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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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(4). 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
References
Publication
Received: 25 August 2008
Revised: 18 May 2010
Accepted: 22 March 2010
Published: 7 June 2010
Proposed: Benson Farb
Seconded: Walter Neumann, Jean-Pierre Otal
Authors
Steven P Kerckhoff
Department of Mathematics
Stanford University
Building 380, Sloan Hall
Stanford, CA 94305
USA
Peter A Storm
Jane Street Capital, LLC
1 New York Plaza, 33rd Floor
New York, NY 10004
USA