Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle

Abstract

We propose a program to study groups acting faithfully on $S^{\mathfrak{1}}$ in terms of numbers of pairwise transverse dense invariant laminations. We give some examples of groups that admit a small number of invariant laminations as an introduction to such groups. The main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on $S^{\mathfrak{1}}$ is conjugate to a Fuchsian group if and only if it admits three very full laminations with a variation on the transversality condition. Some partial results toward a similar characterization of hyperbolic $\mathfrak{3}$–manifold groups that fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated $\mathfrak{3}$–manifolds developed by Thurston, Calegari and Dunfield.