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

Hyungryul Baik

Geometry & Topology 19 (2015) 2081–2115

We propose a program to study groups acting faithfully on S1 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 S1 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 3–manifold groups that fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3–manifolds developed by Thurston, Calegari and Dunfield.

This paper is dedicated to the memory of William Thurston (1946–2012)

Fuchsian group, lamination, circular order, convergence group
Mathematical Subject Classification 2010
Primary: 20H10, 37C85
Secondary: 37E30, 57M60
Received: 9 September 2013
Revised: 12 July 2014
Accepted: 16 September 2014
Published: 29 July 2015
Proposed: Benson Farb
Seconded: Walter Neumann, Danny Calegari
Hyungryul Baik
Mathematisches Institut
Rheinische Friedrich-Wilhelms-Universität Bonn
Endenicher Allee 60
D-53115 Bonn