Algebraic & Geometric Topology

Cosmetic surgery and the link volume of hyperbolic $3$–manifolds

Yo’av Rieck and Yasushi Yamashita

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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.)

Article information

Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3445-3521.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M12: Special coverings, e.g. branched 57M50: Geometric structures on low-dimensional manifolds

link volume hyperbolic volume cosmetic surgery Dehn surgery 3–manifolds hyperbolic manifolds branched covering


Rieck, Yo’av; Yamashita, Yasushi. Cosmetic surgery and the link volume of hyperbolic $3$–manifolds. Algebr. Geom. Topol. 16 (2016), no. 6, 3445--3521. doi:10.2140/agt.2016.16.3445.

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