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The link volume of
$3$–manifolds
Yo’av Rieck and Yasushi Yamashita |
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Algebraic & Geometric Topology 13 (2013) 927–958
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Abstract |
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We view closed orientable –manifolds as covers of branched over hyperbolic links. To a cover , of degree and branched over a hyperbolic link , we assign the complexity . We define an invariant of –manifolds, called the link volume and denoted by , that assigns to a 3-manifold the infimum of the complexities of all possible covers , where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently can be represented as a cover of . We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold , is less than . We prove a structure theorem that is similar to (and uses) the celebrated theorem of Jørgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic –manifold is much bigger than its volume. Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves. |
Keywords
$3$–manifolds, hyperbolic volume, branched covers, knots
and links
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Mathematical Subject Classification 2010
Primary: 57M12, 57M50
Secondary: 57M27
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References |
Publication
Received: 7 May 2012
Revised: 21 September 2012
Accepted: 26 October 2012
Published: 5 April 2013
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Authors |
| Yo’av Rieck | |
| Department of Mathematics University of Arkansas Fayetteville, AR 72701 USA |
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| Yasushi Yamashita | |
| Department of Information and
Computer Sciences Nara Women’s University Nara 630-8506 Japan |
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