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Divisible cube complexes and finite-order automorphisms of RAAGs

Elia Fioravanti

Abstract

We give a geometric characterisation of those groups that arise as fixed subgroups of finite-order untwisted automorphisms of right-angled Artin groups (RAAGs). They are precisely the fundamental groups of a class of compact special cube complexes that we term "divisible". The main corollary is that surface groups arise as fixed subgroups of finite-order automorphisms of RAAGs, as do all commutator subgroups of right-angled Coxeter groups. These appear to be the first examples of such fixed subgroups that are not themselves isomorphic to RAAGs. Using a variation of canonical completions, we also observe that every special group arises as the fixed subgroup of an automorphism of a finite-index subgroup of a RAAG.

Divisible cube complexes and finite-order automorphisms of RAAGs

Abstract

We give a geometric characterisation of those groups that arise as fixed subgroups of finite-order untwisted automorphisms of right-angled Artin groups (RAAGs). They are precisely the fundamental groups of a class of compact special cube complexes that we term "divisible". The main corollary is that surface groups arise as fixed subgroups of finite-order automorphisms of RAAGs, as do all commutator subgroups of right-angled Coxeter groups. These appear to be the first examples of such fixed subgroups that are not themselves isomorphic to RAAGs. Using a variation of canonical completions, we also observe that every special group arises as the fixed subgroup of an automorphism of a finite-index subgroup of a RAAG.
Paper Structure (18 sections, 18 theorems, 4 equations, 2 figures)

This paper contains 18 sections, 18 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cube complexes
  4. Special cube complexes
  5. Fixed subgroups
  6. Automorphisms of RAAGs
  7. Fixed subgroups of finite-order untwisted automorphisms
  8. Divisible cube complexes
  9. The host
  10. The finite ${\rm CAT(0)}$ cube complex $\mathbb{E}$
  11. The 1--skeleton of the host
  12. Completing the host: higher-dimensional cubes
  13. The extended host and its automorphisms
  14. The converse
  15. Automorphisms of finite-index subgroups of RAAGs
  16. ...and 3 more sections

Key Result

Theorem 1

The following properties are equivalent for a group $H$:

Figures (2)

  • Figure 1: A graph $\Lambda$ such that the RAAG $A_{\Lambda}$ admits an involution $\varphi$ with $\mathop{\mathrm{Fix}}\nolimits(\varphi)$ isomorphic to the genus--$2$ surface group.
  • Figure 3: Assorted counterexamples.

Theorems & Definitions (45)

  • Theorem 1
  • Corollary 2
  • Proposition 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 3.1: Strongly divisible
  • ...and 35 more