Abstract
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.
Citation
Yo’av Rieck. Yasushi Yamashita. "The link volume of $3$–manifolds." Algebr. Geom. Topol. 13 (2) 927 - 958, 2013. https://doi.org/10.2140/agt.2013.13.927
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