Exceptional hyperbolic 3-manifolds

TL;DR

This paper refines a conjecture on the classification of thin tubed hyperbolic 3-manifolds and establishes a new geometric property related to closed geodesics and embedded tubes.

Contribution

It corrects and completes a previous conjecture and proves a new geometric property about closed geodesics in hyperbolic 3-manifolds.

Findings

Classification of thin tubed hyperbolic 3-manifolds refined

Established a lower bound on the radius of embedded tubes containing closed geodesics

Identified a specific volume threshold for hyperbolic 3-manifolds

Abstract

We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of radius log(3)/2.