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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). 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