Laminations on the circle and convergence groups

Abstract

A discrete group of circle homeomorphisms is a Fuchsian group if and only if it is a convergence group (this is due to Tukia, Casson-Jungreis, Gabai, ...). We show that the convergence property can also be characterized in terms of invariant laminations on the circle, so this gives a new characterization of Fuchsian groups. We also discuss what can be said about fibered hyperbolic 3-manifold groups. The main motivation of the work is Thurston´s universal circle theory.