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

Alexander Fel'shtyn, Mateusz Slomiany|arXiv (Cornell University)|2026. 03. 09.

Analytic Number Theory Research인용 수 0

한 줄 요약

본 논문은 tame endomorphisms/maps에서 Reidemeister 수와 Nielsen 수의 성장률, 점근적 성질, 및 Gauss congruences를 분석하고 Nielsen coincidence zeta functions의 유리성을 입증한다.

ABSTRACT

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.

연구 동기 및 목표

• Investigate growth rate and asymptotic behavior of Reidemeister numbers {R(φ^n)} for tame endomorphisms of torsion-free nilpotent groups of finite Prüfer rank.
• Extend the study to Reidemeister coincidence numbers {R(φ^n,ψ^n)} for tame pairs of endomorphisms and establish Gauss congruences.
• Analyze Nielsen coincidence numbers {N(f^n,g^n)} and prove rationality of Nielsen coincidence zeta functions and related Gauss congruences.
• Show existence of growth rate for {N(f^n,g^n)} in the setting of tame map pairs on compact nilmanifolds.
• Relate growth rates to eigenvalues on abelian sections and to topological entropy in the dynamical setting.]
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제안 방법

• Provide a group-theoretic framework for twisted conjugacy and coincidences.
• Express R(φ^n) as a product over abelian factors via isolated lower central series and eigenvalues α_k,ξ_{k,i}.
• Use profinite completion and p-adic analysis to derive R(φ^n) formulas and growth rates.
• Derive Gauss congruences for {R(φ^n,ψ^n)} using p-adic and adelic techniques.
• Establish rationality of Nielsen coincidence zeta function and finite Gauss congruences for {N(f^n,g^n)} alongside growth-rate results.
• Relate growth rates to eigenvalues and to topological entropy of dual maps.

실험 결과

연구 질문

• RQ1 What is the growth rate R^∞(φ) of Reidemeister numbers for tame endomorphisms of torsion-free nilpotent groups of finite Prüfer rank?
• RQ2 How can R(φ^n) and R(φ^n,ψ^n) be expressed in terms of eigenvalues on abelian sections, and what are their asymptotics?
• RQ3 Do Gauss congruences hold for the sequences {R(φ^n,ψ^n)} and {R(φ^n)} in the stated setting?
• RQ4 Is the Nielsen coincidence zeta function rational, and do Gauss congruences hold for {N(f^n,g^n)}?
• RQ5 Does the growth rate for {N(f^n,g^n)} exist for tame maps on compact nilmanifolds?

주요 결과

• The growth rate exists for {R(φ^n)} and given by a product of maxima of eigenvalues over abelian factors.
• A closed formula expresses R(φ^n) via eigenvalues ξ_{k,i} of induced endomorphisms on abelian sections.
• Gauss congruences are established for the sequence {R(φ^n,ψ^n)} in the tame pair setting.
• The Nielsen coincidence zeta function is rational and Gauss congruences hold for {N(f^n,g^n)}.
• The growth rate for {N(f^n,g^n)} exists for tame map pairs on compact nilmanifolds.
• Connections are drawn between growth rates and topological entropy via unitary dual maps.