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The Noded Schottky Space
Author(s) -
Hidalgo Rubén A.
Publication year - 1996
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-73.2.385
Subject(s) - riemann surface , genus , mathematics , schottky diode , group (periodic table) , space (punctuation) , rank (graph theory) , point (geometry) , pure mathematics , surface (topology) , combinatorics , topology (electrical circuits) , geometry , physics , computer science , quantum mechanics , zoology , biology , diode , operating system
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).