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

Abstract

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.