Surfaces and Planar Discontinuous Groups Revised and Expanded Transl
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Heiner Zieschang
Elmar Vogt Hans-Dieter Coldewey Surfaces and Planar Discontinuous Groups
Revised and Expanded Translation Translated from the German by J. Stillwell
Springer-Verlag Berlin Heidelberg New York 1980
Authors Heiner Zieschang Universit~t Bochum Institut flit Mathematik Universit~tsstr. 150 4630 Bochum 1 Federal Republic of Germany Elmar Vogt Freie Universit~t Berlin Institut f~Jr Mathematik I H~ittenweg 9 1000 Berlin 33 Federal Republic of Germany Hans-Dieter Coldewey Allescherstr. 40b 8000 M0nchen 71 Federal Republic of Germany Revised and expanded translation of: H. Zieschang/E. Vogt/H.-D. Coldewey, Fl~chen und ebene diskontinuierliche Gruppen (Lecture Notes in Mathematics, vol. 122) published by Springer-Verlag Berlin-Heidelberg-New York, 1970
AMS Subject Classifications (1980): 20 Exx, 20 Fxx, 30 F35, 32 G 15, 51M10, 57 Mxx ISBN 3-540-10024-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10024-5 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 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
In memori~7 Kurt R e i d e m e ~ t e r
Introduction For the two-dimensional manifolds, the surfaces, the classical topological problems - classification and Hmuptver~atun S - have lonS been solved~ and more
delicate questions can be investigated. However, the most interesting side of sur-
face theory is not the topological, but the analytic. Results of the complex ana-
lytic theory, e.g., are often purely topo]osical~ but their proofs ape not, usin S
deep theorems of function theory. This results from a natural and close connection with discontinuous groups of motions in the non-euclidean or euclidean plane. The following lectures deal in the fim'st place with cori~inatorial topological
theorems on surfaces and planar discontinuous gn~oups{ thus we have adopted the concept of the book "EinfL~nruns in die kombinatorische Topologie" by K. Reidemeister.
Admittedly, in chapters 1-5 we have not kept strictly to the combinatorial conception~ but have changed to a~nother category where this seems converient. ~(v i), i = 1,...,m. Exercises:
E 1.13-17
[]
13 1,8
GEOMETRIC
INTERPRETATION
OF THE
NIELSEN
PROPERTY
In this section we give the geometric interpretation of the Nielsen property for generators of a subgroup from [Reidemeister-Brm~dis 1959]. Let S be the fundmmental group of a graph C wlnichhas only one vertex and let
U be the covering complex for the subgroup C. We take the free generators for S to
be those belonging to
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