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

For every finite graph $\Gamma$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_\Gamma$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_\Gamma)$ 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-spheres. The dimension d can be read off from the defining graph $\Gamma$ and is determined by the rank of a certain Coxeter subgroup of $Out^0(A_\Gamma)$. In order to show this, we refine the decomposition sequence for $Out^0(A_\Gamma)$ 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.