Laminar free hyperbolic 3-manifolds

Sergio R. Fenley

We analyse the existence question for essential laminations in 3-manifolds. The purpose of the article is to prove that there are infinitely many closed hyperbolic $3$-manifolds which do not admit essential laminations. This gives a definitive negative answer to a fundamental question posed by Gabai and Oertel when they introduced essential laminations. The proof is obtained by analysing certain group actions on trees and showing that certain 3-manifold groups only have trivial actions on trees. There are corollaries concerning the existence question for Reebless foliations and pseudo-Anosov flows.