Outer space for RAAGs

Abstract

For any right-angled Artin group AΓ, we construct a finite-dimensional space OΓ on which the group Out(AΓ) of outer automorphisms of AΓ acts with finite point stabilizers. We prove that OΓ is contractible, so that the quotient is a rational classifying space for Out(AΓ). The space OΓ blends features of the symmetric space of lattices in Rn with those of outer space for the free group Fn. Points in OΓ 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 AΓ.