Cosmetic surgery and the link volume of hyperbolic 3–manifolds

Rieck, Yo’av; Yamashita, Yasushi

doi:10.2140/agt.2016.16.3445

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

Yo’av Rieck and Yasushi Yamashita

Algebraic & Geometric Topology 16 (2016) 3445–3521

DOI: 10.2140/agt.2016.16.3445

Abstract

We prove that for any $V > 0$ there exists a hyperbolic manifold $M_{V}$ such that $Vol (M_{V}) < 2.03$ and $LinkVol (M_{V}) > V$ . 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:

Let $K$ be a component of a link $L$ in $S^{3}$ . Then “most” slopes on $K$ cannot be completed to a cosmetic surgery on $L$ , unless $K$ becomes a component of a Hopf link.
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.)

Keywords

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

Mathematical Subject Classification 2010

Primary: 57M12, 57M50

References

Publication

Received: 1 October 2015

Revised: 24 February 2016

Accepted: 27 March 2016

Published: 15 December 2016

Authors

Yo’av Rieck
Department of Mathematics University of Arkansas Fayetteville, AR 72701 United States

Yasushi Yamashita
Department of Information and Computer Sciences Nara Women’s University Kitauoya Nishimachi, Nara 630-0586 Japan

Volume 16, issue 6 (2016)