The smallest Haken hyperbolic polyhedra
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- by Christopher K. Atkinson and Shawn Rafalski PDF
- Proc. Amer. Math. Soc. 141 (2013), 1393-1404 Request permission
Abstract:
We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral $3$–orbifold contains an essential $2$–suborbifold, up to a canonical decomposition along essential hyperbolic triangle $2$–suborbifolds.References
- Colin Adams and Eric Schoenfeld, Totally geodesic Seifert surfaces in hyperbolic knot and link complements. I, Geom. Dedicata 116 (2005), 237–247. MR 2195448, DOI 10.1007/s10711-005-9018-z
- Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053–1077. With an appendix by Nathan Dunfield. MR 2328715, DOI 10.1090/S0894-0347-07-00564-4
- E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445–478 (Russian). MR 0259734
- E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
- Christopher K. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), no. 2, 1225–1254. MR 2519588, DOI 10.2140/agt.2009.9.1225
- Christopher K. Atkinson, Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra, Geom. Dedicata 153 (2011), 177–211. MR 2819670, DOI 10.1007/s10711-010-9563-y
- 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 (English, with English and French summaries). MR 2060653
- 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
- William D. Dunbar, Hierarchies for $3$-orbifolds, Topology Appl. 29 (1988), no. 3, 267–283. MR 953958, DOI 10.1016/0166-8641(88)90025-9
- 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, DOI 10.1090/S0894-0347-09-00639-0
- David Gabai, Robert Meyerhoff, and Peter Milley, Mom technology and volumes of hyperbolic 3-manifolds, Comment. Math. Helv. 86 (2011), no. 1, 145–188. MR 2745279, DOI 10.4171/CMH/221
- 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, DOI 10.4007/annals.2009.170.123
- D. Heard, Orb, www.ms.unimelb.edu.au/~snap/orb.html.
- Taiyo Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8 (2008), no. 3, 1523–1565. MR 2443253, DOI 10.2140/agt.2008.8.1523
- T. H. Marshall and Gaven J. Martin, Minimal co-volume hyperbolic lattices. II. Simple torsion in Kleinian groups (2008), Preprint.
- Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
- John Milnor, Collected papers. Vol. 1, Publish or Perish, Inc., Houston, TX, 1994. Geometry. MR 1277810
- Yosuke Miyamoto, Volumes of hyperbolic manifolds with geodesic boundary, Topology 33 (1994), no. 4, 613–629. MR 1293303, DOI 10.1016/0040-9383(94)90001-9
- Shawn Rafalski, Immersed turnovers in hyperbolic 3-orbifolds, Groups Geom. Dyn. 4 (2010), no. 2, 333–376. MR 2595095, DOI 10.4171/GGD/86
- —, Small hyperbolic polyhedra, Pacific J. Math. 255 (2011), no. 1, 191–240.
- John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4
- E. Steinitz, Polyeder und Raumeinteilungen, Enzylk. Math. Wiss. 3 (1922), 1–139.
Additional Information
- Christopher K. Atkinson
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19106
- MR Author ID: 873749
- Email: ckatkin@temple.edu
- Shawn Rafalski
- Affiliation: Department of Mathematics and Computer Science, Fairfield University, Fairfield, Connecticut 06824
- Email: srafalski@fairfield.edu
- Received by editor(s): August 19, 2011
- Published electronically: September 5, 2012
- Communicated by: Michael Wolf
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1393-1404
- MSC (2010): Primary 52B10, 57M50, 57R18
- DOI: https://doi.org/10.1090/S0002-9939-2012-11665-X
- MathSciNet review: 3008886