Volume 14, issue 3 (2010)

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From the hyperbolic $24$–cell to the cuboctahedron

Steven P Kerckhoff and Peter A Storm

Geometry & Topology 14 (2010) 1383–1477
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.

Keywords
hyperbolic manifold, discrete group
Mathematical Subject Classification 2000
Primary: 22E40
Secondary: 20F55, 20H10, 51M99