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

Baik, Hyungryul¹

¹ Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Geometry & topology, Tome 19 (2015) no. 4, pp. 2081-2115.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2081

Classification : 20H10, 37C85, 37E30, 57M60
Keywords: Fuchsian group, lamination, circular order, convergence group

Bibliographie

[1] I Agol, Tameness of hyperbolic $3$–manifolds,

[2] D Calegari, Circular groups, planar groups and the Euler class, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7 (2004) 431

[3] D Calegari, Foliations and the geometry of $3$–manifolds, Oxford University Press (2007)

[4] D Calegari, N M Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149

[5] D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385

[6] J W Cannon, E L Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998) 809

[7] A Casson, D Jungreis, Convergence groups and Seifert fibered $3$–manifolds, Invent. Math. 118 (1994) 441

[8] D Gabai, Convergence groups are Fuchsian groups, Bull. Amer. Math. Soc. 25 (1991) 395

[9] É Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329

[10] J H Hubbard, Teichmüller theory and applications to geometry, topology and dynamics, I: Teichmüller theory, Matrix Editions (2006)

[11] J Manning, The virtual Haken conjecture,

[12] V Markovic, Criterion for Cannon's conjecture, Geom. Funct. Anal. 23 (2013) 1035

[13] R L Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925) 416

[14] W P Thurston, Three-manifolds, foliations and circles, I,

[15] P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1