On virtual indicability and property (T) for outer automorphism groups of RAAGs

TL;DR

This paper establishes conditions on the defining graph of a right-angled Artin group that determine when its automorphism and outer automorphism groups are virtually indicable or have property (T), respectively.

Contribution

It provides new criteria linking graph properties to the (virtual) indicability and property (T) of automorphism groups of RAAGs.

Findings

Identifies graph conditions implying automorphism groups are virtually indicable.

Provides a criterion for outer automorphism groups to have property (T).

Connects graph-theoretic properties with algebraic properties of automorphism groups.

Abstract

We give a condition on the defining graph of a right-angled Artin group which implies its automorphism group is virtually indicable, that is, it has a finite-index subgroup that admits a homomorphism onto $\Z$. We use this as part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.