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

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.

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

“…We then generalised these examples step by step until we arrived at the constructions given below. These computer programs are based on and contain parts of programs written by Benjamin Brück, which can be found on [5] and were used for [15].…”

“…This theorem is later stated in more detail as Theorem 3.1. The question for which graphs Λ we can find such a graph Γ is already mentioned in [15,Question 3.4]. In that paper, Richard D. Wade and Benjamin Brück study a different topic related to the outer automorphism group of rightangled Artin groups.…”