Laminations On The Circle And Hyperbolic Geometry

Abstract

We develop a program studying group actions on the circle with dense invariant laminations. Such actions naturally and frequently arise in the study of hyperbolic manifolds. We use the convergence group theorem to prove that for a discrete torsion-free group acting on the circle, it is topologically conjugate to a ¨ Mobius group if and only if it admits three very-full invariant laminations with a certain transversality condition. We also discuss the case when a group acts on the circle with two very-full invariant laminations with disjoint endpoints. We show that such a group shares many interesting features with the fundamental group of a hyperbolic 3-manifold which fibers over the circle.