Schottky uniformizations of Automorphisms of Riemann surfaces

Original · EN

It is well known that the collection of uniformizations of a closed Riemann surface S is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples (Ω,Γ,P:Ω→ S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω→ S is a regular holomorphic cover map with Γ as its deck group. Let τ:S → S be a conformal (respectively, anticonformal) automorphism of S of finite order n, and let (Ω,Γ,P:Ω→ S) be a Schottky uniformization of S. Assume that τ lifts with respect to the previous Schottky uniformization, that is, there exists a Möbius (respectively, extended Möbius) transformation κ, keeping Ω invariant, with P ∘ κ=τ∘ P. The Kleinian (respectively, extended Kleinian) group K=< Γ, κ> contains Γ as a finite index normal subgroup and K/Γ Zₙ. We provide a structural picture of K in terms of the Klein-Maskit's combination theorems and some basic groups. Some consequences are (i) the determination of the number of topologically different types of such groups (fixed n and the rank of the Schottky normal subgroup) and (ii) for n prime, the number of normal Schottky normal subgroups, up to conjugacy, that K has.