Fuchsian Groups, Circularly Ordered Groups, and Dense Invariant Laminations on the Circle

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We propose a program to study groups acting faithfully on S¹ in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S¹ is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3-manifolds developed by Thurston and Calegari-Dunfield.