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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The closure of a random braid is a hyperbolic link


Author: Jiming Ma
Journal: Proc. Amer. Math. Soc. 142 (2014), 695-701
MSC (2010): Primary 57M25, 57M50, 20F36
Published electronically: October 31, 2013
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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.


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Additional Information

Jiming Ma
Affiliation: School of Mathematical Science, Fudan University, Shanghai, People’s Republic of China, 200433
Email: majiming@fudan.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11775-2
PII: S 0002-9939(2013)11775-2
Keywords: Hyperbolic link, braid group, random walk, Dehn surgery
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
Article copyright: © Copyright 2013 American Mathematical Society