Geometry & Topology

From the hyperbolic $24$–cell to the cuboctahedron

Steven P Kerckhoff and Peter A Storm

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

hyperbolic manifold discrete group


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.

Export citation


  • J W Anderson, A brief survey of the deformation theory of Kleinian groups, from: “The Epstein birthday schrift”, (I Rivin, C Rourke, C Series, editors), Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 23–49
  • E M Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. $($N.S.$)$ 83 (125) (1970) 256–260
  • M Boileau, B Leeb, J Porti, Geometrization of $3$–dimensional orbifolds, Ann. of Math. $(2)$ 162 (2005) 195–290
  • A Borel, Introduction aux groupes arithmétiques, Publ. Institut Math. Univ. Strasbourg, XV. Actualités Sci. Ind. 1341, Hermann, Paris (1969)
  • B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245–317
  • R D Canary, On the Laplacian and the geometry of hyperbolic $3$–manifolds, J. Differential Geom. 36 (1992) 349–367
  • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Math. Soc. Japan, Tokyo (2000) With a postface by S Kojima
  • H S M Coxeter, Twelve geometric essays, Southern Illinois Univ. Press, Carbondale, Ill. (1968)
  • H S M Coxeter, Regular polytopes, third edition, Dover, New York (1973)
  • H S M Coxeter, Regular complex polytopes, second edition, Cambridge Univ. Press (1991)
  • H Garland, M S Raghunathan, Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups, Ann. of Math. $(2)$ 92 (1970) 279–326
  • P de la Harpe, An invitation to Coxeter groups, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (E Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publ., River Edge, NJ (1991) 193–253
  • G D Mostow, Strong rigidity of locally symmetric spaces, Annals Math. Studies 78, Princeton Univ. Press (1973)
  • G Prasad, Strong rigidity of ${\bf Q}$–rank $1$ lattices, Invent. Math. 21 (1973) 255–286
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse Math. ihrer Grenzgebiete 68, Springer, New York (1972)
  • A Selberg, On discontinuous groups in higher-dimensional symmetric spaces, from: “Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960)”, Tata Inst., Bombay (1960) 147–164
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • E B Vinberg, On groups of unit elements of certain quadratic forms, Math. USSR Sbornik 16 (1972) 17–35 Translated by D E Brown
  • E B Vinberg, O V Shvartsman, Discrete groups of motions of spaces of constant curvature, from: “Geometry, II”, Ency. Math. Sci. 29, Springer, Berlin (1993) 139–248
  • H C Wang, Topics on totally discontinuous groups, from: “Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970)”, (W M Boothby, G L Weiss, editors), Pure and Appl. Math. 8, Dekker, New York (1972) 459–487
  • A Weil, On discrete subgroups of Lie groups. II, Ann. of Math. $(2)$ 75 (1962) 578–602
  • A Weil, Remarks on the cohomology of groups, Ann. of Math. $(2)$ 80 (1964) 149–157
  • D Witte Morris, Introduction to arithmetic groups, Preprint (2003) Available at \setbox0\makeatletter\@url {\unhbox0