Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

Abstract

For every finite graph Γ $\Gamma$ , we define a simplicial complex associated to the outer automorphism group of the right-angled Artin group (RAAG) A Γ $A_\Gamma$ . These complexes are defined as coset complexes of parabolic subgroups of Out 0 ( A Γ ) $\operatorname{Out}^0(A_\Gamma )$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen–Macaulay and in particular homotopy equivalent to a wedge of d $d$ -spheres. The dimension d $d$ can be read off from the defining graph Γ $\Gamma$ and is determined by the rank of a certain Coxeter subgroup of Out 0 ( A Γ ) $\operatorname{Out}^0(A_\Gamma )$ . To show this, we refine the decomposition sequence for Out 0 ( A Γ ) $\operatorname{Out}^0(A_\Gamma )$ established by Day–Wade, generalise a result of Brown concerning the behaviour of coset posets under short exact sequences and determine the homotopy type of free factor complexes associated to relative automorphism groups of free products.