On the first cohomology of automorphism groups of graph groups

TL;DR

This paper investigates the first cohomology of automorphism groups of right-angled Artin groups, identifying conditions on the defining graph that determine when these groups virtually surject onto the integers.

Contribution

It introduces conditions (B1) and (B2) on the graph to characterize when automorphism groups are virtually indicable or have property (T), linking graph properties to group cohomology and linearity.

Findings

Conditions (B1) and (B2) determine the (virtual) indicability of automorphism groups.

Condition (B2) is equivalent to the associated matrix group having Kazhdan's property (T).

Automorphism groups virtually surject onto Z under certain graph conditions.

Abstract

We study the (virtual) indicability of the automorphism group $Aut(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$ associated to a simplicial graph $\Gamma$. First, we identify two conditions -- denoted (B1) and (B2) -- on $\Gamma$ which together imply that $H^1(G, Z)=0$ for certain finite-index subgroups $G<Aut(A_\Gamma)$. On the other hand we will show that (B2) is equivalent to the matrix group ${\mathcal H} = {\rm Im}(Aut(A_\Gamma) \to Aut(H_1(A_\Gamma))) <GL(n,Z)$ not being virtually indicable, and also to $\mathcal H$ having Kazhdan's property (T). As a consequence, $Aut(A_\Gamma)$ virtually surjects onto $Z$ whenever $\Gamma$ does not satisfy (B2). In addition, we give an extra property of $\Gamma$ ensuring that $Aut(A_\Gamma)$ and $Out(A_\Gamma)$ virtually surject onto $Z$. Finally, in the appendix we offer some remarks on the linearity problem for $Aut(A_\Gamma)$.