Automorphism Groups of Compact Bordered Klein Surfaces A Combinatori
This research monograph provides a self-contained approach to the problem of determining the conditions under which a compact bordered Klein surface S and a finite group G exist, such that G acts as a group of automorphisms in S. The cases dealt with here
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1439 Emilio Bujalance Jose J. Etayo Jose M. Gamboa Grzegorz Gromadzki
Automorphism Groups of Compact Bordered Klein Surfaces A Combinatorial Approach
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Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and F. Takens
1439 Emilio Bujalance Jose J. Etayo Jose M. Gamboa Grzegorz Gromadzki
Automorphism Groups of Compact Bordered Klein Surfaces A Combinatorial Approach
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Authors
Emilio Bujalance Dpto. de Maternaticas Fundamentales Facultad de Ciencias Universidad a Distancia (UNED) 28040 Madrid, Spain Jose Javier Etayo Jose Manuel Gamboa Dpto. de Algebra Facultad de Matematicas Universidad Complutense 28040 Madrid, Spain Grzegorz Gromadzki Institute of Mathematics, WSP Chodkiewicza 30 85-064 Bydgoszcz, Poland
Mathematics Subject Classification (1980): 14H; 20H; 30F ISBN 3-540-52941-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52941-1 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 Printed on acid-free paper
To Raquel (and our daughters Raquel, M.Teresa, Carla and Lidia) To Almudena To Alicia To Terenia
INTRODUCTION
Classical results on automorphism groups of complex algebraic curves. Given a complex algebraic curve by means of its polynomial equations, it is very difficult to get information about its birational automorphisms, unless the curve is either rational or elliptic. For curves C of genus Schwarz proved in 1879 the finiteness of the group Aut(C) of automorphisms of C, [111]. Afterwards Hurwitz, applying his famous ramification formula, showed that IAut(C) I -s 84(P-l), [69]. By means of classical methods of complex algebraic geometry, Klein showed that IAut(C)1 s48 if p=2. Then, Gordan proved that IAut(C)I -s 120 for p=4, [51], and Wiman, who carefully studied the cases 2sps6, in particular established that IAut(C)I s 192 for p=5 and IAut(C)1 i(Ui)S:;C+}. Each (Ui'¢i) is said to be a chart and we say that this chart is positive if ¢i(Ui) s:; C +. The transition functions of 1: are the homeomorphisms ¢ .. =¢.¢.-1 :¢.(U. nu.) ------;.¢.(U.nu.). IJ
1 J
J
1
J
1
1
J
Simple arguments show that ¢.(u.nu.) is open in C (resp J 1 J same is true for ¢.(U. nu.). 1 1 J The orientability of S is defined as for a real identification of C with R
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