  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.
