Díaz Sánchez, Raquel and Garijo, Ignacio and Hidalgo, Rubén A.
(2011)
*Uniformization of conformal involutions on stable Riemann surfaces.*
Israel Journal of Mathematics , 186
(1).
pp. 297-331.
ISSN 0021-2172

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## Abstract

Let S be a closed Riemann surface of genus g. It is well known that there are Schottky groups producing uniformizations of S (Retrosection Theorem). Moreover, if τ: S → S is a conformal involution, it is also known that there is a Kleinian group K containing, as an index two subgroup, a Schottky group G that uniformizes S and so that K/G induces the cyclic group 〈τ〉. Let us now assume S is a stable Riemann surface and τ: S → S is a conformal involution. Again, it is known that S can be uniformized by a suitable noded Schottky group, but it is not known whether or not there is a Kleinian group K, containing a noded Schottky group G of index two, so that G uniformizes S and K/G induces 〈τ〉. In this paper we discuss this existence problem and provide some partial answers: (1) a complete positive answer for genus g ≤ 2 and for the case that S/〈τ〉 is of genus zero; (2) the existence of a Kleinian group K uniformizing the quotient stable Riemann orbifold S/〈τ〉. Applications to handlebodies with orientation-preserving involutions are also provided.

Item Type: | Article |
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Uncontrolled Keywords: | Riemann surfaces; conformal involutions |

Subjects: | Sciences > Mathematics > Geometry |

ID Code: | 15719 |

References: | I. Agol, Tameness of hyperbolic 3-manifolds, arXiv:math.GT/0405568 (2004). L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Annals of Mathematics 72 (1960), 385–404. L. Ahlfors and L. Sario, Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1960. L. Bers, Automorphic forms for Schottky groups, Advances in Mathematics 16 (1975), 332–361. D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, Journal of the American Mathematical Society 19 (2006), 385–446. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publications Mathématiques. Institut de Hautes Études Scientifiques 36 (1969), 75–110. P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus, Publications Mathématiques. Institut de Hautes Études Scientifiques 36 (1970), 75–109. R. Dáz, I. Garijo, G. Gromadzki and R. A. Hidalgo, Structure of Whittaker groups and applications to conformal involutions on handlebodies, Topology and its Applications 157 (2010), 2347–2361. L. Gerritzen and F. Herrlich, The extended Schottky space, Journal für die Reine und Angewandte Mathematik 389 (1988), 190–208. R. A. Hidalgo, On Γ-hyperelliptic Schottky groups, Notas de la Sociedad de Matematica de Chile 8 (1989), 27–36. R. A. Hidalgo, The noded Schottky space, London Mathematical Society (3) 73 (1996), 385–403. R. A. Hidalgo, Noded Fuchsian groups, Complex Variables 36 (1998), 45–66. R. A. Hidalgo, Cyclic extensions of Schottky uniformizations, Annales Academi Scientiarium Fennicae 29 (2004), 329–344. R. A. Hidalgo, Schottky uniformizations of automorphisms of Riemann surfaces, preprint. R. A. Hidalgo and B. Maskit, On neoclassical Schottky groups, Transactions of the American Mathematical Society 358 (2006), 4765–4792. J. Igusa. Arithmetic variety of moduli for genus two, Annals of Mathematics 72 (1960), 612–649. L. Keen, On hyperelliptic Schottky groups, Annales Academi Scientiarium Fennicea series A.I. Mathematica 5 (1980), 165–174. P. Koebe, Über die Uniformisierung der Algebraischen Kurven II, Mathematische Annalen 69 (1910), 1–81. I. Kra and B. Maskit, Pinched two component Kleinian groups, in Analysis and Topology, World Scientific Press, River Edge, NJ, 1998, pp. 425–465. B. Maskit, A characterization of Schottky groups, Journal d’Analyse Mathématique 19 (1967), 227–230. B. Maskit, On free Kleinian groups, Duke Mathematical Journal 48 (1981), 755–765. B. Maskit, Parabolic elements in Kleinian groups, Annals of Mathematics 117 (1983), 659–668. B. Maskit, Kleinian Groups, G.M.W. 287, Springer-Verlag, Berlin, 1988. B. Maskit, On Klein’s combination theorem IV, Transactions of the American Mathematical Society 336 (1993), 265–294. S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Wiley, New York, 1988. H. E. Rauch, The singularities of the modulus space, American Mathematical Society. Bulletin 68 (1962), 390–394. S. Wolpert, The geometry of the moduli space of Riemann surfaces, Bulletin (New Series) of the American Mathematical Society 11 (1984), 189–191. |

Deposited On: | 21 Jun 2012 11:23 |

Last Modified: | 06 Feb 2014 10:30 |

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