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

Abstract

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.

27 pages, 10 figures

References in corpus (1)

• Tameness of hyperbolic 3-manifolds

Cited by in corpus (3)

• Spaces of invariant circular orders of groups
• On laminar groups, Tits alternatives, and convergence group actions on $S^2$
• Characterization of Fuchsian groups as laminar groups & the structure theorem of hyperbolic 2-orbifolds