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Profinite properties of Coxeter groups

Sam Hughes, Philip Möller, Olga Varghese

TL;DR

This work develops a comprehensive profinite-analytic program for Coxeter groups, establishing Serre goodness and proving that many structural and cohomological invariants are detectable from the profinite completion $\widehat{W}$. By combining profinite Bass–Serre theory with Coxeter combinatorics, the authors show that various splittings and decompositions are preserved under profinite completion and prove wide families of profinite rigidity or almost rigidity results (e.g., for Gromov-hyperbolic FC-type, extra large type, odd/complete Coxeter groups, and low virtual cohomological dimension cases). They also derive that invariants such as FA, the number of ends, Schur multipliers, and certain $\ell^2$-invariants are profinite data for good residually finite Coxeter groups, with an Atiyah-conjecture appendix heightening the connection between analysis and profinite structure. The results significantly advance understanding of which group-theoretic features of Coxeter groups are encoded in finite quotients, with implications for the isomorphism problem and classification within families of Coxeter groups. The appendix extends these themes to the Atiyah conjecture and $\ell^2$-invariants, showing robust profinite invariance in this setting and linking to virtual fibering questions in Coxeter groups.

Abstract

We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the profinite completion. As an application we prove a number of families Coxeter groups are profinitely rigid amongst Coxeter groups. We also prove that Gromov-hyperbolic FC type, large type, and odd Coxeter groups are almost profinitely rigid amongst Coxeter groups. In the appendix, Sam Fisher and Sam Hughes show that the Atiyah Conjecture holds for all Coxeter groups, and that $\ell^2$-Betti numbers and their positive characteristic analogues are profinite invariants of Coxeter groups and of virtually compact special groups.

Profinite properties of Coxeter groups

TL;DR

This work develops a comprehensive profinite-analytic program for Coxeter groups, establishing Serre goodness and proving that many structural and cohomological invariants are detectable from the profinite completion . By combining profinite Bass–Serre theory with Coxeter combinatorics, the authors show that various splittings and decompositions are preserved under profinite completion and prove wide families of profinite rigidity or almost rigidity results (e.g., for Gromov-hyperbolic FC-type, extra large type, odd/complete Coxeter groups, and low virtual cohomological dimension cases). They also derive that invariants such as FA, the number of ends, Schur multipliers, and certain -invariants are profinite data for good residually finite Coxeter groups, with an Atiyah-conjecture appendix heightening the connection between analysis and profinite structure. The results significantly advance understanding of which group-theoretic features of Coxeter groups are encoded in finite quotients, with implications for the isomorphism problem and classification within families of Coxeter groups. The appendix extends these themes to the Atiyah conjecture and -invariants, showing robust profinite invariance in this setting and linking to virtual fibering questions in Coxeter groups.

Abstract

We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the profinite completion. As an application we prove a number of families Coxeter groups are profinitely rigid amongst Coxeter groups. We also prove that Gromov-hyperbolic FC type, large type, and odd Coxeter groups are almost profinitely rigid amongst Coxeter groups. In the appendix, Sam Fisher and Sam Hughes show that the Atiyah Conjecture holds for all Coxeter groups, and that -Betti numbers and their positive characteristic analogues are profinite invariants of Coxeter groups and of virtually compact special groups.
Paper Structure (35 sections, 81 theorems, 63 equations, 7 figures, 3 tables)

This paper contains 35 sections, 81 theorems, 63 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on Coxeter groups
  4. Coxeter--Dynkin diagrams
  5. Parabolic subgroups
  6. The poset of conjugacy classes of finite subgroups
  7. Profinite completions
  8. Gromov-hyperbolic Coxeter groups
  9. Goodness and applications
  10. Coxeter groups are good
  11. Some easy applications
  12. Schur multipliers
  13. Conjugacy classes of finite subgroups
  14. Product decompositions
  15. Coxeter groups of low cohomological dimension
  16. ...and 20 more sections

Key Result

Lemma 1

Let ${\mathcal{C}}$ be a class of finitely presented residually finite groups. If $G$ in ${\mathcal{C}}$ has $|{\mathcal{G}}_{\mathcal{C}}(G)|=1$, then the isomorphism problem for $G$ is solvable in ${\mathcal{C}}$.

Figures (7)

  • Figure 1: $\Delta(p,q,r)$.
  • Figure 2: Coxeter--Dynkin diagram of type $\mathtt{X}_n$ where $\mathtt{X}_n$ has $n$ vertices. Note that $m\geq3$ and $\mathtt{A}_2=\mathtt{I}_2(3),$$\mathtt{B}_2=\mathtt{I}_2(4)$, and $\mathtt{G}_2=\mathtt{I}_2(6)$.
  • Figure 3: Two Coxeter diagrams giving non-isomorphic Coxeter groups with isomorphic posets of finite subgroups up to conjugacy.
  • Figure 4: Two triangles glued together along an edge.
  • Figure 5: Tiny house.
  • ...and 2 more figures

Theorems & Definitions (161)

  • Lemma
  • Theorem A
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 151 more