Outer space for RAAGs

Abstract

For any right-angled Artin group $A_{\Gamma }$ we construct a finite-dimensional space ${\mathcal{O}}_{\Gamma }$ on which the group $\text{Out}(A_{\Gamma })$ of outer automorphisms of $A_{\Gamma }$ acts with finite point stabilizers. We prove that ${\mathcal{O}}_{\Gamma }$ is contractible, so that the quotient is a rational classifying space for $\text{Out}(A_{\Gamma })$. The space ${\mathcal{O}}_{\Gamma }$ 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 ${\mathcal{O}}_{\Gamma }$ 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 $A_{\Gamma }$.

61 pages, 17 figures. Modified statement of the main theorem, changed exposition, added a figure