Complutense University Library

Examples of pleating varieties for twice punctured tori


Díaz Sánchez, Raquel and Series, Caroline (2004) Examples of pleating varieties for twice punctured tori. Transactions of the American Mathematical Society, 356 (2). pp. 621-658. ISSN 0002-9947

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


We give an explicit description of some pleating varieties (sets with a fixed set of bending lines in the convex hull boundary) in the quasi-Fuchsian space of the twice punctured torus. In accordance with a conjecture of the second author, we show that their closures intersect Fuchsian space in the simplices of minima introduced by Kerckhoff. All computations are done using complex Fenchel-Nielsen coordinates for quasi-Fuchsian space referred to a maximal system of curves.

Item Type:Article
Uncontrolled Keywords:Kleinian groups; Fuchsian groups and their generalizations
Subjects:Sciences > Mathematics > Geometry
ID Code:15711

F. Bonahon and J-P. Otal. Laminations mesurées de plissage des variétés hyperboliques de dimension 3, preprint, 2001.

R. D. Canary, D. B. A. Epstein and P. Green. Notes on notes of Thurston. In D. B. A. Epstein, editor, "Analytical and Geometric Aspects of Hyperbolic Space", LMS Lecture Notes 111, 3–92. Cambridge University Press, 1987.

R. Díaz and C. Series. Limits of lines of minima in Thurston's boundary of Teichmüller space, Algebraic and Geometric Topology 3, 207–234, 2003.

D. B. A. Epstein and A. Marden. Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In D. B. A. Epstein, editor, "Analytical and Geometric Aspects of Hyperbolic Space", LMS Lecture Notes 111, 112–253. Cambridge University Press, 1987.

A. Fahti, P. Laudenbach, and V. Poénaru. Travaux de Thurston sur les surfaces, Astérisque 66--67. Société Mathématique de France, 1979.

F. Gardiner and L. Keen. Holomorphic motions and quasi-Fuchsian manifolds, Contemp. Math. 240, 159–173, 1999

P.A. Griffiths and J. Harris. Principles of Algebraic geometry. Wiley, 1978.

R.D. Horowitz. Characters of free groups represented in the two dimensional special linear group, Comm. Pure Appl. Math. 25, 635–649, 1972.

L. Keen and C. Series. Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32, 719–749, 1993.

1L. Keen and C. Series. Continuity of convex hull boundaries, Pacific J. Math. 168(1), 183–206, 1995.

L. Keen and C. Series. How to bend pairs of punctured tori. In J. Dodziuk and L. Keen, editors, "Lipa's Legacy", Contemp. Math. 211, 359–388, 1997

L. Keen and C. Series. Pleating invariants for punctured torus groups, Topology, 2003.

L. Keen and C. Series. The Riley slice of Schottky space, Proceedings of the London Mathematical Society 3, 72–90, 1994.

S. Kerckhoff. The Nielsen realization problem, Ann. of Math. 117, 235–265, 1983.

S. Kerckhoff. Lines of Minima in Teichmüller space, Duke Math J. 65, 187–213, 1992.

Y. Komori and C. Series. Pleating coordinates for the Earle embedding, Ann. de la Fac. des Sciences de Toulouse, Vol. X, 69–105, 2001

C. Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41(1), 173–188, 1994

I. Kra. On lifting Kleinian groups to SL(2,C) . In: "Differential Geometry and Complex Analysis", I. Chavel and H. Farkas, editors, 181–193. Springer-Verlag, 1985.

C. Series. Lectures on pleating coordinates for once punctured tori, In Hyperbolic Spaces and Related topics, RIMS Kokyuroku 1104, Kyoto, 30–108, 1999.

C. Series. On Kerckhoff Minima and Pleating Loci for quasi-Fuchsian Groups, Geometriae Dedicata 88, 211–237, 2001.

C. Series, Limits of quasifuchsian groups with small bending, preprint 2002. arXiv:mathGT/0209190

S. P. Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, International J. Math. 5(2), 239–251, 1994.

W. Thurston. Earthquakes in two-dimensional hyperbolic geometry. In D. B. A. Epstein, editor, "Low-dimensional Topology and Kleinian Groups", LMS Lecture Notes 112, 91–112. Cambridge University Press, 1987

W. Thurston. Three-dimensional Geometry and Topology, Vol. 1. Princeton U.P., 1997.

Deposited On:21 Jun 2012 09:16
Last Modified:06 Feb 2014 10:30

Repository Staff Only: item control page