Revista Matemática Iberoamericana


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Volume 26, Issue 2, 2010, pp. 639–649
DOI: 10.4171/RMI/612

Published online: 2010-08-31

Lowest uniformizations of closed Riemann orbifolds

Rubén A. Hidalgo[1]

(1) Universidad Técnica Federico Santa María, Valparaíso, Chile

A Kleinian group containing a Schottky group as a finite index subgroup is called a Schottky extension group. If $\Omega$ is the region of discontinuity of a Schottky extension group $K$, then the quotient $\Omega/K$ is a closed Riemann orbifold; called a Schottky orbifold. Closed Riemann surfaces are examples of Schottky orbifolds as a consequence of the Retrosection Theorem. Necessary and sufficient conditions for a Riemann orbifold to be a Schottky orbifold are due to M. Reni and B. Zimmermann in terms of the signature of the orbifold. It is well known that the lowest uniformizations of a closed Riemann surface are exactly those for which the Deck group is a Schottky group. In this paper we extend such a result to the class of Schottky orbifolds, that is, we prove that the lowest uniformizations of a Schottky orbifold are exactly those for which the Deck group is a Schottky extension group.

Keywords: Orbifolds, Schottky groups, Kleinian groups

Hidalgo Rubén: Lowest uniformizations of closed Riemann orbifolds. Rev. Mat. Iberoamericana 26 (2010), 639-649. doi: 10.4171/RMI/612