I introduce a new way to look at the group actions on the circle via the number of transverse dense invariant laminations. As a motivational example, we characterize the fuchsian groups in terms of the invariant laminations. Having infinitely many invariant laminations with some additional assumptions on the laminations guarantees that the given group is fuchsian. From the ideas developed in the proof, we can also prove that having three invariant laminations with stronger assumptions gives the same result. The key ingredient is the convergence action. The development of the theory was motivated by Thurston's universal circle construction for tautly foliated 3-manifold groups. It is also related to the lamination approach of the study on complex polynomials.

I will emphasize the dynamical point of view of the theory more.