Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold dna .B nnamkcE
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desiveR Translation and Expanded
Translated from the German yb .J Stillwell
galreV-regnirpS
Berlin Heidelberg New kroY 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 rJ~f Mathematik I
H~ittenweg 9
1000 Berlin 33
Federal Republic of Germany
Hans-Dieter Coldewey
Allescherstr. 40b
8000 M0nchen 17
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 ,51
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 si subject to All copyright. rights era whether reserved, eht whole or
of part the concerned, is material specifically those of reprinting, translation,
re-use of illustrations, broadcasting, reproduction yb photocopying machine or
similar means, dna storage ni data Under § banks. 54 of the Copyright German
waL where copies era for made other than a use, private eef to the is payable
amount the publisher, of eht eef to eb determined yb with the publisher. agreement
© by galreV-regnirpS Berlin Heidelberg 1980
Printed ni Germany
Printing dna binding: Beltz Offsetdruck, Hemsbach/Bergstr.
012345-0413/1412
In memori~7 Kurt Reideme~ter
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 con-
cept 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 concep-
tion~ but have changed to a~nother category where this seems converient. <nus we
are far ±born striving for the purity of the above book, and .e o~roup . S theoretic
theorems are sometimes proved seome~grically, and vice versa.
In chapter 6 we consider Riemann surfaces and give an elementary foundation
of a theory due to 0. Teichm@ller for the solution of the modular problem. In chap-
ter 7 we prove the triangulation theorem d'ra the Hauptverm~dtung for surfaces% these
theorems help explain why, in 2-dimensional topolo~, the cornbinatorial (or PL)
theory covers most interesting questions.
In order to read the main parts of the first five chapters only basic knowledge
of group theory is necessary; the combinatorial S-tour theory and surface theory is
developed from scratch. In some of the later sections in the chapters we use some
results from algebraic topology for convenience. However these sections ~rac be
omitted at a first readirk3. For chapters 6 and 7 a basic knowledge of general l~ra
algebraic topology, non-euclidean geometry and complex analysis is necessary~ we
~ope that we have described a simple approach to the problems considered. I have
not tried to list all the mathema~tical work hha< is conmec-ted with die combinatorial
theory of surfaces and to value a]] articles appropriately.
Vl
This English version of "Fl~chen und ebene diskontinuierliche Gruppen" con-
rains substantial additions to the original text; among them:
- A closer consideration of the Reidemeister-Schreier method in section 2.1 which
leads to an algebraic proof of the Riemann-Hurwitz formula in 4.14.
Proofs of the Grushko theorem and the Freiheitssatz in 2.9 and 2.11.
-
Construction of mappings between s~faces wi~n given degrees ~n 3.5.
-
- The theory of intersection numbers for curves on surfaces and Hopf's theory of
mappings between surfaces in 3.6 and 3.7.
- The classification of finitely generated discontinuous groups of the plane which
do not necessarily have compact fundamental region in 4.11.
On decompositions of planar, groups Ln 4.12.
-
- On the action of fJite groups on surfaces in 4.15. Generalized versions of the
Baer and Nielsen theorems in 5.14-16.
- Proofs of the Sch~nflies theorem, the triangulation theorem and the Hauptvermu-
tung for surfaces in chapter .7
In completing the lectures on surfaces ! have been lucky to have a big number
of helpers. As already said in the Gernmn version 1970 my thanks go to A.B.
Soss~nskij, A.V. Cernavskij and A,M. }i~cbeath for suggestions and criticism.
El~mr Vogt made the notes of my lectures in Frankfurt, and Dieter Coldewey worked
with me on sections 6.9-11. My thanks go also to E. Gramberg, .F Hillefeld, R.
Keller, N. Peczynski, W. Reiwer dn~a .B ~mn:s.~remmniZ for their discussions on the
different problems. In addition I wish to thank Frau Faber for typing the original
German text and Frau Schwarz for typing the English variant, W. Bitter for drawing
the figures and .G Krause and .B Wicha for reading the proofs. Considerable help
in the completion of the book was given to me by John Stillwell who not only trans-
lated the origJmml text : but helped to make it more understandable and to insert
new parts.
