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arxiv: 1906.09959 · v1 · pith:A3D3PE43new · submitted 2019-06-21 · 🧮 math.GR · math.DS· math.RT

Dynamical zeta functions of Reidemeister type and representations spaces

Alexander Fel'shtyn , Malwina Zietek This is my paper

Pith reviewed 2026-05-25 18:33 UTC · model grok-4.3

classification 🧮 math.GR math.DSmath.RT
keywords Reidemeister zeta functiondynamical zeta functionsAbelian groupsautomorphismsrepresentation spacesmapping toriPólya-Carlson dichotomyrationality
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The pith

The Reidemeister zeta function for a large class of automorphisms of Abelian groups is either rational or has the unit circle as a natural boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that the Reidemeister zeta function satisfies a Pólya-Carlson dichotomy for a large class of automorphisms of Abelian groups: it is either rational or has the unit circle as a natural boundary. It also establishes rationality and functional equations for dynamical representation theory zeta functions counting fixed irreducible representations under several classes of groups. Connections are made between these zeta functions and the Reidemeister torsions of mapping tori, plus behavior under restriction of the endomorphism to subgroups and quotients. A sympathetic reader would care because the results classify the analytic behavior of counting functions that arise in algebraic dynamics and tie them directly to topological invariants.

Core claim

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. 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.

What carries the argument

The Reidemeister zeta function, which generates the count of fixed points of endomorphism iterates up to conjugation, together with its link to dynamical representation theory zeta functions on representation spaces.

If this is right

  • The zeta functions are rational and satisfy functional equations inside the stated classes of groups.
  • The zeta functions connect directly to Reidemeister torsions of the associated mapping tori.
  • The connection between Reidemeister and representation zeta functions persists under restriction to subgroups and quotients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dichotomy and rationality results may extend to additional classes of groups or endomorphisms beyond those treated here.
  • These zeta functions could provide new ways to compute or relate torsions in concrete mapping tori examples.
  • The representation-space approach might yield similar analytic classifications for dynamical systems on non-Abelian groups under suitable restrictions.

Load-bearing premise

The endomorphism or automorphism must belong to the large class of automorphisms of Abelian groups or one of the several unspecified classes of groups for the rationality, dichotomy, and functional equation statements to hold.

What would settle it

An explicit automorphism of an Abelian group inside the considered class for which the Reidemeister zeta function is neither rational nor has the unit circle as a natural boundary would disprove the claimed dichotomy.

read the original abstract

In this paper we continue to study the Reidemeister zeta function. We prove P\'olya -- 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript continues the study of Reidemeister zeta functions associated to endomorphisms and automorphisms of groups. It proves a Pólya-Carlson dichotomy (rationality versus natural boundary) for the Reidemeister zeta function for a large class of automorphisms of Abelian groups. Rationality and functional equations are established for dynamical representation theory zeta functions (counting fixed irreducible representations under iteration) for several classes of groups. Connections are derived between these zeta functions and Reidemeister torsions of the corresponding mapping tori, together with relations between the two families of zeta functions under restriction of an endomorphism to a subgroup or passage to a quotient group.

Significance. If the stated theorems hold, the work supplies explicit analytic criteria and functional equations for a family of dynamical zeta functions in group theory, together with direct links to topological invariants (Reidemeister torsion) and to representation-theoretic counting functions. The restriction/quotient compatibility relations unify constructions that had previously been treated separately. These results extend the existing literature on rationality of Reidemeister zeta functions and provide concrete tools for analyzing mapping tori and dynamical systems on groups.

minor comments (2)
  1. The introduction would benefit from a brief, explicit characterization (even if only by reference to a later theorem) of the 'large class of automorphisms of Abelian groups' to which the dichotomy applies, rather than leaving the scope entirely to the statement of the main theorem.
  2. Notation for the dynamical representation theory zeta function is introduced without an immediate low-dimensional example; adding one (e.g., for a cyclic group) would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper states and proves the Pólya-Carlson dichotomy, rationality, and functional equations directly for explicitly scoped classes of automorphisms of abelian groups and several listed classes of groups. The mapping-torus torsion connection and the restriction/quotient relations are obtained from the same explicit constructions and definitions supplied in the text. No equation or central claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; prior work is referenced only as background while the new results rest on independent arguments within the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the results rest on standard tools from complex analysis for zeta functions and basic facts from group theory and topology; no free parameters, ad-hoc axioms, or new entities are indicated.

axioms (2)
  • standard math Standard properties of analytic continuation and rationality criteria for zeta functions in complex analysis
    Invoked to establish the Pólya-Carlson dichotomy and functional equations.
  • standard math Basic facts about Reidemeister torsion and mapping tori in algebraic topology
    Used to establish the stated connections between zeta functions and torsions.

pith-pipeline@v0.9.0 · 5652 in / 1384 out tokens · 27929 ms · 2026-05-25T18:33:47.110165+00:00 · methodology

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Reference graph

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