Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groups

TL;DR

This paper investigates the higher algebraic fibering properties of right-angled Coxeter groups with finite building complexes, demonstrating that their random subcomplexes are highly connected and that their commutator subgroups have significant finiteness properties.

Contribution

It introduces new probabilistic and topological methods to analyze the fibering properties of RACGs with finite building complexes, including examples with specific finiteness and hyperbolicity properties.

Findings

Random subcomplexes are highly connected

The commutator subgroup admits an epimorphism to Z with a finitely presented kernel

Examples of RACGs with kernels of type F2 but not FP3

Abstract

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\mathbb{Z}$ whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type $\textrm{F}_2$ but not $\textrm{FP}_3$, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz-Norin-Wise involving Bestvina-Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.