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2016 Cosmetic surgery and the link volume of hyperbolic $3$–manifolds
Yo’av Rieck, Yasushi Yamashita
Algebr. Geom. Topol. 16(6): 3445-3521 (2016). DOI: 10.2140/agt.2016.16.3445


We prove that for any V > 0 there exists a hyperbolic manifold MV such that Vol(MV ) < 2.03 and LinkVol(MV ) > V . This was conjectured by the authors in [Algebr. Geom. Topol. 13 (2013) 927–958, Conjecture 1.3].

The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain:

  1. Let K be a component of a link L in S3. Then “most” slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link.

  2. Let X be a manifold and ϵ > 0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than ϵ. (Note that it is not true that there are only finitely many fillings fulfilling this condition.)


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Yo’av Rieck. Yasushi Yamashita. "Cosmetic surgery and the link volume of hyperbolic $3$–manifolds." Algebr. Geom. Topol. 16 (6) 3445 - 3521, 2016.


Received: 1 October 2015; Revised: 24 February 2016; Accepted: 27 March 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1358.57008
MathSciNet: MR3584264
Digital Object Identifier: 10.2140/agt.2016.16.3445

Primary: 57M12 , 57M50

Keywords: 3–manifolds , branched covering , cosmetic surgery , Dehn surgery , hyperbolic manifolds , hyperbolic volume , link volume

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 6 • 2016
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