Geometry & Topology

From the hyperbolic $24$–cell to the cuboctahedron

Steven P Kerckhoff and Peter A Storm

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 14, Number 3 (2010), 1383-1477.

Dates
Received: 25 August 2008
Revised: 18 May 2010
Accepted: 22 March 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732229

Digital Object Identifier
doi:10.2140/gt.2010.14.1383

Mathematical Reviews number (MathSciNet)
MR2653730

Zentralblatt MATH identifier
1213.57023

Subjects
Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 51M99: None of the above, but in this section

Keywords
hyperbolic manifold discrete group

Citation

Kerckhoff, Steven P; Storm, Peter A. From the hyperbolic $24$–cell to the cuboctahedron. Geom. Topol. 14 (2010), no. 3, 1383--1477. doi:10.2140/gt.2010.14.1383. https://projecteuclid.org/euclid.gt/1513732229


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