×

Guts and volume for hyperbolic 3-orbifolds with underlying space \(S^3\). (English) Zbl 1394.57023

In this paper, the authors study volumes of hyperbolic \(3\)-orbifolds. Let \({\mathcal{O}}\) be a hyperbolic \(3\)-orbifold with underlying space the \(3\)-sphere, and let \(S\) be a closed, incompressible \(2\)-suborbifold of \({\mathcal{O}}\). The topology of \(S\) and the way it is embedded in \({\mathcal{O}}\) has an effect on the volume of \({\mathcal{O}}\). To analyse that effect, the authors define a certain pared orbifold called \(\text{guts}(S)\). They describe the possibilities for \(\text{guts}(S)\), in the case \(S\) has the \(2\)-sphere as underlying space and four cone points. They then give a complete description of the topological structure of \({\mathcal{O}}\) when the guts are empty. When the guts are non-empty, they obtain lower bounds for the volume of \({\mathcal{O}}\) in terms of the topology of the guts.

MSC:

57R18 Topology and geometry of orbifolds
57M50 General geometric structures on low-dimensional manifolds
57N16 Geometric structures on manifolds of high or arbitrary dimension
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agol, Ian, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Am. Math. Soc., 138, 10, 3723-3732 (2010) · Zbl 1203.57006
[2] Agol, Ian; Storm, Peter A.; Thurston, William, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J. Am. Math. Soc., 20, 4, 1053-1077 (2007) · Zbl 1155.58016
[3] Atkinson, Christopher K.; Rafalski, Shawn, The smallest Haken hyperbolic polyhedra, Proc. Am. Math. Soc., 141, 4, 1393-1404 (2013) · Zbl 1271.52010
[4] Boileau, Michel; Maillot, Sylvain; Porti, Joan, Three-Dimensional Orbifolds and Their Geometric Structures, Panoramas et Synthèses, vol. 15 (2003), Société Mathématique de France: Société Mathématique de France Paris · Zbl 1058.57009
[5] Boileau, Michel; Porti, Joan, Geometrization of 3-Orbifolds of Cyclic Type, Astérisque, vol. 272, 208 (2001), Appendix A by Michael Heusener and Porti · Zbl 0971.57004
[6] Bonahon, F.; Siebenmann, L. C., The characteristic toric splitting of irreducible compact 3-orbifolds, Math. Ann., 278, 1-4, 441-479 (1987) · Zbl 0629.57007
[7] Canary, Richard D.; McCullough, Darryl, Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups, Mem. Am. Math. Soc., 172, 812 (2004), xii+218 · Zbl 1062.57027
[8] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P., Three-Dimensional Orbifolds and Cone-Manifolds, MSJ Memoirs, vol. 5 (2000), Mathematical Society of Japan: Mathematical Society of Japan Tokyo, With a postface by Sadayoshi Kojima · Zbl 0955.57014
[9] Gabai, David; Meyerhoff, Robert; Milley, Peter, Minimum volume cusped hyperbolic three-manifolds, J. Am. Math. Soc., 22, 4, 1157-1215 (2009) · Zbl 1204.57013
[10] Gabai, David; Meyerhoff, Robert; Milley, Peter, Mom technology and volumes of hyperbolic 3-manifolds, Comment. Math. Helv., 86, 1, 145-188 (2011) · Zbl 1207.57023
[11] Gehring, Frederick W.; Martin, Gaven J., Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group, Ann. Math. (2), 170, 1, 123-161 (2009) · Zbl 1171.30014
[12] Jaco, William H.; Shalen, Peter B., Seifert fibered spaces in 3-manifolds, Mem. Am. Math. Soc., 21, 220 (1979), viii+192 · Zbl 0415.57005
[13] Johansson, K., Homotopy Equivalences of 3-Manifolds with Boundary, Lecture Notes in Mathematics, vol. 761 (1979) · Zbl 0412.57007
[14] Kapovich, Michael, Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics (2009), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, Reprint of the 2001 edition · Zbl 1180.57001
[15] T.H. Marshall, Gaven J. Martin, Minimal co-volume hyperbolic lattices. II. Simple torsion in Kleinian groups, Preprint, 2008.; T.H. Marshall, Gaven J. Martin, Minimal co-volume hyperbolic lattices. II. Simple torsion in Kleinian groups, Preprint, 2008. · Zbl 1252.30030
[16] Milnor, John, The Schläfli Differential Formula, Collected Papers, vol. 1, Geometry (1994), Publish or Perish Inc. · Zbl 0857.01015
[17] Miyamoto, Yosuke, Volumes of hyperbolic manifolds with geodesic boundary, Topology, 33, 4, 613-629 (1994) · Zbl 0824.53038
[18] Morgan, John W., On Thurston’s uniformization theorem for three-dimensional manifolds, (The Smith Conjecture. The Smith Conjecture, New York, 1979. The Smith Conjecture. The Smith Conjecture, New York, 1979, Pure Appl. Math., vol. 112 (1984), Academic Press: Academic Press Orlando, FL), 37-125 · Zbl 0599.57002
[19] Otal, Jean-Pierre, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque, vol. 235 (1996), x+159 · Zbl 0855.57003
[20] Otal, Jean-Pierre, Thurston’s hyperbolization of Haken manifolds, (Surveys in Differential Geometry, Vol. III. Surveys in Differential Geometry, Vol. III, Cambridge, MA, 1996 (1998), Int. Press: Int. Press Boston, MA), 77-194 · Zbl 0997.57001
[21] Rafalski, Shawn, Immersed turnovers in hyperbolic 3-orbifolds, Groups Geom. Dyn., 4, 2, 333-376 (2010) · Zbl 1194.57024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.