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Reidemeister numbers for arithmetic Borel subgroups in type A

Paula Macedo Lins de Araujo, Yuri Santos Rego

TL;DR

This work develops a uniform framework for determining when upper triangular matrix groups over integral domains have the Reidemeister infinity property. Central to the approach is a decomposition of automorphisms of the unitriangular group $\mathbf{U}_n(R)$ (Levchuk’s theorem) into inner, diagonal, central, extremal, flip, and ring-induced components, allowing a reduction to a small, explicit automorphism subset. The authors introduce a key criterion: if for every ring automorphism $\alpha$ the additive Reidemeister number $R(\alpha_{\mathrm{add}})$ and the flip-induced $R(\tau_{\alpha})$ are infinite, then $\mathbf{B}_n(R)$ and $\mathbb{P}\mathbf{B}_n(R)$ have $R_\infty$ for all $n\ge4$, with wide-ranging applications to $S$-arithmetic groups over rings like $\mathbb{Z}[t]$, $\mathbb{Z}[t,t^{-1}]$, and rings of integers $\mathcal{O}_\mathbb{K}$. The paper also exhibits positive-characteristic counterexamples where $R_\infty$ fails, clarifying the limits of the criterion. By constructing explicit families of $S$-arithmetic groups $\Gamma_{n,p}$, the authors demonstrate both $R_\infty$ behavior and non-$R_\infty$ phenomena, highlighting the nuanced interaction between ring structure, arithmeticity, and twisted conjugacy. The results illuminate how Reidemeister numbers for soluble arithmetic groups diverge from their linear-algebraic counterparts and set the stage for further investigations into twisted conjugacy in other Lie types and arithmetic contexts.

Abstract

The Reidemeister number $R(\varphi)$ of a group automorphism $\varphi \in \mathrm{Aut}(G)$ encodes the number of orbits of the $\varphi$-twisted conjugation action of $G$ on itself, and the Reidemeister spectrum of $G$ is defined as the set of Reidemeister numbers of all of its automorphisms. We obtain a sufficient criterion for some groups of triangular matrices over integral domains to have property $R_\infty$, which means that their Reidemeister spectrum equals $\{\infty\}$. Using this criterion, we show that Reidemeister numbers for certain soluble $S$-arithmetic groups behave differently from their linear algebraic counterparts -- contrasting with results of Steinberg, Bhunia, and Bose.

Reidemeister numbers for arithmetic Borel subgroups in type A

TL;DR

This work develops a uniform framework for determining when upper triangular matrix groups over integral domains have the Reidemeister infinity property. Central to the approach is a decomposition of automorphisms of the unitriangular group (Levchuk’s theorem) into inner, diagonal, central, extremal, flip, and ring-induced components, allowing a reduction to a small, explicit automorphism subset. The authors introduce a key criterion: if for every ring automorphism the additive Reidemeister number and the flip-induced are infinite, then and have for all , with wide-ranging applications to -arithmetic groups over rings like , , and rings of integers . The paper also exhibits positive-characteristic counterexamples where fails, clarifying the limits of the criterion. By constructing explicit families of -arithmetic groups , the authors demonstrate both behavior and non- phenomena, highlighting the nuanced interaction between ring structure, arithmeticity, and twisted conjugacy. The results illuminate how Reidemeister numbers for soluble arithmetic groups diverge from their linear-algebraic counterparts and set the stage for further investigations into twisted conjugacy in other Lie types and arithmetic contexts.

Abstract

The Reidemeister number of a group automorphism encodes the number of orbits of the -twisted conjugation action of on itself, and the Reidemeister spectrum of is defined as the set of Reidemeister numbers of all of its automorphisms. We obtain a sufficient criterion for some groups of triangular matrices over integral domains to have property , which means that their Reidemeister spectrum equals . Using this criterion, we show that Reidemeister numbers for certain soluble -arithmetic groups behave differently from their linear algebraic counterparts -- contrasting with results of Steinberg, Bhunia, and Bose.
Paper Structure (20 sections, 16 theorems, 83 equations, 1 table)

This paper contains 20 sections, 16 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Auxiliary results, and structure of the groups
  3. The theorem of Levchuk
  4. Conjugation by elements of $\mathbf{U}_n(R)$ and $\mathbf{D}_n(R)$
  5. Central automophisms
  6. Extremal automorphisms
  7. Flip automorphism
  8. Group automorphisms induced by ring automorphisms
  9. Proof of Theorem \ref{['thm:Levchuk']}
  10. Restatement and proof of Theorem \ref{['thm:Additive']}
  11. A subset of automorphisms of $\mathbf{U}_n(R)$
  12. Concluding the proof of Theorem \ref{['thm:Additive']}
  13. Applications of Theorem \ref{['thm:Additive']}
  14. First case: polynomials in one variable R = R0[t]
  15. The case of Laurent polynomials R = R0[t,1/t]
  16. ...and 5 more sections

Key Result

Theorem 1.1

For each characteristic $p$ (zero or prime), there is a global field $\mathbb{K}_p$ of characteristic $\mathop{\mathrm{char}}\nolimits(\mathbb{K}_p)=p$ and $S$-arithmetic subgroups $\Gamma_{n,p} \leq \mathbb{P}\mathbf{B}_n(\mathbb{K}_p)$ that satisfy the following: If $p \geq 5$, the groups $\Gamma_{n,p}$ can be chosen to be finitely generated. (If $\mathrm{char}(\mathbb{K}_{p}) = p = 0$, the gro

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: See FelshtynTroitskyCrelle
  • Lemma 2.2: See Daciberg, WongCrelle, DacibergWongCrelle
  • Lemma 2.3: See Romankov, KarelDacibergAbelian
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1: Levchuk LevchukOriginal
  • Remark 3.2
  • ...and 14 more