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

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.