Volume 21, issue 6 (2017)

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Quasi-isometric classification of right-angled Artin groups, I: The finite out case

Jingyin Huang

Geometry & Topology 21 (2017) 3467–3537
Abstract

Let $G$ and ${G}^{\prime }$ be two right-angled Artin groups. We show they are quasi-isometric if and only if they are isomorphic, under the assumption that the outer automorphism groups $Out\left(G\right)$ and $Out\left({G}^{\prime }\right)$ are finite. If we only assume $Out\left(G\right)$ is finite, then ${G}^{\prime }$ is quasi-isometric to $G$ if and only if ${G}^{\prime }$ is isomorphic to a subgroup of finite index in $G$. In this case, we give an algorithm to determine whether $G$ and ${G}^{\prime }$ are quasi-isometric by looking at their defining graphs.

Keywords
quasi-isometric classification, right-angled Artin groups, extension complexes, generalized star extension
Mathematical Subject Classification 2010
Primary: 20F65, 20F67, 20F69