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Effective bilipschitz
bounds on drilling and filling
David Futer, Jessica S Purcell and Saul Schleimer |
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Geometry & Topology 26 (2022) 1077–1188
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Abstract |
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We prove explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic –manifold and the thick part of any of its long Dehn fillings. Given a bilipschitz constant and a thickness constant , we quantify how long a Dehn filling suffices to guarantee a –bilipschitz map on –thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff’s theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation. Our quantitative results have two immediate applications. First, we relate the Margulis number of to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed –manifold whose Margulis number is less than . Combined with Shalen’s upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of –manifolds whose Margulis numbers are below . Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold , we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on . This reduces the cosmetic surgery conjecture on to an explicit finite search. |
Keywords
cone manifold, hyperbolic manifold, Dehn filling, Dehn
surgery, cosmetic surgery, Margulis number
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Mathematical Subject Classification
Primary: 30F40, 57K10, 57K32
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Supplementary material |
References |
Publication
Received: 7 August 2019
Revised: 15 February 2021
Accepted: 16 April 2021
Published: 3 August 2022
Proposed: David Gabai
Seconded: Mladen Bestvina, Stavros Garoufalidis
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Authors |
| David Futer | |
| Department of Mathematics Temple University Philadelphia, PA United States |
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| https://math.temple.edu/~dfuter | |
| Jessica S Purcell | |
| School of Mathematics Monash University Clayton VIC Australia |
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| http://users.monash.edu/~jpurcell/ | |
| Saul Schleimer | |
| Department of Mathematics University of Warwick Coventry United Kingdom |
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| http://homepages.warwick.ac.uk/~masgar/ | |
