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Uniformization of conformal involutions on stable Riemann surfaces

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
Uncontrolled Keywords:Riemann surfaces; conformal involutions
Subjects:Sciences > Mathematics > Geometry
ID Code:15719
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