The structure of extended function groups
Conformal (respectively, anticonformal) automorphisms of the Riemann sphere are provided by the Möbius (respectively, extended Möbius) transformations. A Kleinian group (respectively, an extended Kleinian group) is a discrete group of Möbius transformations (respectively, a discrete group of Möbius and extended Möbius transformations, necessarily containing extended ones).
A function group (respectively, an extended function group) is a finitely generated Kleinian group (respectively, a finitely generated extended Kleinian group) with an invariant connected component of its region of discontinuity.
A structural decomposition of function groups, in terms of the Klein- Maskit combination theorems, was provided by Maskit in the middle of the 70’s. One should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. The aim of this paper is to state and provide a proof of such a decomposition structural picture.
L. V. Ahlfors, “Finitely generated Kleinian groups”, Amer. J. of Math., vol. 86, pp. 413–429, 1964.
L. V. Ahlfors, “Correction to “Finitely generated Kleinian groups”. Amer. J. Math., vol. 87, p. 759, 1965.
A. Haas, “Linearization and mappings onto pseudocircle domains”, Trans. Amer. Math. Soc., vol. 282, no. 1, pp. 415–429, 1984.
R. A. Hidalgo and B. Maskit, “A Note on the Lifting of Automorphisms” in Geometry of Riemann Surfaces, Lecture Notes of the London Mathematics Society, vol. 368, New York: Cambridge University Press, 2009, pp. 260–267.
B. Maskit, “A theorem on planar covering surfaces with applications to 3-manifolds”, Ann. of Math. vol. 81, no. 2, pp. 341–355, 1965.
B. Maskit, “Construction of Kleinian groups,” in Proceedings of the Conference on Complex Analysis, Minneapolis: Springer-Verlag, 1965, pp. 281–296.
B. Maskit, “On Klein’s combination theorem III”, in Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies, vol. 66, Princeton: Princeton Univ. Press, 1971, pp. 297–316;
B. Maskit, “On Klein’s combination theorem. IV”, Trans. Amer. Math. Soc., vol. 336, no. 1, pp. 265–294, 1993.
B. Maskit, “Decomposition of certain Kleinian groups”, Acta Math., vol. 130, pp. 243–263, 1973.
B. Maskit, “On the classification of Kleinian Groups I. Koebe groups”, Acta Math., vol. 135, pp. 249–270, 1975.
B. Maskit, “On the classification of Kleinian Groups II. Signatures”, Acta Math., vol. 138, no. 1-2, pp. 17–42, 1976.
B. Maskit, “On extended quasifuchsian groups”, Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 15, no. 1, pp. 53–64, 1990.
B. Maskit, Kleinian Groups, Grundlehren der Mathematischen Wissenschaften, Berlin: Springer-Verlag, 1988.
K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Mathe- matical Monographs, New York: The Clarendon Press, Oxford University Press, 1998.
W. Meeks and S.-T. Yau, “The equivariant Dehn’s lemma and loop theorem”, Comment. Math. Helvetici, vol. 56, no. 2, pp. 225–239, 1981.
W. Meeks and S.-T. Yau, “The equivariant loop theorem for three-dimensional manifolds and a review of the existence theorems for minimal surfaces”, in The Smith conjecture (New York, 1979), Pure Appl. Math., vol. 112, Orlando, FL: Academic Press, 1984, pp.153–163.
H. Poincaré, “Mémoire: Les groupes Kleinéens”, Acta Math., vol. 3, pp. 193–294, 1982.
H. Poincaré, “Sur l’uniformisation des fonctions analytiques”, Acta Math., vol. 31, no. 1, pp. 1–63, 1908.
A. Selberg, “On discontinuous groups in higher dimensional symmetric spaces” in Contributions to function theory, Internat. Colloq. Function Theory, Bombay: Tata Institute of Fundamental Research, 1960, pp. 147–164.
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