Outer space for RAAGs (2007.09725v2)

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 $A_{\Gamma}$.