Revista Matemática Iberoamericana

Lowest uniformizations of closed Riemann orbifolds

Rubén A. Hidalgo

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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.

Article information

Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 639-649.

First available in Project Euclid: 4 June 2010

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Zentralblatt MATH identifier

Primary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 30F40: Kleinian groups [See also 20H10]

orbifolds Schottky groups Kleinian groups


Hidalgo, Rubén A. Lowest uniformizations of closed Riemann orbifolds. Rev. Mat. Iberoamericana 26 (2010), no. 2, 639--649.

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