For every finite graph Γ, we define a simplicial complex associated to
the outer automorphism group of the RAAG AΓ. These complexes are
defined as coset complexes of parabolic subgroups of Out0(AΓ) 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 Γ and is determined by the rank of a certain Coxeter
subgroup of Out0(AΓ). In order to show this, we refine the
decomposition sequence for Out0(AΓ) 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 relative free factor
complexes associated to Fouxe-Rabinovitch groups.Comment: 56 pages, 5 figure