Outer space for untwisted automorphisms of right-angled Artin groups

Abstract

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