## Geometry & Topology

### From the hyperbolic $24$–cell to the cuboctahedron

#### 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 $Isom(ℍ4)$. 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
Revised: 18 May 2010
Accepted: 22 March 2010
First available in Project Euclid: 20 December 2017

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

#### 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

#### References

• 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 http://msri.org/publications/books/gt3m/ {\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 http://people.uleth.ca/~dave.morris {\unhbox0