Higher generating subgroups and Cohen–Macaulay complexes

Abstract

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi subgroups and describe higher generating families of parabolic subgroups in Aut(Fn).

Homotopy type of the complex of free factors of a free group

“…The results there are proved in the more general setting of automorphism groups of free products. These are used to determine the homotopy type of an analogue of the free factor complex for automorphisms of right‐angled Artin groups in [4].…”

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

“…Theorem B is now an easy corollary of Cohen–Macaulayness of $$\operatorname{CC}(O ,\, \mathcal {P}(O))$$ and the results of [9]. Corollary The family $$\mathcal {P}_m(O)$$ of rank‐$$m$$ parabolic subgroups of $$O$$ is $$m$$‐generating, the corresponding coset complex $$\operatorname{CC}(O ,\, \mathcal {P}_m(O))$$ is $$m$$‐spherical.…”

“…As an application of Theorem A and the results of [9], we obtain higher generating families of subgroups of $$O$$ in the sense of Abels–Holz (for the definition, see Section 3.1.2). Theorem The family $$\mathcal {P}_m(O)$$ of rank‐$$m$$ parabolic subgroups of $$O$$ is $$m$$‐generating.…”

“…An immediate consequence of this is that $$\operatorname{CC}(O ,\, \mathcal {P}(O))$$ is a chamber complex , that is, that each pair $$\sigma ,\tau$$ of facets of $$\operatorname{CC}(O ,\, \mathcal {P}(O))$$ can be connected by a sequence of facets $$\sigma = \tau _1,\ldots , \tau _k=\tau$$ such that for all $$1\leqslant i \leqslant k$$, the intersection of $$\tau _i$$ and $$\tau _{i+1}$$ is a face of codimension 1 (see [4, Proposition 11.7; 9, Remark 2.8]).…”

Calculating the virtual cohomological dimension of the automorphism group of a RAAG

“…In this direction, Brück [6, Section 7] uses careful choices of restriction maps to construct a decomposition tree for where the leaves can be described quite explicitly. As a trade‐off, the leaves that appear in the decomposition tree of Brück are slightly more general (there are groups generated by partial conjugations that are not necessarily of type (), (), or ()).…”