Outer space for RAAGs

TL;DR

This paper constructs a finite-dimensional, contractible space for right-angled Artin groups that acts as a classifying space for their outer automorphism groups, blending features of symmetric spaces and Outer space.

Contribution

It introduces a new space, $ ext{O}_ ext{ extGamma}$, with a proper action by $ ext{Out}(A_ ext{ extGamma})$, providing a geometric model similar to Outer space for free groups.

Findings

$ ext{O}_ ext{ extGamma}$ is contractible.

$ ext{O}_ ext{ extGamma}$ has finite-dimensionality.

The space acts with finite point stabilizers.

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 $A_{\Gamma}$.