Abstract
LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed.
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References
W. Abikoff,Augmented Teichmüller spaces, Bull. Amer. Math. Soc.82 (1976), 333–334.
W. Abikoff,Degenerating families of Riemann surfaces, Ann. of Math. (2)105 (1977), 29–44.
L. V. Ahlfors,Finitely generated Kleinian groups, Amer. J. Math.86 (1964), 413–429; correction, ibid Amer. J. Math.87 (1965), 759.
L. Bers,On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math. (2)91 (1970), 570–600.
L. Bers,Spaces of Kleinian groups, inSeveral Complex Variables, Maryland 1970, Lecture notes in Math.155, 1970, pp. 9–34.
P. Deligne and D. Mumford,The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math.36 (1969), 75–109.
J. Gilman and P. Waterman,Classical two-parabolic T-Schottky groups, J. Anal. Math.98 (2006), 1–42.
R. A. Hidalgo,The noded Schottky space, Proc. London Math. Soc.73 (1996), 385–403.
R. A. Hidalgo,On noded Fuchsian groups, Complex Variables36 (1998), 45–66.
R. A. Hidalgo and B. Maskit,On neoclassical Schottky groups, Trans. Amer. Math. Soc., to appear.
J. James and R. C. Penner,Riemann's moduli space and the symmetric groups, Contemp. Math.150 (1993), 247–290.
L. Keen, B. Maskit and C. Series,Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets, J. Reine Angew. Math.436 (1993), 209–219.
I. Kra,On spaces of Kleinian groups, Comment. Math. Helv.47 (1972), 53–69.
I. Kra and B. Maskit,The deformation space of a Kleinian group, Amer. J. Math.103 (1981), 1065–1102.
I. Kra and B. Maskit,Pinching two component Kleinian groups, Analysis and Topology, World Scientific Press, 1998, pp. 425–465.
O. Lehto,Univalent Functions and Techmüller Spaces, Springer-Verlag, New York, 1987.
A. Marden,Schottky groups and circles, inContributions to Analysis, Academic Press, New York and London, 1974, pp. 273–276.
B. Maskit,Kleinian groups, Springer-Verlag, Berlin, Heidelberg, New York, 1988.
B. Maskit,On boundaries of Teichmüller spaces II, Ann. of Math. (2)91 (1970), 607–639.
B. Maskit,Self-maps on Kleinian groups, Amer. J. of Math. Vol. XCIII3 (1971), 840–856.
B. Maskit,Parabolic elements in Kleinian groups, Annals of Math. (2)117 (1983), 659–668.
R. L. Moore,Concerning upper semi-continuous collection of continua, Trans. Amer. Math. Soc.27 (1925), 412–428.
D. Mumford,Curves and their Jacobians, Univ. Michigan Press, Ann Arbor, 1975.
S. Nag,The Complex Analytic Theory of Teichmüller Spaces, Wiley Interscience, New York, 1988.
K. Ohshika,Geometrically finite Kleinian groups and parabolic elements, Proc. Edinburgh Math. Soc. (2)41 (1998), 141–159.
D. Sullivan,On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics, Ann. of Math. Studies97 (1980), 465–496.
Hiro-o Yamamoto,An example of a non-classical Schottky group, Duke Math. J.63 (1991), 193–197.
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Partially supported by Projects Fondecyt 1030252, 1030373, 1040333, Projects UTFSM 12.05.21, 12.05.23 and by grant of the University of Bergen.
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Hidalgo, R.A., Vasil'ev, A. Noded Teichmüller spaces. J. Anal. Math. 99, 89–107 (2006). https://doi.org/10.1007/BF02789443
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DOI: https://doi.org/10.1007/BF02789443