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Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

Manuel Wiedmer

Abstract

We prove that every right-angled Artin group occurs as a finite-index subgroup of the outer automorphism group of another right-angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^N$ for some $N$. For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day-Wade and Wade-Brück about when this group is a right-angled Artin group and when it has finite index.

Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

Abstract

We prove that every right-angled Artin group occurs as a finite-index subgroup of the outer automorphism group of another right-angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to for some . For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day-Wade and Wade-Brück about when this group is a right-angled Artin group and when it has finite index.
Paper Structure (10 sections, 11 theorems, 11 equations, 4 figures, 1 table)

This paper contains 10 sections, 11 theorems, 11 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Right-angled Artin groups
  5. Pure symmetric outer automorphisms
  6. Finite-index condition for PSO(RAAG)
  7. Condition for when PSO(RAAG) is a right-angled Artin group
  8. Right-angled Artin groups as finite-index subgroups of Out(RAAG)
  9. Conclusion and outlook
  10. Small graphs

Key Result

Theorem A

For any graph $\Lambda$, the right-angled Artin group $A_\Lambda$ is a finite-index subgroup of the outer automorphism group of some other right-angled Artin group $A_\Gamma$.

Figures (4)

  • Figure 3.1: Construction of $\Gamma$
  • Figure 3.2: $\Gamma - \mathop{\mathrm{st}}\nolimits(w)$ for some important vertices $w \in V(\Gamma)$
  • Figure 3.3: Construction of $\Gamma'$
  • Figure A.1: Proving Theorems \ref{['thm:anyraagisfinindsubgrofoutraag']} and \ref{['thm:anyraagisfinindsubgrofoutraagnographauto']} for the small graphs

Theorems & Definitions (23)

  • Theorem A
  • Definition A
  • Remark
  • Theorem A
  • Definition A
  • Definition A
  • Remark
  • Remark
  • Theorem A: DayWade
  • Remark
  • ...and 13 more