  We propose a program to study groups acting faithfully on S^1 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 S^1 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.
