On the first cohomology of automorphism groups of graph groups

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• 作者：Javier Aramayona ; ^{aramayona@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Conchita Martí ; nez-Pé ; rez
• 关键词：Right-angled Artin group ; Automorphism ; Virtually indicable ; Property T
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：452
• 期：Complete
• 页码：17-41
• 全文大小：488 K

文摘

We study the (virtual) indicability of the automorphism group Aut(A_Γ)$\mathrm{Aut}(A_{\mathrm{\Gamma }})$ of the right-angled Artin group A_Γ$A_{\mathrm{\Gamma }}$ associated to a simplicial graph Γ. First, we identify two conditions – denoted (B1) and (B2) – on Γ which together imply that H¹(G,Z)=0$H^1(G,\mathbb{Z})=0$ for certain finite-index subgroups G<Aut(A_Γ)$G<\mathrm{Aut}(A_{\mathrm{\Gamma }})$. On the other hand we will show that (B2) is equivalent to the matrix group H=Im(Aut(A_Γ)→Aut(H₁(A_Γ)))<GL(n,Z)$\mathcal{H}=\mathrm{Im}(\mathrm{Aut}(A_{\mathrm{\Gamma }})\to \mathrm{Aut}(H_1(A_{\mathrm{\Gamma }})))<\mathrm{GL}(n,\mathbb{Z})$ not being virtually indicable, and also to H$\mathcal{H}$ having Kazhdan's property (T). As a consequence, Aut(A_Γ)$\mathrm{Aut}(A_{\mathrm{\Gamma }})$ virtually surjects onto Z$\mathbb{Z}$ whenever Γ does not satisfy (B2). In addition, we give an extra property of Γ ensuring that Aut(A_Γ)$\mathrm{Aut}(A_{\mathrm{\Gamma }})$ and Out(A_Γ)$\mathrm{Out}(A_{\mathrm{\Gamma }})$ virtually surject onto Z$\mathbb{Z}$. Finally, in the appendix we offer some remarks on the linearity problem for Aut(A_Γ)$\mathrm{Aut}(A_{\mathrm{\Gamma }})$.