A Crash Course on Kleinian Groups Lectures given at a special sessio
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San Francisco, CA, USA 1974
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Springer
Lecture Notes in Mathematics Edited by A Dold and B. Eckmann
400 A Crash Course on Kleinian Groups Lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco
Edited by Lipman Bers and Irwin Kra
Springer-Verlag Berlin . Heidelberg· NewYork 1974
Lipman Bers Columbia University, Morningside Heights, New York, NY/USA Irwin Kra SUNY at Stony Brook, Stony Brook. New York, NY/USA
Library of Congress Cataloging in Publication Data
American Mathematical Society. A crash course on K1einian groups, San Francisco,
197~
(Lecture notes in mathematics, 400) l. K1einian groups. I. Bers, Lipman, ed. II. Kra, Irwin, ed. III. Title. IV. Series: Lecture notes in mathematics (Berlin, 4oo) QA3.I28 no. 400 [QA331] 510'.8s [512'.55] 74-13853
AMS Subject Classifications (1970): Primary: 30-02, 32G15 Secondary: 30A46, 30A58, 30A60 ISBN 3-540-06840-6 Springer-verlag Berlin· Heidelberg· New York ISBN 0-387-06840-6 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
To Lars
v.
Ahlfors
PREFACE It has recently become customary to have "special sessions" at meetings of the AMS,
consisting of short invited
lectures, and intended for groups of specialists.
At the
Annual Winter Meeting at San Francisco, we tried to have a special session addressed to non-specialists.
The lecturers
were asked to prepare in advance texts of their talks, and these were distributed at the meeting.
These texts, slightly
revised, are collected in the present fascicule.
(We also
included an abstract of a forthcoming paper by H. Masur.) The present "crash course" does not intend to do more than to give a reader an introductory survey of some topics which became important in the modern theory of Kleinian groups.
The references to literature, though by no
means complete, should enable anyone interested in more detailed information to obtain same. Lars Ahlfors, who played a decisive part in the recent revival of Kleinian groups, could not be present at San Francisco.
It is fitting to dedicate this modest effort to
him. L.B.
I.K.
CONTENTS
Chapter l What is a. Kleinian group? by Lipman Bers
l
Chapter 2 Quasiconformal mappings and uniformiza.tion by C. J. Earle
15
Chapter 3 Automorphic forms and Eichler cohomology by Frederick P. Gardiner
24
Chapter 4 Deformation spaces by Irwin Kra.
48
Chapter 5 Metrics on Teichmtlller space by H. L. Royden
71
Chapter 6 Mo
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