The Noded Schottky Space

We introduce coordinates, given by fixed points, for the marked Schottky space S g of genus g ⩾ 2. With these coordinates, different to the ones given by Bers, Sato and Gerritzen, we can think of the space S g as an open subset of C×C 3g−4 . A partial closure of S g , denoted by NS g and called the marked noded Schottky space of genus g , is considered. Each point in NS g corresponds to a geometrically finite free group of rank g , called a (marked) noded Schottky group of genus g , Conversely, each such group corresponds to a point in NS g . We have that each noded Schottky group of genus g uniformizes a stable Riemann surface of genus g . Moreover, we show that every stable Riemann surface is uniformized by such a group (retrosection theorem with nodes).