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A note on virtual duality and automorphism groups of right-angled Artin groups

Richard D. Wade, Benjamin Brück

TL;DR

The paper demonstrates that not all outer automorphism groups of right-angled Artin groups are virtual duality groups, by leveraging the Brady–Meier criterion that links duality to Cohen–Macaulay flag complexes and a Join Lemma construction that yields finite-index RAAG subgroups inside Out$(A_ abla)$. It provides a concrete graph $ abla$ whose associated join $ abla_1 abla_2$ yields a RAAG $A_{ abla}$ whose flag complex fails to be Cohen–Macaulay, thereby obstructing duality for $A_{ abla}$ and, consequently, for ${ m Out}(A_ abla)$ as a virtual duality group. The Aut case is treated similarly, and the discussion extends to obstructions via Fouxe-Rabinovitch groups and commensurability considerations, including computer-assisted constructions by Brück. The work highlights that duality properties of RAAG automorphism groups are delicate and intimately tied to combinatorial topology of flag complexes, with implications for potential bordifications and broader questions in geometric group theory.

Abstract

A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen--Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

A note on virtual duality and automorphism groups of right-angled Artin groups

TL;DR

The paper demonstrates that not all outer automorphism groups of right-angled Artin groups are virtual duality groups, by leveraging the Brady–Meier criterion that links duality to Cohen–Macaulay flag complexes and a Join Lemma construction that yields finite-index RAAG subgroups inside Out. It provides a concrete graph whose associated join yields a RAAG whose flag complex fails to be Cohen–Macaulay, thereby obstructing duality for and, consequently, for as a virtual duality group. The Aut case is treated similarly, and the discussion extends to obstructions via Fouxe-Rabinovitch groups and commensurability considerations, including computer-assisted constructions by Brück. The work highlights that duality properties of RAAG automorphism groups are delicate and intimately tied to combinatorial topology of flag complexes, with implications for potential bordifications and broader questions in geometric group theory.

Abstract

A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen--Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

Paper Structure

This paper contains 10 sections, 9 theorems, 7 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proof of Theorem A
  3. Right-angled Artin groups
  4. Flag complexes, the Cohen--Macaulay condition, and duality for RAAGs.
  5. Finiteness conditions for ${\rm{Out}}(A_\Gamma)$ and the Join Lemma
  6. The proof of Theorem A and the Aut case
  7. Further discussion
  8. Obstructions to duality: Fouxe-Rabinovitch groups
  9. Commensurability problems
  10. Computer-assisted construction of further examples, by Benjamin Brück

Key Result

Theorem 1

Let $\Gamma$ be the graph given in Figure fig. Then ${\rm{Out}}(A_\Gamma)$ is not a virtual duality group.

Figures (3)

  • Figure 1: A graph $\Gamma=\Gamma_1 \sqcup \Gamma_2$ such that ${\rm{Out}}(A_\Gamma)$ is not a virtual duality group. The grey triangles are added to show the flag complex $\hat{\Gamma}$ determined by $\Gamma$.
  • Figure 2: Three points in the spine of the relative outer space for $A_\Gamma=\mathbb{Z}^2 \ast \mathbb{Z}^3 \ast \mathbb{Z}^4$, whose ${\rm{Out}}(A_\Gamma)$-stabilizers are isomorphic to $\mathbb{Z}^2$, $\mathbb{Z}^3$, and $\mathbb{Z}^4$, respectively.
  • Figure 3: Two graphs $\Gamma_i$ with 9 vertices such that ${\rm{Out}}(A_{\Gamma_i})$ is not a virtual duality group. The top row shows the defining graphs $\Gamma_i$, the bottom row shows graphs $\Theta_i$ such that ${\rm{PSO}}(A_{\Gamma_i})\cong A_{\Theta_i}$.

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 1.1: Join Lemma
  • Definition 2.1: Cohen--Macaulay complexes
  • Theorem 2.2: BM, Theorem C
  • Lemma 2.3
  • proof
  • Proposition 2.4: CF, Section 6
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 8 more