Infinitely many commensurability classes of compact Coxeter polyhedra in $\mathbb{H}^4$ and $\mathbb{H}^5$

TL;DR

This paper demonstrates that specific families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space lead to infinitely many distinct classes of reflection groups, expanding understanding of hyperbolic geometry and group theory.

Contribution

It establishes the existence of infinitely many commensurability classes of reflection groups arising from Coxeter polyhedra in higher-dimensional hyperbolic spaces.

Findings

Infinitely many commensurability classes in $ ext{H}^4$ and $ ext{H}^5$

Construction of families of Coxeter polyhedra by Makarov

Implications for hyperbolic reflection groups

Abstract

We prove that certain families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups in these dimensions.