Trees / Edition 1

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The seminal ideas of this book played a key role in the development of group theory since the 70s. Several generations of mathematicians learned geometric ideas in group theory from this book. In it, the author proves the fundamental theorem for the special cases of free groups and tree products before dealing with the proof of the general case. This new edition is ideal for graduate students and researchers in algebra, geometry and topology.

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Editorial Reviews

From the Publisher

From the reviews:

"In the case of an author like Serre, there is almost no need to underline the book's qualities of elegance and precision: over and above these, it provides abundant links to other topics, particularly by means of the remarks and the numerous exercises which, for the most part, are not easy to solve, but genuinely augment the content of the book. The greatest quality of this book however, is, in my opinion, that one finds in it many new and interesting ideas of very considerable substance, but presented in their very simplest form."
H. Behr, Frankfurt in: Jahresbericht der DMV, (84/3) 1982

From the reviews: "Serre's notes on groups acting on trees have appeared in various forms (all in French) over the past ten years and they have had a profound influence on the development of many areas, for example, the theory of ends of discrete groups. This fine translation is very welcome and I strongly recommend it as an introduction to an important subject. In Chapter I, which is self-contained, the pace is fairly gentle. The author proves the fundamental theorem for the special cases of free groups and tree products before dealing with the (rather difficult) proof of the general case." (A.W. Mason in Proceedings of the Edinburgh Mathematical Society 1982)

"The book under review is the … second printing of the English edition of 1980 … . The seminal ideas of the book played an important role in the development of group theory since the seventies … . Nowadays the book already can be called classical. … Several generations of mathematicians learned geometric ideas in group theory from this stimulating book; without doubt, the new edition will be useful for graduate students and researchers working in algebra, geometry, and topology." (Oleg V. Belegradek, Zentralblatt MATH, Vol. 1013, 2003)

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Product Details

  • ISBN-13: 9783540442370
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 1/17/2003
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 1st ed. 1980. Corr. 2nd printing 2002
  • Edition number: 1
  • Pages: 144
  • Product dimensions: 0.44 (w) x 9.21 (h) x 6.14 (d)

Table of Contents

Ch. I Trees and Amalgams 1
1 Amalgams 1
1.1 Direct limits 1
1.2 Structure of amalgams 2
1.3 Consequences of the structure theorem 5
1.4 Constructions using amalgams 8
1.5 Examples 11
2 Trees 13
2.1 Graphs 13
2.2 Trees 17
2.3 Subtrees of a graph 21
3 Trees and free groups 25
3.1 Trees of representatives 25
3.2 Graph of a free group 26
3.3 Free actions on a tree 27
3.4 Application: Schreier's theorem 29
App Presentation of a group of homeomorphisms 30
4 Trees and amalgams 32
4.1 The case of two factors 32
4.2 Examples of trees associated with amalgams 35
4.3 Applications 36
4.4 Limit of a tree of groups 37
4.5 Amalgams and fundamental domains (general case) 38
5 Structure of a group acting on a tree 41
5.1 Fundamental group of a graph of groups 41
5.2 Reduced words 45
5.3 Universal covering relative to a graph of groups 50
5.4 Structure theorem 54
5.5 Application: Kurosh's theorem 56
6 Amalgams and fixed points 58
6.1 The fixed point property for groups acting on trees 58
6.2 Consequences of property (FA) 59
6.3 Examples 60
6.4 Fixed points of an automorphism of a tree 61
6.5 Groups with fixed points (auxiliary results) 64
6.6 The case of SL[subscript 3](Z) 67
Ch. II SL[subscript 2] 69
1 The tree of SL[subscript 2] over a local field 69
1.1 The tree 69
1.2 The groups GL(V) and SL(V) 74
1.3 Action of GL(V) on the tree of V; stabilizers 76
1.4 Amalgams 78
1.5 Ihara's theorem 82
1.6 Nagao's theorem 85
1.7 Connection with Tits systems 89
2 Arithmetic subgroups of the groups GL[subscript 2] and SL[subscript 2] over a function field of one variable 96
2.1 Interpretation of the vertices of [Gamma]\X as classes of vector bundles of rank 2 over C 96
2.2 Bundles of rank 1 and decomposable bundles 99
2.3 Structure of [Gamma]\X 103
2.4 Examples 111
2.5 Structure of [Gamma] 117
2.6 Auxiliary results 120
2.7 Structure of [Gamma]: case of a finite field 124
2.8 Homology 125
2.9 Euler-Poincare characteristic 131
Bibliography 137
Index 141
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