On the first cohomology of automorphism groups of graph groups

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 H = Im(Aut(A(Gamma)) -> Aut(H-1(A(Gamma)))) < GL(n, Z) not being virtually indicable, and also to 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)). (C) 2015 Elsevier Inc. All rights reserved.