Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume
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Abstract:
The work of Jørgensen and Thurston shows that there is a finite number $N(v)$ of orientable hyperbolic $3$-manifolds with any given volume $v$. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume $v$ can grow at least factorially fast with $v$. A similar statement holds for closed hyperbolic $3$-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of $N(v)$ in terms of $v$ for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with $v$. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.References
- 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
- Richard E. Bedient, Double branched covers and pretzel knots, Pacific J. Math. 112 (1984), no. 2, 265–272. MR 743984, DOI 10.2140/pjm.1984.112.265
- Mikhail Belolipetsky, Tsachik Gelander, Alexander Lubotzky, and Aner Shalev, Counting arithmetic lattices and surfaces, Ann. of Math. (2) 172 (2010), no. 3, 2197–2221. MR 2726109, DOI 10.4007/annals.2010.172.2197
- Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992. MR 1219310, DOI 10.1007/978-3-642-58158-8
- M. Boileau and L. Siebenmann, A planar classification of pretzel knots and Montesinos knots, Publications Mathématiques d’Orsay, Université de Paris-Sud (1980).
- F. Bonahon, Involutions et fibrés de seifert dans les variétés de dimension $3$, Thèse de $3$ème cycle, University of Paris XI, Orsay (1979).
- Gerhard Burde and Heiner Zieschang, Knots, 2nd ed., De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR 1959408
- M. Burger, T. Gelander, A. Lubotzky, and S. Mozes, Counting hyperbolic manifolds, Geom. Funct. Anal. 12 (2002), no. 6, 1161–1173. MR 1952926, DOI 10.1007/s00039-002-1161-1
- Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks, A census of cusped hyperbolic $3$-manifolds, Math. Comp. 68 (1999), no. 225, 321–332. With microfiche supplement. MR 1620219, DOI 10.1090/S0025-5718-99-01036-4
- S. Carlip, Dominant topologies in Euclidean quantum gravity, Classical Quantum Gravity 15 (1998), no. 9, 2629–2638. Topology of the Universe Conference (Cleveland, OH, 1997). MR 1649663, DOI 10.1088/0264-9381/15/9/010
- Eric Chesebro and Jason DeBlois, Algebraic invariants, mutations, and commensurability of link complements, ArXiv e-prints (2012), 1202.0765 .
- J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
- Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
- Roberto Frigerio, Bruno Martelli, and Carlo Petronio, Dehn filling of cusped hyperbolic $3$-manifolds with geodesic boundary, J. Differential Geom. 64 (2003), no. 3, 425–455. MR 2032111
- David Futer, Efstratia Kalfagianni, and Jessica Purcell, Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics, vol. 2069, Springer, Heidelberg, 2013. MR 3024600, DOI 10.1007/978-3-642-33302-6
- David Futer and Jessica S. Purcell, Links with no exceptional surgeries, Comment. Math. Helv. 82 (2007), no. 3, 629–664. MR 2314056, DOI 10.4171/CMH/105
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
- Michael Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 40–53. MR 636516
- Joel Hass and Peter Scott, The existence of least area surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 310 (1988), no. 1, 87–114. MR 965747, DOI 10.1090/S0002-9947-1988-0965747-6
- C. Hodgson and H. Masai, On the number of hyperbolic 3-manifolds of a given volume, ArXiv e-prints (2012), 1203.6551 .
- Craig D. Hodgson and Jeffrey R. Weeks, Symmetries, isometries and length spectra of closed hyperbolic three-manifolds, Experiment. Math. 3 (1994), no. 4, 261–274. MR 1341719, DOI 10.1080/10586458.1994.10504296
- Louis H. Kauffman and Sofia Lambropoulou, Classifying and applying rational knots and rational tangles, Physical knots: knotting, linking, and folding geometric objects in $\Bbb R^3$ (Las Vegas, NV, 2001) Contemp. Math., vol. 304, Amer. Math. Soc., Providence, RI, 2002, pp. 223–259. MR 1953344, DOI 10.1090/conm/304/05197
- Thilo Kuessner, Mutation and recombination for hyperbolic 3-manifolds, J. Gökova Geom. Topol. GGT 5 (2011), 20–30. MR 2872549
- Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
- Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067, DOI 10.2140/pjm.1984.111.209
- Daniel Ruberman, Mutation and volumes of knots in $S^3$, Invent. Math. 90 (1987), no. 1, 189–215. MR 906585, DOI 10.1007/BF01389038
- William Thurston, Geometry and topology of 3-manifolds, lecture notes, Princeton University (1978).
- Norbert J. Wielenberg, Hyperbolic $3$-manifolds which share a fundamental polyhedron, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 505–513. MR 624835
Additional Information
- Christian Millichap
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
- Email: christian.millichap@gmail.com
- Received by editor(s): September 3, 2012
- Received by editor(s) in revised form: November 4, 2013
- Published electronically: January 16, 2015
- Communicated by: Daniel Ruberman
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2201-2214
- MSC (2010): Primary 52A22; Secondary 46B09
- DOI: https://doi.org/10.1090/S0002-9939-2015-12395-7
- MathSciNet review: 3314126