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The smallest Haken hyperbolic polyhedra
Authors:
Christopher K. Atkinson and Shawn Rafalski
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1393-1404
MSC (2010):
Primary 52B10, 57M50, 57R18
Posted:
September 5, 2012
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References |
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Additional Information
Abstract: We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral  -orbifold contains an essential  -suborbifold, up to a canonical decomposition along essential hyperbolic triangle  -suborbifolds.
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- 1.
- Colin Adams and Eric Schoenfeld, Totally geodesic Seifert surfaces in hyperbolic knot and link complements. I, Geom. Dedicata 116 (2005), 237-247. MR 2195448 (2006j:57008)
- 2.
- Ian Agol, Peter A. Storm, and William Thurston, Lower bounds on volumes of hyperbolic Haken
 -manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053-1077. MR 2328715 (2008i:53086)
- 3.
- E. M. Andreev, On convex polyhedra in Lobachevski spaces, Math. USSR Sbornik 10 (1970), no. 3, 413-440. MR 0259734 (41:4367)
- 4.
- -, On convex polyhedra of finite volume in Lobachevski space, Math. USSR Sbornik 12 (1970), no. 2, 255-259. MR 0273510 (42:8388)
- 5.
- Christopher K. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), no. 2, 1225-1254. MR 2519588 (2010k:57035)
- 6.
- -, Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra, Geom. Dedicata 153 (2011), 177-211. MR 2819670
- 7.
- Michel Boileau, Sylvain Maillot, and Joan Porti, Three-dimensional orbifolds and their geometric structures, Panoramas et Synthèses [Panoramas and Syntheses], vol. 15, Société Mathématique de France, Paris, 2003. MR 2060653 (2005b:57030)
- 8.
- Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, vol. 5, Mathematical Society of Japan, Tokyo, 2000, With a postface by Sadayoshi Kojima. MR 1778789 (2002c:57027)
- 9.
- William D. Dunbar, Hierarchies for
 -orbifolds, Topology Appl. 29 (1988), no. 3, 267-283. MR 953958 (89h:57008)
- 10.
- David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157-1215. MR 2525782 (2011a:57031)
- 11.
- -, Mom technology and volumes of hyperbolic
 -manifolds, Comment. Math. Helv. 86 (2011), no. 1, 145-188. MR 2745279
- 12.
- Frederick W. Gehring and Gaven J. Martin, Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group, Ann. of Math. (2) 170 (2009), no. 1, 123-161. MR 2521113 (2010h:57029)
- 13.
- D. Heard, Orb, www.ms.unimelb.edu.au/~snap/orb.html.
- 14.
- Taiyo Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8 (2008), no. 3, 1523-1565. MR 2443253 (2009k:57025)
- 15.
- T. H. Marshall and Gaven J. Martin, Minimal co-volume hyperbolic lattices. II. Simple torsion in Kleinian groups (2008), Preprint.
- 16.
- Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135 (90a:30132)
- 17.
- John Milnor, The Schläfli differential formula, Collected papers. Vol. 1, Geometry, Publish or Perish Inc., 1994. MR 1277810 (95c:01043)
- 18.
- Yosuke Miyamoto, Volumes of hyperbolic manifolds with geodesic boundary, Topology 33 (1994), no. 4, 613-629. MR 1293303 (95h:57014)
- 19.
- Shawn Rafalski, Immersed turnovers in hyperbolic
 -orbifolds, Groups Geom. Dyn. 4 (2010), no. 2, 333-376. MR 2595095 (2011a:57036)
- 20.
- -, Small hyperbolic polyhedra, Pacific J. Math. 255 (2011), no. 1, 191-240.
- 21.
- John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730 (95j:57011)
- 22.
- E. Steinitz, Polyeder und Raumeinteilungen, Enzylk. Math. Wiss. 3 (1922), 1-139.
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Additional Information
Christopher K. Atkinson
Affiliation:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19106
Email:
ckatkin@temple.edu
Shawn Rafalski
Affiliation:
Department of Mathematics and Computer Science, Fairfield University, Fairfield, Connecticut 06824
Email:
srafalski@fairfield.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11665-X
PII:
S 0002-9939(2012)11665-X
Keywords:
Hyperbolic polyhedra,
3–dimensional Coxeter polyhedra,
hyperbolic orbifold,
polyhedral orbifold,
hyperbolic volume,
Haken orbifold
Received by editor(s):
August 19, 2011
Posted:
September 5, 2012
Communicated by:
Michael Wolf
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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