeister classes and twisted Burnside theorem

This is a (mostly expository) paper on Reidemeister classes, twisted Burnside-Frobenius theory, congruences, R-infinity property and all that. It was written in 2005 and published in 2008. We post it as it was, only the bibliography data is updated. For some of the recent progress see e.g. arXiv:0903.4533, arXiv:0903.3455, arXiv:0802.2937, arXiv:0712.2601, arXiv:0704.3411, arXiv:math/0703744, arXiv:math/0606725, arXiv:math/0606764, arXiv:0805.1371 and references there.

math.GR↗