Outer space for untwisted automorphisms of right-angled Artin groups

Abstract

For a right-angled Artin group $A_{\Gamma }$, the untwisted outer automorphism group $U\left(A_{\Gamma }\right)$ is the subgroup of $Out\left(A_{\Gamma }\right)$ generated by all of the Laurence–Servatius generators except twists (where a twist is an automorphism of the form $v\mapsto vw$ with $vw=wv$). We define a space ${\Sigma }_{\Gamma }$ on which $U\left(A_{\Gamma }\right)$ acts properly and prove that ${\Sigma }_{\Gamma }$ is contractible, providing a geometric model for $U\left(A_{\Gamma }\right)$ and its subgroups. We also propose a geometric model for all of $Out\left(A_{\Gamma }\right)$, defined by allowing more general markings and metrics on points of ${\Sigma }_{\Gamma }$.