Weil-Petersson translation length and manifolds with many fibered fillings

TL;DR

This paper establishes a link between Weil-Petersson translation lengths of pseudo-Anosov mapping classes and the structure of 3-manifolds with many fibered fillings, identifying finite sets of curves and resulting hyperbolic manifolds.

Contribution

It proves that bounded normalized Weil-Petersson translation length implies a finite set of curves leading to a finite list of hyperbolic 3-manifolds after drilling.

Findings

Finite set of transverse, level curves in mapping tori with bounded translation length

Drilling these curves yields a finite collection of hyperbolic 3-manifolds

New estimates for Weil-Petersson translation length of compositions and powers

Abstract

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.