Outer space for RAAGs

Abstract

For any right-angled Artin group $A_{\mathrm{\Gamma }}$, we construct a finite-dimensional space ${\mathcal{O}}_{\mathrm{\Gamma }}$ on which the group $Out(A_{\mathrm{\Gamma }})$ of outer automorphisms of $A_{\mathrm{\Gamma }}$ acts with finite point stabilizers. We prove that ${\mathcal{O}}_{\mathrm{\Gamma }}$ is contractible, so that the quotient is a rational classifying space for $Out(A_{\mathrm{\Gamma }})$. The space ${\mathcal{O}}_{\mathrm{\Gamma }}$ blends features of the symmetric space of lattices in ${\mathbb{R}}^n$ with those of outer space for the free group $F_n$. Points in ${\mathcal{O}}_{\mathrm{\Gamma }}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{\mathrm{\Gamma }}$.