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

2021/07/22 by Eduard Schesler, Matthew C. B. Zaremsky, Schesler, Eduard +1
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2107.10958

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 ℤ whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type \textrmF₂ but not \textrmFP₃, 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.

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