From the hyperbolic 24-cell to the cuboctahedron

TL;DR

This paper constructs a family of 4-dimensional hyperbolic orbifolds by deforming a hyperbolic 24-cell, leading to new examples of discrete groups with interesting geometric properties, using an explicit algebraic and geometric approach.

Contribution

It introduces a novel deformation method for hyperbolic 24-cell orbifolds, producing infinite covolume groups and finite covolume Coxeter groups in four dimensions.

Findings

Constructed explicit examples of hyperbolic 4-orbifolds

Generated infinite covolume, rigid discrete groups

Produced finite covolume Coxeter groups as images

Abstract

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry group of hyperbolic 4-space. It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling.