Thurston's Bounded image theorem

TL;DR

This paper provides the first published proof of Thurston's bounded image theorem, a crucial step in the uniformisation of Haken manifolds, utilizing recent advances in Kleinian group theory.

Contribution

It offers the first known proof of Thurston's original bounded image theorem, extending the understanding of Kleinian groups and 3-manifold uniformisation.

Findings

Proof of Thurston's bounded image theorem established

Relies on recent developments in Kleinian group theory

Advances the understanding of Haken manifold uniformisation

Abstract

Thurston's bounded image theorem is one of the key steps in his proof of the uniformisation theorem for Haken manifolds. Thurston never published its proof, and no proof has been known up to today, although a proof of its weaker version, called the bounded orbit theorem is known. In this paper, we give a proof of the original bounded image theorem, relying on recent development of Kleinian group theory.