On the first cohomology of automorphism groups of graph groups

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 $Aut(A_\Gamma)$.