Virtually Fibering Right-Angled Coxeter Groups

TL;DR

This paper investigates the algebraic structure of certain right-angled Coxeter groups, demonstrating the existence of finite index subgroups with specific quotient and kernel properties using Morse theory and combinatorial methods.

Contribution

It introduces new conditions under which right-angled Coxeter groups have subgroups with finitely generated kernels and describes examples where these conditions succeed or fail.

Findings

Existence of finite index subgroups with quotients to Z and finitely generated kernels.

Application of Bestvina-Brady Morse theory to Coxeter groups.

Examples include right-angled reflection groups in H^4 with specific fundamental domains.

Abstract

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb H^4$ with fundamental domain the $120$-cell or the $24$-cell.