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

TL;DR

This paper characterizes Fuchsian groups acting on the circle through the existence of three specific dense invariant laminations, linking geometric group actions to lamination structures and extending to hyperbolic 3-manifold groups.

Contribution

It provides a new characterization of Fuchsian groups via invariant laminations and explores partial results for hyperbolic 3-manifold groups, connecting circle actions to 3-manifold topology.

Findings

Fuchsian groups are characterized by three very-full laminations with transversality.

Groups conjugate to Fuchsian groups admit specific lamination configurations.

Partial results relate hyperbolic 3-manifold groups to lamination structures.

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.