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

TL;DR

This paper investigates hyperbolic surface bundles over the circle as branched double covers of the 3-sphere, analyzing their entropy behavior and providing a new construction method linking braids and monodromies.

Contribution

It demonstrates that the minimal entropy of such bundles scales like 1/g and offers an alternative construction connecting braids to monodromies in branched covers.

Findings

Minimal entropy scales as 1/g for hyperbolic surface bundles

Provides a new construction linking braids and monodromies

Shows surface bundles can be obtained as branched double covers over links

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.