Pseudo-periodic Maps and Degeneration of Riemann Surfaces
The first part of the book studies pseudo-periodic maps of a closed surface of genus greater than or equal to two. This class of homeomorphisms was originally introduced by J. Nielsen in 1944 as an extension of periodic maps. In this book, the conjugacy c
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For further volumes: http://www.springer.com/series/304
2030
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Yukio Matsumoto Jos´e Mar´ıa Montesinos-Amilibia
Pseudo-periodic Maps and Degeneration of Riemann Surfaces
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Yukio Matsumoto Gakushuin University Department of Mathematics Mejiro 1-5-1 171-8588 Toshima-ku Tokyo Japan [email protected]
Jos´e Mar´ıa Montesinos-Amilibia Universidad Complutense Facultad de Matem´aticas Departamento de Geometr´ıa y Topolog´ıa Plaza de las Ciencias 3 28040 Madrid Spain [email protected]
ISBN 978-3-642-22533-8 e-ISBN 978-3-642-22534-5 DOI 10.1007/978-3-642-22534-5 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: 2011934808 Mathematics Subject Classification (2010): 14-XX, 57-XX c Springer-Verlag Berlin Heidelberg 2011 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: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated with respect and affection to the memory of Professor Itiro Tamura (1926–1991)
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Preface
In 1944, Nielsen introduced a certain type of mapping classes of a surface which were called by him surface transformation classes of algebraically finite type, [53]. He introduced this type of mapping classes as a generalization of the mapping classes of finite order. By the celebrated Nielsen Theorem [52], the latter classes contain surface homeomorphisms of finite order (For a generalization, see Kerckhoff [30]). A mapping class of algebraically finite type does not necessarily contain a homeomorphism of finite order, but using Nielsen’s theorem [52], one can show that it contains a homeomorphism f satisfying the following conditions (in what follows f will be an orientation-preserving homeomorphism of a closed, connected, oriented surface of genus g, ˙g ): 1. There exists a disjoint union of simple closed curves (which will be called cut curves) C D C1 [ C2 [ [ Cr on ˙g such that f .C / D C , and 2. the restriction of f to the complement of C , f j.˙g C / W ˙g C ! ˙g C is isotopic to a periodic map, namely a homeomorphism of finite order. (Cf. [53, Sect. 14], [22]). In the present
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