Effective hyperbolization of 3-manifolds

Abstract

After a brief survey of the study of low-dimensional manifolds -- spaces locally modelled on the topological space R^n for n<5 -- we will discuss the idea of geometrization. The latter refers to a program, initiated by Thurston, of studying 3-dimensional manifolds by means of eight geometric model spaces, which include the Euclidean space and hyperbolic space.

It turns out that most 3-dimensional manifolds carry a hyperbolic structure, i.e. they are locally modelled on hyperbolic space. A precise version of this statement was proven by Maher invoking topological work of Hempel and Perelman's celebrated analysis results on Ricci-flow. We will discuss an approach to hyperbolization of most 3-dimensional manifolds that circumvents invoking the Ricci flow. As an upshot, with this approach one can control many geometric quantities such as injectivity radius, diameter, and volume of hyperbolic structures. We will provide ``hands-on'' examples of 3-manifolds that can be hyperbolized with this scheme.

Based on joint work in progress with A. Sisto and G. Viaggi.