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Homogeneity in Coxeter groups and split crystallographic groups

Simon André, Gianluca Paolini

TL;DR

The paper analyzes model-theoretic homogeneity in Coxeter and crystallographic groups. It proves AE-homogeneity for irreducible affine Coxeter groups and, more broadly, for many torsion-generated hyperbolic groups, while showing that split crystallographic groups can be non-homogeneous and that some hyperbolic Coxeter groups fail $ ext{EAE}$-homogeneity. A key theme is the interplay between algebraic rigidity (via Stallings/JSJ theory) and profinite rigidity, leading to results on profinite and AE-type rigidity for abelian-by-finite groups and affine Coxeter groups. The work also develops criteria under which homogeneity is preserved in direct products and provides explicit counterexamples to delineate the limits of these phenomena. Overall, the paper deepens understanding of how first-order types, automorphism orbits, and profinite data interact in structured groups like Coxeter and crystallographic groups, with implications for both model theory and group theory.

Abstract

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not $\mathrm{EAE}$-homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. We also prove that finitely generated abelian-by-finite groups are homogeneous if and only if they are profinitely homogeneous, i.e., any tuple of words from the group is profinitely rigid. We use this to deduce that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context.

Homogeneity in Coxeter groups and split crystallographic groups

TL;DR

The paper analyzes model-theoretic homogeneity in Coxeter and crystallographic groups. It proves AE-homogeneity for irreducible affine Coxeter groups and, more broadly, for many torsion-generated hyperbolic groups, while showing that split crystallographic groups can be non-homogeneous and that some hyperbolic Coxeter groups fail -homogeneity. A key theme is the interplay between algebraic rigidity (via Stallings/JSJ theory) and profinite rigidity, leading to results on profinite and AE-type rigidity for abelian-by-finite groups and affine Coxeter groups. The work also develops criteria under which homogeneity is preserved in direct products and provides explicit counterexamples to delineate the limits of these phenomena. Overall, the paper deepens understanding of how first-order types, automorphism orbits, and profinite data interact in structured groups like Coxeter and crystallographic groups, with implications for both model theory and group theory.

Abstract

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not -homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. We also prove that finitely generated abelian-by-finite groups are homogeneous if and only if they are profinitely homogeneous, i.e., any tuple of words from the group is profinitely rigid. We use this to deduce that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context.

Paper Structure

This paper contains 36 sections, 58 theorems, 25 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation
  4. Homogeneity and rigidity in split crystallographic groups
  5. Preliminaries on crystallographic groups
  6. Type-determinacy and almost-homogeneity in crystallographic groups
  7. Non-homogeneous crystallographic groups
  8. First counterexample (proof of Theorem \ref{['counterexample_theorem2']})
  9. Second counterexample
  10. Homogeneity and profinite homogeneity in finitely generated abelian-by-finite groups
  11. Homogeneity in affine Coxeter groups
  12. Preliminary results
  13. Reflection-preserving morphisms
  14. Maximal parabolic finite subgroups
  15. Extending automorphisms of maximal finite subgroups
  16. ...and 21 more sections

Key Result

Theorem 1.2

Affine Coxeter groups, hyperbolic even Coxeter groups and hyperbolic one-ended Coxeter groups are homogeneous (in fact, $\mathrm{AE}$-homogeneous).

Figures (14)

  • Figure 1: The irreducible affine Coxeter groups and their Coxeter graphs. The grey boxes show the corresponding finite Coxeter groups. The groups $\tilde{A}_n,\tilde{B}_n,\tilde{C}_n$ on the left-hand side are defined for $n\geqslant 2$ and the group $\tilde{D}_n$ is defined for $n\geqslant 4$.
  • Figure 2: Coxeter group $D_n$, with $n\geqslant 4$.
  • Figure 3: Affine Coxeter group $\tilde{B}_n$, with $n\geqslant 3$ (when $n=3$, the edge connecting $s_3$ and $s_4$ is labelled with $4$).
  • Figure 4: Affine Coxeter group $\tilde{D}_n$, with $n\geqslant 4$.
  • Figure 5: The affine Coxeter group $\tilde{C}_2$.
  • ...and 9 more figures

Theorems & Definitions (132)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 122 more