Transactions of the American Mathematical Society

Author(s): Raquel Díaz; Caroline Series
Journal: Trans. Amer. Math. Soc. 356 (2004), 621-658.
MSC (2000): Primary 30F40, 20H10, 32G15
Posted: September 22, 2003
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Abstract: 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.

1.: F. Bonahon and J-P. Otal.
Laminations mesurées de plissage des variétés hyperboliques de dimension 3,
preprint, 2001.
2.: 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. MR 89e:57008
3.: 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.
4.: 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. MR 89c:52014
5.: A. Fahti, P. Laudenbach, and V. Poénaru.
Travaux de Thurston sur les surfaces, Astérisque 66-67.
Société Mathématique de France, 1979. MR 82m:57003
6.: F. Gardiner and L. Keen.
Holomorphic motions and quasi-Fuchsian manifolds,
Contemp. Math. 240, 159-173, 1999. MR 2000k:30023
7.: P.A. Griffiths and J. Harris.
Principles of Algebraic geometry.
Wiley, 1978. MR 80b:14001
8.: R.D. Horowitz.
Characters of free groups represented in the two dimensional special linear group,
Comm. Pure Appl. Math. 25, 635-649, 1972. MR 47:3542
9.: L. Keen and C. Series.
Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori,
Topology 32, 719-749, 1993. MR 95g:32030
10.: L. Keen and C. Series.
Continuity of convex hull boundaries,
Pacific J. Math. 168(1), 183-206, 1995. MR 96d:30055
11.: 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. MR 98m:30063
12.: L. Keen and C. Series.
Pleating invariants for punctured torus groups,
Topology, 2003.
13.: L. Keen and C. Series.
The Riley slice of Schottky space,
Proceedings of the London Mathematical Society 3, 72-90, 1994. MR 95j:32033
14.: S. Kerckhoff.
The Nielsen realization problem,
Ann. of Math. 117, 235-265, 1983. MR 85e:32029
15.: S. Kerckhoff.
Lines of Minima in Teichmüller space,
Duke Math J. 65, 187-213, 1992. MR 93b:32027
16.: Y. Komori and C. Series.
Pleating coordinates for the Earle embedding,
Ann. de la Fac. des Sciences de Toulouse, Vol. X, 69-105, 2001.
17.: C. Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41(1), 173-188, 1994. MR 96g:30079
18.: I. Kra.
On lifting Kleinian groups to $SL(2,\mathbb C)$ .
In: ``Differential Geometry and Complex Analysis", I. Chavel and H. Farkas, editors, 181-193. Springer-Verlag, 1985. MR 86h:30078
19.: C. Series.
Lectures on pleating coordinates for once punctured tori,
In Hyperbolic Spaces and Related topics, RIMS Kokyuroku 1104, Kyoto, 30-108, 1999. MR 2000m:57019
20.: C. Series.
On Kerckhoff Minima and Pleating Loci for quasi-Fuchsian Groups,
Geometriae Dedicata 88, 211-237, 2001. MR 2002j:30066
21.: C. Series, Limits of quasifuchsian groups with small bending, preprint 2002. arXiv:mathGT/0209190
22.: S. P. Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, International J. Math. 5(2), 239-251, 1994. MR 94m:32030
23.: 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. MR 88m:57015
24.: W. Thurston.
Three-dimensional Geometry and Topology, Vol.1.
Princeton U.P., 1997. MR 97m:57016

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Raquel Díaz
Affiliation: Departamento de Geometría y Topología, Facultad Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: radiaz@mat.ucm.es

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