#### Volume 19, issue 4 (2015)

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Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle

### Hyungryul Baik

Geometry & Topology 19 (2015) 2081–2115
##### Abstract

We propose a program to study groups acting faithfully on ${S}^{\mathfrak{1}}$ in terms of numbers of pairwise transverse dense invariant laminations. We give some examples of groups that admit a small number of invariant laminations as an introduction to such groups. The main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on ${S}^{\mathfrak{1}}$ is conjugate to a Fuchsian group if and only if it admits three very full laminations with a variation on the transversality condition. Some partial results toward a similar characterization of hyperbolic $\mathfrak{3}$–manifold groups that fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated $\mathfrak{3}$–manifolds developed by Thurston, Calegari and Dunfield.

 This paper is dedicated to the memory of William Thurston (1946–2012)
##### Keywords
Fuchsian group, lamination, circular order, convergence group
##### Mathematical Subject Classification 2010
Primary: 20H10, 37C85
Secondary: 37E30, 57M60