On hyperbolic surface bundles over the circle as branched double covers of the 3-sphere

Hirose, Susumu; Kin, Eiko

doi:10.1090/proc/14825

On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere
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by Susumu Hirose and Eiko Kin

Proc. Amer. Math. Soc. 148 (2020), 1805-1814

DOI: https://doi.org/10.1090/proc/14825

Published electronically: January 15, 2020

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Abstract:

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma’s theorem when closed $3$-manifolds are branched double covers of the $3$-sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set.

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