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Rotation groups, mediangle graphs, and periagroups: a unified point of view on Coxeter groups and graph products of groups

Anthony Genevois

Abstract

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we propose a combinatorial geometry (mediangle graphs) to study them. As an application, we give natural and unified proofs for several results that hold for both Coxeter groups and graph products of groups.

Rotation groups, mediangle graphs, and periagroups: a unified point of view on Coxeter groups and graph products of groups

Abstract

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we propose a combinatorial geometry (mediangle graphs) to study them. As an application, we give natural and unified proofs for several results that hold for both Coxeter groups and graph products of groups.
Paper Structure (39 sections, 52 theorems, 38 equations)

This paper contains 39 sections, 52 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Rotation groups.
  3. Periagroups.
  4. Mediangle graphs.
  5. Discussion about Theorem \ref{['thm:BigIntro']}.
  6. Applications.
  7. Organisation of the article.
  8. Acknowledgements.
  9. Basis definitions about graphs
  10. Mediangle graphs
  11. Convexity
  12. Long convex cycles
  13. Hyperplanes
  14. Projections
  15. Angles between hyperplanes
  16. ...and 24 more sections

Key Result

Theorem 1.1

If $(G \curvearrowright X, \mathcal{R})$ is a rotation system and $o \in X$ a basepoint, then $G$ is a periagroup with $\mathcal{R}_o$ as a basis and $X$ is a mediangle graph isomorphic to $\mathrm{Cayl}(G,\mathcal{R}_0)$. Conversely, if $G$ is a periagroup with $\mathcal{B}$ as a basis, then $Y:= \

Theorems & Definitions (116)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 106 more