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) AnormalΓ$A_\Gamma$. These complexes are defined as coset complexes of parabolic subgroups of Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$ 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 Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$. To show this, we refine the decomposition sequence for Out0(AnormalΓ)$\operatorname{Out}^0(A_\Gamma )$ 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.

A note on virtual duality and automorphism groups of right-angled Artin groups

“…Recently, Brück [11] has constructed an complex which is a hybrid of the free factor complex and the Tits building for , although has larger -stabilizers than one might hope for (for instance, is trivial for the graph from our main theorem). Brück showed that these complexes are Cohen–Macaulay, so one hope is that could be used to reduce problems about duality to the behaviour of -stabilizers in .…”

Right‐angled Artin groups as finite‐index subgroups of their outer automorphism groups

“…We then generalised these examples step by step until we arrived at the constructions given below. These computer programs are based on and contain parts of programs written by Benjamin Brück, which can be found on [5] and were used for [15].…”

“…Similarly, their outer automorphism groups $$\operatorname{Out}(A_\Gamma)$$ may be viewed as interpolating between $$\operatorname{GL}_n(\mathbb {Z})$$ and $$\operatorname{Out}(F_n)$$. For examples of how this has been used, one can consider [4, 6, 9, 10, 11].…”

“…Similarly, their outer automorphism groups Out(𝐴 Γ ) may be viewed as interpolating between GL 𝑛 (ℤ) and Out(𝐹 𝑛 ). For examples of how this has been used, one can consider [4,6,9,10,11].…”