From the hyperbolic $24$–cell to the cuboctahedron

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 $Isom\left({ℍ}^4\right)$. 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.