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

Baik, Hyungryul

doi:10.2140/gt.2015.19.2081

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

Hyungryul Baik

Geometry & Topology 19 (2015) 2081–2115

DOI: 10.2140/gt.2015.19.2081

Abstract

We propose a program to study groups acting faithfully on $S^{1}$ 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 $S^{1}$ 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 $3$ –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)

Keywords

Fuchsian group, lamination, circular order, convergence group

Mathematical Subject Classification 2010

Primary: 20H10, 37C85

Secondary: 37E30, 57M60

References

Publication

Received: 9 September 2013

Revised: 12 July 2014

Accepted: 16 September 2014

Published: 29 July 2015

Proposed: Benson Farb

Seconded: Walter Neumann, Danny Calegari

Authors

Hyungryul Baik
Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany

Volume 19, issue 4 (2015)