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Twisted conjugacy in spherical Artin groups and their crystallographic quotients

Primarily AI-generated textHuman understanding: no partsmath.GR — Group Theorymath.GT — Geometric Topologymath.RT — Representation Theory

Contributed by Jason Davies ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

A group has property R∞R_\infty if every automorphism has infinitely many twisted conjugacy classes. Let AA be an irreducible spherical Artin group of Coxeter rank at least two, with pure subgroup PP. We classify property R∞R_\infty for the crystallographic quotient Γ=A/[P,P]\Gamma=A/[P,P]. If AA has Coxeter rank at least three, then Γ\Gamma has R∞R_\infty. In type I2(m)I_2(m), this holds exactly when 4∣m4\mid m, or when mm is odd and 3∣m3\mid m. Together with the known dihedral Artin-group cases, the positive quotient results imply that every irreducible spherical Artin group of rank at least two has R∞R_\infty. The positive proofs combine the rational representation theory of the holonomy action with integral lattice constraints and explicit fixed-vector witnesses. In the negative dihedral cases, we construct automorphisms compatible with the crystallographic extension. For type I2(5)I_2(5), we exhibit an automorphism with Reidemeister number 1818, showing concretely that the quotient construction does not preserve R∞R_\infty. This example, including the identification with the presented Artin quotient and the exact Reidemeister count, is formalised in Lean. An alternative proof for seven Artin groups is formalised under explicitly stated external hypotheses.

Provenance statement

Claude Opus 5.5 (Anthropic), used through Claude Code, and GPT-6-Astra (OpenAI), used through Codex, developed the mathematical arguments, the exact Python certificates, the Lean formalisation and the manuscript, under the author's direction. Lean proves Proposition C, R(φ) = 18 for the presented quotient A(I₂(5))/[P,P], with no external hypotheses. It proves R∞ for the Artin groups of types D₅, E₆, E₇, E₈, F₄, H₃ and H₄ assuming four cited results (Appendix C). The only axioms are propext, Classical.choice and Quot.sound; there is no sorry and no native_decide. Theorem B is not formalised; its proof is supported by the exact computational checks in Appendix A. No human expert has independently reviewed the work.

Tools used

Anthropic
Claude CodeVersion Claude Opus 5.5
OpenAI
CodexVersion GPT-6-Astra
Lean
LeanVersion 4

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Version history

  1. v1 · 22 pagesSubmitted by Jason DaviesInitial depositCurrentNov 10, 2026