Existence of ruled wrappings in hyperbolic $3$–manifolds

Abstract

We present a short elementary proof of an existence theorem of certain $CAT\left(-1\right)$–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. Finally, we will give a short proof of the conjecture along their ideas.