Volume 11, issue 4 (2007)

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Automorphisms of $2$–dimensional right-angled Artin groups

Ruth Charney, John Crisp and Karen Vogtmann

Geometry & Topology 11 (2007) 2227–2264
 arXiv: math/0610980v2
Abstract

We study the outer automorphism group of a right-angled Artin group ${A}_{\Gamma }$ in the case where the defining graph $\Gamma$ is connected and triangle-free. We give an algebraic description of $Out\left({A}_{\Gamma }\right)$ in terms of maximal join subgraphs in $\Gamma$ and prove that the Tits’ alternative holds for $Out\left({A}_{\Gamma }\right)$. We construct an analogue of outer space for $Out\left({A}_{\Gamma }\right)$ and prove that it is finite dimensional, contractible, and has a proper action of $Out\left({A}_{\Gamma }\right)$. We show that $Out\left({A}_{\Gamma }\right)$ has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound.

Keywords
right-angled Artin groups, outer automorphisms, outer space
Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 20F65, 20F28