CONTENTS
Introduction
Interdependence of Topics
.1 FREE GROUPS AND GRAPHS 1
1,1 Free Groups 1
1.2 Word and Conjugacy Problems 3
1.3 Graphs 4
1.4 The Fundamental Group of a Grail 6
1.5 Coverings of Graphs 7
1.6 The Reidemeister-Schreier Met~md for Subgroups 9
1.7 The Nielsen }~thod 10
1.8 Geometric Interpretation of the Nielsen Property 13
1.9 Automorphisms of a Free Group of Finite Rank 14
Exercises 16
.2 2-DI}<NSIONAL COMPLEXESAND COMBINATORIAL PRESE~]ZTIONS OF GROUPS 19
2.1 Tietze's meroeq~'~ 19
2.2 The Re~demeister-Sd~reier donlteM 24
2.3 Free Products with Amalgamation 31
2.4 2-dimensional Complexes 37
2.5 Coverings 39
2.6 Proof of the Kurosh Theorem 43
2.7 Homology groups 44
2.8 The First Homotopy Group and the Fundamental Group, Seifert-
Van Kampen Theorem. Pasting Complexes Together 47
2.9 Grus~<o's Theorem 48
2.10 On the Nielsen }½thod in Free Products 51
2.11 On Groups with one Defining Relation 45
2.12 Remarks 95
Exercises 61
.3 SURFACES 64
3.1 Definitions 64
3.2 Classification of Finite Stmfaces 68
3.3 Kneser's Formula 73
IIIV
3.4 Coveriz~gs of Surfaces 97
3.5 Types of Simple Closed Curves on Surfaces 84
3.6 Intersection Numbers of Cm~ves 87
3.7 On Continuous Mappings Bet~4een Surfaces 94
3.8 Remarks 101
Exercises 102
4. PLANAR DISCO~flN]JOUS SPILORG 106
4.1 Planar Ne~s 106
4.2 Automorphisms of a Planar Net 109
4.3 Automorphism Groups of Planar Hets 112
4.4 Fundamental Domains 115
4.5 The ~ZLgebraic Structure of Planar Discontinmous Groups 116
4.6 Classification of Plaz~ Discontinuous Groups 120
4.7 Existence Proof 122
4.8 On the Algebraic Structure of Planar Groups 125
4.9 The Word and Conjugacy Problems 128
4.10 Surface Subgroups of Finite Index 132
4.11 On Planar Groups with Non-Compact Fundamental Domain 137
4.12 On Decompositions of Discontinuous Groups of the Plane 140
4.13 Planar Group Presentations and Diagrams 142
4.14 Subgroups and the Rienmn~-Hu~itz Formula 150
4.15 Finite Groups Acting on Closed Surfaces 160
4.16 On the Rank of Planar Discontinuous Groups 170
Exercises 174
.5 AU~fOMORPHISMS OF R_aq/ALP GROUPS 178
5.1 Preliminary Considerations 178
5.2 Binary Products 179
5.3 Homotopic Binary Products 184
5.4 Free Generators of the Group of Relations 190
5.5 The Happin~s ~trix 192
5.6 The Dehn-Nielsen Theor{~ 194
5.7 Nielsen's Theorem for Bounded Surfaces 196
5.8 Automorphisms of Plans, Groups 199
5.9 Combinatorial Isotopy 204
5.10 A solution of the Word and Conjugacy Problem in FurJmmental
Groups of Surfaces 205
5.11 Isotopy of Free Homotopic Simple Closed Curves 206
5.12 Basepoint Preserving Isotopies of Simple Closed Curves 211
XI
5.13 Irmer Autommrphisms and Isotopies 214
5.14 el~l Baer Theor~ for Planar Discontinuous Groups 216
5.15 On the Mapping Class Group 227
5.16 Coverings ~nd %he I~pping Class Group 232
Exercises 239
6. ON THE COMPLEX ANALYTIC TIEORY OF RII~ANN SURFACES AND PLAff~.
DISCONTI}~JOUS GROLPS
6.1 Introduction 242
6.2 Structures on Surfaces 242
6.3 The Non-Euclidean Plane 244
6.4 Planar Discontinuous Groups 247
6.5 On the Hodular Problem 250
6.6 TeichmLtller Theory and its Consequences 251
6.7 A Basis for a Teicbm~ler Theory 255
6.8 The Classical Foundation 263
6.9 Teichmiiller Spaces 265
6.10 Some Lenmas from Hyperbolic Geometry 286
6.11 Canonical Polygops 273
6.12 Automorphisms of Finite red~h£ 283
Exercises 285
.7 ON T}~ TOPOLOGICAL THEORY OF SURFACES 287
7.1 Homological Properties of Subsets of the Plane 287
7.2 Local Path Cor~ectivity 291
7.3 Construction of a Crosscut 294
7.4 The Sch~nflies Theorem 299
7.5 Triangulation and Haup~tvermutung for Surfaces 304
Exercises 308
Books on surface theory and related subjects 310
Bibliography 311
List of }btation 330
Index 331
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1, FREE GROUPS AND GRAPHS
1,1 FR-EE GROUPS
We consider 1-dimensional complexes (graphs) and their fundamental sroups which
have an interesting combinatorial property: they are the so-called free groups. We
lead into the problems of combinatorial group theory and develop the methods of Niel-
sen and Reidem~ister-Seb~ier. Basic for our approach is the close connection of
complexes with groups and coverings with sub,ups, which is important for later
chapters. Here we begin this theory assuming only the definition of a S~oup.
1.1.1 Definition. Given a system of symbols (Si)iE I we call each e~ression of the
W=S I s SS2 ... S n s , .s : i i
al a2 en
a word in eht Si, and the empty word is denoted by "1". The product of words is de-
fined by writing the symbols of one after those of the other. We call ~Jo words
W = WIS[S~eW 2 and V = WIW 2 elementarily equivalent. Two words W and V are called
equivalent when there is a finite sequence of words W = W1,W2,...,W m = V where the
i W and Wi+ 1 are elementarily equivalent to each other. The equivalence class of the
word W is denoted by [W]. We define the product of two equivalence classes ]W[ and
]V[ by
[w] • [v] : [~].
The product is well-defined; for if W : WI,W2~...,W n W' : and V : V1,...,V m : 'V are
two sequences of elementarily equivalent words then WV W = iV ,WnV ~W2V1,... I ,1
WV2~... ,WV m = W'V' is likewise such a sequence. The set of equivalence classes con-
stitutes a group under this multiplication, the identity element of which is the
class of the empty word. The inverse element of
1~ ]nS
]W[ : [S ... S is [W -1] : S[ -s n ... S Sl]. -
al an an ~1
