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

Abstract

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 $Ou{t}^0(A_{\Gamma })$ 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 $Ou{t}^0(A_{\Gamma })$. In order to show this, we refine the decomposition sequence for $Ou{t}^0(A_{\Gamma })$ 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.

56 pages, 5 figures; v2: version accepted at Journal of the London Mathematical Society, incorporating referee suggestions

References in corpus (6)

• Algebraic laminations for free products and arational trees
• Relative automorphism groups of right-angled Artin groups
• On the bordification of outer space
• Homotopy type of the complex of free factors of a free group
• On the finiteness length of some soluble linear groups
• Higher generating subgroups and Cohen-Macaulay complexes

Cited by in corpus (5)

• Calculating the virtual cohomological dimension of the automorphism group of a RAAG
• Homotopy type of the complex of free factors of a free group
• Right-angled Artin groups as finite-index subgroups of their outer automorphism groups
• A note on virtual duality and automorphism groups of right-angled Artin groups
• Connectivity of partial basis complexes of freely decomposable groups