Braids, entropies and fibered 2-fold branched covers of 3-manifolds

TL;DR

This paper investigates the minimal entropy of pseudo-Anosov monodromies in hyperbolic fibered 3-manifolds, showing that for infinitely many manifolds, this entropy scales inversely with the genus of the fiber surface.

Contribution

It establishes the existence of infinitely many 3-manifolds where the minimal entropy of hyperbolic surface bundle covers decreases proportionally to 1/g.

Findings

Existence of infinitely many 3-manifolds with minimal entropy ~ 1/g

Minimal entropy is comparable to the inverse of genus g

Focus on 2-fold branched covers and pseudo-Anosov monodromies

Abstract

It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.