Existence of ruled wrappings in hyperbolic 3-manifolds

TL;DR

This paper provides an elementary proof for the existence of specific CAT(-1)-surfaces in open hyperbolic 3-manifolds, simplifying key steps in the proof of Marden's Tameness Conjecture.

Contribution

It introduces a new, simplified proof technique for the existence of ruled surfaces, replacing a key lemma in the original proof of Marden's Tameness Conjecture.

Findings

Elementary proof of existence of CAT(-1)-surfaces

Replacement of a key lemma in the original proof

Simplification of the proof of Marden's Tameness Conjecture

Abstract

We present a short elementary proof of an existence theorem of certain CAT(-1)-surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem.