On Teichm\"uller spaces of Koebe groups

Abstract

In this paper we parametrize the Teichm\"uller spaces of constructible Koebe groups, that is Kleinian group that arise as covering of $2-$orbifolds determined by certain normal subgroups of their fundamental groups. We also study the covering spaces of the Teichm\"uller spaces of those Koebe groups. Finally we prove an isomorphisms theorem similar to the Bers-Greenberg theorem for Fuchsian groups. Our method yields a technique to compute explictly generators of Koebe groups, possibly by programming a computer.