From the hyperbolic 24–cell to the cuboctahedron

Kerckhoff, Steven P¹ ; Storm, Peter A²

¹ Department of Mathematics, Stanford University, Building 380, Sloan Hall, Stanford, CA 94305, USA
² Jane Street Capital, LLC, 1 New York Plaza, 33rd Floor, New York, NY 10004, USA

Geometry & topology, Tome 14 (2010) no. 3, pp. 1383-1477.

Résumé

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.

Détail

DOI : 10.2140/gt.2010.14.1383

Keywords: hyperbolic manifold, discrete group

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