Abstract
We show how Kerckhoff's results on minima of length functions on Teichmüller space can be used to analyse the possible bending loci of the boundary of the convex hull for quasi-Fuchsian groups near to the Fuchsian locus.
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References
Abikoff, W.: The Real Analytic Theory of Teichmüller Space, Lecture Notes in Maths. 820, Springer, New York, 1976.
Bonahon, F. and Otal, J-P.: Laminations mesurées de plissage des variétés hyperboliques de dimension 3, Preprint, 2001.
Canary, R.D., Epstein, D.B.A. and Green, P.: Notes on notes of Thurston, In: D.B.A. Epstein (ed), Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Notes Ser.111,Campbridge Univ.Press, 1987, pp. 3–92.
Diaz, R. and Series, C.: Examples of pleating varieties for the twice punctured torus, Warwick preprint.
Epstein, D.B.A. and Marden, A.: Convex hulls in hyperbolic space,a theorem of Sullivan,and easured pleated surfaces, In: D.B.A. Epstein (ed), Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Notes Ser.111, Cambridge Univ.Press, 1987, pp. 112–253.
Fahti, A., Laudenbach, P. and Poénaru, V.: Travaux de Thurston sur les surfaces, Astérisque 66–67, SociétéMathéatique de France, 1979.
Fenchel, W.: Elementary Geometry in Hyperbolic Space, De Gruyter Stud. Math. 11, De Gruyter, New York, 1989.
Imayoshi, Y. and Taniguchi, M.: An Introduction to Teichmüller spaces, Springer, Tokyo 1992.
Keen, L. and Series, C.: Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32(4) (1993), 719–749.
Keen, L. and Series, C.: Continuity of convex hull boundaries, Pacific J. Math. 168(1) (1995), 183–206.
Keen, L. and Series, C.: How to bend pairs of punctured tori. In: J. Dodziuk and L. Keen (eds), Lipa's Legacy, Contep. Math. 211, Aer. Math. Soc. Providence, 1997, pp.359–388.
Keen, L. and Series, C.: Pleating invariants for punctured torus groups,Warwick preprint, 1998.
Kerckhoff, S.: The Nielsen realization problem, Ann. of Math. 117(2) (1983), 235–265.
Kerckhoff, S.: Earthquakes are analytic, Comment. Math. Helv. 60 (1985), 17–30.
Kerckhoff, S.: Lines of Minima in Teichmüller space, Duke Math. J. 65 (1992), 187–213.
Kourouniotis, C.: Complex length coordinates for quasi-Fuchsian groups, Mathematika, 41(1) (1994), 173–188.
Marden, A.: The geometry of ¢nitely generated Kleinian groups, Ann. Math. 99 (1974), 607–639.
Otal, J.P.: Le théorème d' hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235, SociétéMath.France,(1996).
Otal,J.P.: Sur le coeur convexe d' une variétéhyperbolique de dimension 3,Preprint.
Papadopoulos, A.: Geometric intersection functions and Hamiltonian flows on the space of Measured Foliations on a surface, Pacific J. Math. 124 (1986), 375–402.
Penner, R.C. with Harer, J.: Combinatorics of Train Tracks, Ann. Math. Stud. 125, Princeton Univ. Press, 1992.
Series, C.: An extension of Wolpert's derivative formula, Pacific J. Math. (2000), 223–239.
Series, C.: Lectures on pleating coordinates for once punctured tori, In: Hyperbolic Spaces and Related Topics, RIMS Kokyuroku 1104, Kyoto, 1999,pp. 30–108.
Tan, S.P.: Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, Internat. J. Math. 5(2),(1994), 239–251.
Thurston, W.: Earthquakes in two-di ensional hyperbolic geometry, In: D.B.A. Epstein (ed), Low-dimensional Topology and Kleinian Groups, London Math. Soc., Lecture Notes Ser.112,Cambridge Univ. Press, 1987, pp. 91–112.
Thurston, W.: Geometry and topology of three-manifolds, Princeton lecture notes, 1979.
Wolpert, S.: The Fenchel Nielsen deformation, Ann. Math. 115(3) (1982), 501–528.
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Series, C. On Kerckhoff Minima and Pleating Loci for Quasi-Fuchsian Groups. Geometriae Dedicata 88, 211–237 (2001). https://doi.org/10.1023/A:1013171204254
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DOI: https://doi.org/10.1023/A:1013171204254