A sufficiently complicated noded Schottky group of rank three

  • Rubén A. Hidalgo Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile.
Keywords: Riemann surfaces, Schottky groups


In 1974, Marden proved the existence of non-classical Schottky groups by a theoretical and non-constructive argument. Explicit examples are only known in rank two; the first one by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to construct non-classical Schottky groups in any rank. The method assumes the knowledge of certain algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups. The aim of this paper is to provide explicitly a sufficiently complicated noded Schottky group of rank three and explain how to use it to construct explicit non-classical Schottky groups.


A. I. Bobenko. Schottky uniformization and finite-gap integration, Soviet Math. Dokl. 36 No. 1 (1988), 38–42 (transl. from Russian: Dokl. Akad. Nauk SSSR, 295, No.2 (1987)).

J. Button. All Fuchsian Schottky groups are classical Schottky groups. Geometry & Topology Monographs. Volume 1: The Epstein birthday schrift (1998), 117–125.

V. Chuckrow. Schottky groups and limits of Kleinian groups. Bull. Amer. Math. Soc. 73 No. 1 (1967), 139–141.

R. A. Hidalgo. The noded Schottky space. London Math. Soc. 73 (1996), 385–403

R. A . Hidalgo. Noded Fuchsian groups. Complex Variables 36 (1998), 45–66.

R. A. Hidalgo. Towards a proof of the classical Schottky uniformization conjecture https://arxiv.org/pdf/1709.09515.pdf

R. A. Hidalgo and B. Maskit. On neoclassical Schottky groups. Trans. of the Amer. Math. Soc. 358 (2006), 4765-4792.

Y. Hou. On smooth moduli space of Riemann surfaces. (2016). https://arxiv.org/pdf/1610.03132.pdf

Y. Hou. The classification of Kleinian groups of Hausdorff dimensions at most one. (2016). https://arxiv.org/pdf/1610.03046.pdf

T. Jørgensen and A. Marden. Algebraic and geometric convergence of Kleinian groups. Math. Scand. 66 (1990), 47–72.

P. Koebe. Uber die Uniformisierung der Algebraischen Kurven II. Math. Ann. 69 (1910), 1–81.

I. Kra and B. Maskit. Pinched two component Kleinian groups. In Analysis and Topology, pages 425–465. World Scientific Press, 1998.

C. McMullen. Complex Dynamics and Renormalization. Annals of Mathematical Studies 135, Princeton University Press, (1984).

A. Marden. Schottky groups and circles. In Contributions to Analysis (a collection of papers dedicated to Lipman Bers), 273–278, Academic Press, 1974.

B. Maskit. On the classification of Kleinian Groups I. Koebe groups. Acta Math. 135 (1975), 249–270.

B. Maskit. On the classification of Kleinian Groups II. Signatures. Acta Math. 138 (1976), 17–42.

B. Maskit. Remarks on m-symmetric Riemann surfaces. Contemporary Math. 211 (1997), 433–445.

B. Maskit. On Klein’s combination theorem IV. Trans. Amer. Math. Soc. 336 (1993),265–294.

B. Maskit. On free Kleinian groups. Duke Math. J. 48 (1981),755–765.

B. Maskit. Parabolic elements in Kleinian groups. Annals of Math. 117 (1983), 659–668.

N. Purzitsky. Two-Generator Discrete Free Products. Math. Z. 126 (1972), 209–223.

H. Sato. On a paper of Zarrow. Duke Math. J. 57 (1988), 205–209.

M. Seppälä. Myrberg’s numerical uniformization of hyperelliptic curves. Ann. Acad. Scie. Fenn. Math. 29 (2004), 3–20.

J. P. Williams. Classical and non-classical Schottky groups. Doctoral Thesis, University of Southampton, School of Mathematics (2009), 125pp.

Hiro-o Yamamoto. Sqeezing deformations in Schottky spaces. J. Math. Soc. Japan 31 (1979), 227–243.

Hiro-o Yamamoto. An example of a non-classical Schottky group. Duke Math. J. 63 (1991), 193–197.

R. Zarrow. Classical and non-classical Schottky groups. Duke Math. J. 42 (1975), 717–724.

How to Cite
Hidalgo, R. A. (2020). A sufficiently complicated noded Schottky group of rank three. CUBO, A Mathematical Journal, 22(1), 39–53. https://doi.org/10.4067/S0719-06462020000100039