Trees / Edition 1

Trees / Edition 1

by J. Stilwell, Jean-Pierre Serre
     
 

ISBN-10: 3540442375

ISBN-13: 9783540442370

Pub. Date: 01/17/2003

Publisher: Springer Berlin Heidelberg

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

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Overview

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.

Product Details

ISBN-13:
9783540442370
Publisher:
Springer Berlin Heidelberg
Publication date:
01/17/2003
Series:
Springer Monographs in Mathematics Series
Edition description:
1st ed. 1980. Corr. 2nd printing 2002
Pages:
144
Product dimensions:
0.44(w) x 9.21(h) x 6.14(d)

Table of Contents

Introduction
Ch. ITrees and Amalgams1
1Amalgams1
1.1Direct limits1
1.2Structure of amalgams2
1.3Consequences of the structure theorem5
1.4Constructions using amalgams8
1.5Examples11
2Trees13
2.1Graphs13
2.2Trees17
2.3Subtrees of a graph21
3Trees and free groups25
3.1Trees of representatives25
3.2Graph of a free group26
3.3Free actions on a tree27
3.4Application: Schreier's theorem29
AppPresentation of a group of homeomorphisms30
4Trees and amalgams32
4.1The case of two factors32
4.2Examples of trees associated with amalgams35
4.3Applications36
4.4Limit of a tree of groups37
4.5Amalgams and fundamental domains (general case)38
5Structure of a group acting on a tree41
5.1Fundamental group of a graph of groups41
5.2Reduced words45
5.3Universal covering relative to a graph of groups50
5.4Structure theorem54
5.5Application: Kurosh's theorem56
6Amalgams and fixed points58
6.1The fixed point property for groups acting on trees58
6.2Consequences of property (FA)59
6.3Examples60
6.4Fixed points of an automorphism of a tree61
6.5Groups with fixed points (auxiliary results)64
6.6The case of SL[subscript 3](Z)67
Ch. IISL[subscript 2]69
1The tree of SL[subscript 2] over a local field69
1.1The tree69
1.2The groups GL(V) and SL(V)74
1.3Action of GL(V) on the tree of V; stabilizers76
1.4Amalgams78
1.5Ihara's theorem82
1.6Nagao's theorem85
1.7Connection with Tits systems89
2Arithmetic subgroups of the groups GL[subscript 2] and SL[subscript 2] over a function field of one variable96
2.1Interpretation of the vertices of [Gamma]\X as classes of vector bundles of rank 2 over C96
2.2Bundles of rank 1 and decomposable bundles99
2.3Structure of [Gamma]\X103
2.4Examples111
2.5Structure of [Gamma]117
2.6Auxiliary results120
2.7Structure of [Gamma]: case of a finite field124
2.8Homology125
2.9Euler-Poincare characteristic131
Bibliography137
Index141

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