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 $F_2$ but not ${FP}_3$, 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.

中文翻译：

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有限建筑物的随机子复形，以及直角 Coxeter 群的换向器子群的纤维化

本文的主题是直角 Coxeter 群 (RACG) 的更高虚拟代数纤维化特性，特别关注那些定义旗帜复合体为有限建筑的群。我们证明了特定类别的有限建筑物，它们的随机诱导子复合体具有许多强大的特性，最突出的是它们是高度连接的。由此我们能够推断出 RACG 的换向器子群，定义标志复形为特定类型的有限建筑物，承认外同态$\mathbb{Z}$其核具有很强的拓扑有限性。我们还使用我们的技术来展示内核类型的示例$F_{\mathrm{2个}}$但不是${计划生育}_{\mathrm{3个}}$，以及 RACG 是双曲线且核是有限生成且非双曲线的示例。我们使用的关键工具是 Jankiewicz-Norin-Wise 提出的一种方法的概括，涉及应用于 RACG 的戴维斯复合体的 Bestvina-Brady 离散莫尔斯理论，以及一些概率论证。