Symmetries of Compact Riemann Surfaces
This monograph deals with symmetries of compact Riemann surfaces. A symmetry of a compact Riemann surface S is an antianalytic involution of S. It is well known that Riemann surfaces exhibiting symmetry correspond to algebraic curves which can be defined
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2007
Emilio Bujalance · Francisco Javier Cirre Jos´e Manuel Gamboa · Grzegorz Gromadzki
Symmetries of Compact Riemann Surfaces
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Emilio Bujalance
Jos´e Manuel Gamboa
Facultad de Ciencias, UNED Matem´aticas Fundamentales C/ Senda del Rey 9 28040 Madrid Spain [email protected]
Universidad Complutense Madrid Facultad de Matem´aticas, UCM ´ Departamento de Algebra Plaza de las Ciencias 3 28040 Madrid Spain [email protected]
Francisco Javier Cirre Facultad de Ciencias, UNED Matem´aticas Fundamentales C/ Senda del Rey 9 28040 Madrid Spain [email protected]
Grzegorz Gromadzki University of Gdansk Department of Mathematics Wita Stwosza 57 80-952 Gdansk Poland [email protected]
ISBN: 978-3-642-14827-9 e-ISBN: 978-3-642-14828-6 DOI: 10.1007/978-3-642-14828-6 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010935187 Mathematics Subject Classification (2010): 30F, 14H, 20H, 20D, 57M c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
´ To Alvaro and my grandchildren To my family ´ To Bel´en, Alvaro and Irene To the memory of my parents
Preface
The content of this monograph is situated in the intersection of important branches of mathematics like the theory of one complex variable, algebraic geometry, low dimensional topology and, from the point of view of the techniques used, combinatorial group theory. The main tool comes from the Uniformization Theorem for Riemann surfaces, which relates the topology of Riemann surfaces and holomorphic or antiholomorphic actions on them to the algebra of classical cocompact Fuchsian groups or, more generally, non-euclidean crystallographic groups. Foundations of this relationship were established by A. M. Macbeath in the early sixties and developed later by, among others, D. Singerman. Another important result in Riemann surface theory is the connection between Riemann surfaces and their symmetries with complex algebraic curves and their real forms. Namely, there is a well known functorial bijective correspondence between compact R
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