The closure of a random braid is a hyperbolic link
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- by Jiming Ma PDF
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Abstract:
Let $\mu$ be a probability distribution on the braid group $B_{n}$ with $n \geq 3$ strands. We observe that for a random walk $\omega _{n,k}$ of length $k$ on $B_{n}$, the probability that the closure $\widehat {\omega _{n,k}}$ is a hyperbolic link in $S^{3}$ converges to 1 as $k$ tends to infinity. Moreover, under a mild assumption on $\mu$, we prove the probability that the closure $\widehat {\omega _{n,k}}$ is a hyperbolic knot which has no non-trivial exceptional surgeries is larger than zero for $k$ large enough. The proofs combine several recent deep results.References
- Colin C. Adams, The knot book, W. H. Freeman and Company, New York, 1994. An elementary introduction to the mathematical theory of knots. MR 1266837
- Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI 10.2140/gt.2000.4.431
- Persi Diaconis and Mehrdad Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 2, 159–179. MR 626813, DOI 10.1007/BF00535487
- Yuanan Diao, Nicholas Pippenger, and De Witt Sumners, On random knots, Random knotting and linking (Vancouver, BC, 1993) Ser. Knots Everything, vol. 7, World Sci. Publ., River Edge, NJ, 1994, pp. 187–197. MR 1474271, DOI 10.1142/9789812796172_{0}013
- 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
- David Futer and Saul Schleimer, Cusp geometry of fibered 3-manifolds, arXiv: math.GT/1108.5748, to appear in Amer. J. Math.
- Craig D. Hodgson and Steven P. Kerckhoff, The shape of hyperbolic Dehn surgery space, Geom. Topol. 12 (2008), no. 2, 1033–1090. MR 2403805, DOI 10.2140/gt.2008.12.1033
- Douglas Jungreis, Gaussian random polygons are globally knotted, J. Knot Theory Ramifications 3 (1994), no. 4, 455–464. MR 1304395, DOI 10.1142/S0218216594000332
- Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613
- Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI 10.1007/s002220000047
- Martin Lustig and Yoav Moriah, Horizontal Dehn surgery and genericity in the curve complex, J. Topol. 3 (2010), no. 3, 691–712. MR 2684517, DOI 10.1112/jtopol/jtq022
- Martin Lustig and Yoav Moriah, Are large distance Heegaard splittings generic?, J. Reine Angew. Math. 670 (2012), 93–119. With an appendix by Vaibhav Gadre. MR 2982693, DOI 10.1515/crelle.2011.154
- Joseph Maher, Random Heegaard splittings, J. Topol. 3 (2010), no. 4, 997–1025. MR 2746344, DOI 10.1112/jtopol/jtq031
- Joseph Maher, Linear progress in the complex of curves, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2963–2991. MR 2592943, DOI 10.1090/S0002-9947-10-04903-2
- A. V. Malyutin, Quasimorphisms, random walks, and transient subsets in countable groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 390 (2011), no. Teoriya Predstavleniĭ, Dinamicheskie Sistemy, Kombinatornye Metody. XX, 210–236, 309–310 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 181 (2012), no. 6, 871–885. MR 2870236, DOI 10.1007/s10958-012-0721-7
- Bruno Martelli and Carlo Petronio, Dehn filling of the “magic” 3-manifold, Comm. Anal. Geom. 14 (2006), no. 5, 969–1026. MR 2287152
- Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). MR 1402300
- Carlo Petronio and Joan Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, Expo. Math. 18 (2000), no. 1, 1–35. MR 1751141
- Jessica S. Purcell, Cusp shapes of hyperbolic link complements and Dehn filling, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Stanford University. MR 2705870
- Igor Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353–379. MR 2401624, DOI 10.1215/00127094-2008-009
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
- William P. Thurston, Hyperbolic structures on 3-manifolds, II: surface groups and 3-manifolds which fiber over the circle, arXiv: math.GT/9801045, 1998.
- Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR 1743100, DOI 10.1017/CBO9780511470967
Additional Information
- Jiming Ma
- Affiliation: School of Mathematical Science, Fudan University, Shanghai, People’s Republic of China, 200433
- Email: majiming@fudan.edu.cn
- Received by editor(s): September 18, 2011
- Received by editor(s) in revised form: February 10, 2012, and March 25, 2012
- Published electronically: October 31, 2013
- Additional Notes: The author was supported in part by NSFC 10901038 and NSFC 11371094
- Communicated by: Daniel Ruberman
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 695-701
- MSC (2010): Primary 57M25, 57M50, 20F36
- DOI: https://doi.org/10.1090/S0002-9939-2013-11775-2
- MathSciNet review: 3134009