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

TL;DR

This paper explores the duality properties of right-angled Artin groups (RAAGs), providing examples where the outer automorphism group is not a virtual duality group, thus addressing a question in geometric group theory.

Contribution

It demonstrates that some RAAGs have outer automorphism groups that are not virtual duality groups, expanding understanding of automorphism group structures in RAAGs.

Findings

Identifies RAAGs with non-virtual duality outer automorphism groups

Provides examples based on the Cohen--Macaulay property of the flag complex

Includes computer-assisted search for additional examples

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, Br\"uck describes how he used a computer-assisted search to find further examples.