Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-09-01T17:02:59.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Schottky uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms

Published online by Cambridge University Press:  18 May 2009

Rubén A. Hidalgo
Rubén A. Hidalgo
Affiliation:
Departamento de Mathemáticas, UTFSM, Valparaiso, Chile
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let us consider a pair (S, H) consisting of a closed Riemann surface S and an Abelian group H of conformal automorphisms of S. We are interested in finding uniformizations of S, via Schottky groups, which reflect the action of the group H. A Schottky uniformization of a closed Riemann surface S is a triple (Ώ, G, π:Ώ→S) where G is a Schottky group with Ώ as its region ofdiscontinuity and π:Ώ→S is a holomorphic covering with G ascovering group. We look for a Schottky uniformization (Ώ, G, π:Ώ→S) of S such that for each transformation h in H there exists an automorphisms t of Ώ satisfying h ∘ π = π ∘ t.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 17 - 32
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Ahlfors, L. V. and Sario, L., Riemann surfaces (Princeton Univ. Press, 1960).CrossRefGoogle Scholar
2.Bers, L., Automorphic forms for Schottky groups, Adv. in Math. 16 (1975), 332–361.CrossRefGoogle Scholar
3.Chuckrow, V., On Schottky groups with applications to Kleinian groups. Ann. of Math. 88 (1968), 47–61.CrossRefGoogle Scholar
4.Farkas, H. and Kra, I., Riemann surfaces (Springer-Verlag, 1980).CrossRefGoogle Scholar
5.Hidalgo, R., On γ-hyperelliptic Riemann surfaces, Preprintreihe Sonderforschungsbereich 343 “Discrete Structuren in der Mathematik” 91–045.Google Scholar
6.Hidalgo, R., On γ-hyperelliptic Schottky groups, Notas Soc. Mat. de Chile 8 (1989), 27–36.Google Scholar
7.Hidalgo, R., On Schottky groups with automorphisms (Thesis Ph.D. in Mathematics, S.U.N.Y. at Stony Brook, 1991).Google Scholar
8.Hidalgo, R., On Schottky groups with automorphisms, Preprintreihe Sonderforschungsbereich 343 “Discrete Structuren in der Mathematik”, 91–088.Google Scholar
9.Hidalgo, R., Schottky uniformizations of closed Riemann surfaces with the Klein groups as conformal group of symmetries, Preprintreihe Sonderforschungsbereich 343 “Discrete Structuren in der Mathematik”, 91–087.Google Scholar
10.Hidalgo, R., Closed Riemann surfaces with Dihedral groups of Conformal Automorphisms, Preprintreihe Sonderforschungsbereich 343 “Discrete Structuren in der Mathematik”, 91–086.Google Scholar
11.Keen, L., On hyperellptic Schottky groups, Ann. Acad. Sci. Fenn., series A.I. Mathematica 5 (1980), 165–174.CrossRefGoogle Scholar
12.Klein, F., Neue Beiträge zur Riemann'schen Funktionentheorie, Math. Ann. 21 (1883), 141–218.CrossRefGoogle Scholar
13.Koebe, P., Über die Uniformisierung der Algebraischen Kurven II, Math. Ann. 69 (1910), 1–81.CrossRefGoogle Scholar
14.Maskit, B., Kleinian groups (Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, 1988).Google Scholar
15.Maskit, B., A characterization of Schottky groups, J. d'Analyse Math. 19 (1967), 227–230.CrossRefGoogle Scholar
16.Maskit, B., Canonical domains on Riemann surfaces, Proc. of the Amer. Math. Soc. 106 (1989), 713–721.CrossRefGoogle Scholar
17.Maskit, B., On the classification of Kleinian groups I and II, Acta Math. 135 (1975), 249–270 and 138 (1977), 17–42.CrossRefGoogle Scholar