Kleinian Groups
The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian
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Editors M. Artin S. S. Chern J. M. Frohlich E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. MacLane C. C. Moore J. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai J. Tits M. Waldschmidt S. Watanabe Managing Editors M. Berger
B. Eckmann
S. R. S. Varadhan
Bernard Maskit
Kleinian Groups With 67 Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Bernard Maskit Dept. of Mathematics SUNY at Stony Brook Stony Brook, NY 11794 USA
Mathematics Subject Classification (1980): 30F40
ISBN-13: 978-3-642-64878-6 e-ISBN-13: 978-3-642-61590-0 DOl: 10.1007/978-3-642-61590-0
Library of Congress Cataloging-in-Publication Data Maskit, Bernard. Kleinian groups. (Grundlehren der mathematischen Wissenschaften ; 287) Bibliography: p. Includes index. 1. Kleinian groups. I. Title. II. Series. QA331.M418 1987 515 87-20632 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 microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988
Softcover reprint of the hardcover 1st edition 1988 Typesetting: Asco Trade Typesetting Ltd., Hong Kong
2141/3140-543210
To Wilma
Introduction
The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geometry, and there is now an active sc
