Uniformization of conformal involutions on stable Riemann surfaces

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.