J-P. Serre (Fields Medal 1954) is one of the most exceptional mathematicians of our age and also one of the most talented expositors of high-level mathematics
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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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)
I. Trees and Amalgams.- §1 Amalgams.- 1.1 Direct limits.- 1.2 Structure of amalgams.- 1.3 Consequences of the structure theorem.- 1.4 Constructions using amalgams.- 1.5 Examples.- §2 Trees.- 2.1 Graphs.- 2.2 Trees.- 2.3 Subtrees of a graph.- §3 Trees and free groups.- 3.1 Trees of representatives.- 3.2 Graph of a free group.- 3.3 Free actions on a tree.- 3.4 Application: Schreier´s theorem.- Appendix: Presentation of a group of homeomorphisms.- §4 Trees and amalgams.- 4.1 The case of two factors.- 4.2 Examples of trees associated with amalgams.- 4.3 Applications.- 4.4 Limit of a tree of groups.- 4.5 Amalgams and fundamental domains (general case).- §5 Structure of a group acting on a tree.- 5.1 Fundamental group of a graph of groups.- 5.2 Reduced words.- 5.3 Universal covering relative to a graph of groups . ..- 5.4 Structure theorem.- 5.5 Application: Kurosh´s theorem.- §6 Amalgams and fixed points.- 6.1 The fixed point property for groups acting on trees.- 6.2 Consequences of property (FA).- 6.3 Examples.- 6.4 Fixed points of an automorphism of a tree.- 6.5 Groups with fixed points (auxiliary results).- 6.6 The case of SL3(Z).- II. SL2.- §1 The tree of SL2 over a local field.- 1.1 The tree.- 1.2 The groups GL(V) and SL(V).- 1.3 Action of GL(V) on the tree of V; stabilizers.- 1.4 Amalgams.- 1.5 Ihara´s theorem.- 1.6 Nagao´s theorem.- 1.7 Connection with Tits systems.- §2 Arithmetic subgroups of the groups GL2 and SL2 over a function field of one variable.- 2.1 Interpretation of the vertices of FX as classes of vector bundles of rank over C 96.- 2.2 Bundles of rank and decomposable bundles 99.- 2.3 Structure of ?X.- 2.4 Examples.- 2.5 Structure of ?.- 2.6 Auxiliary results.- 2.7 Structure of ?: case of a finite field.- 2.8 Homology.- 2.9 Euler-Poincaré characteristic.
The present book is an English translation of "Arbres, Amalgames, SL(2)", published in 1977 by J-P.Serre, and written with the collaboration of H.Bass. The first chapter describes the "arboreal dictionary" between graphs of groups and group actions on trees. The second chapter gives applications to the Bruhat-Tits tree of SL(2) over a local field.
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)
I. Trees and Amalgams.-
1 Amalgams.-
2 Trees.-
3 Trees and free groups.-
4 Trees and amalgams.-
5 Structure of a group acting on a tree.-
6 Amalgams and fixed points.- II. SL2.-
1 The tree of SL2 over a local field.-
2 Arithmetic subgroups of the groups GL2 and SL2 over a function field of one variable.
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)
Über den Autor
Inhaltsverzeichnis
I. Trees and Amalgams.- §1 Amalgams.- 1.1 Direct limits.- 1.2 Structure of amalgams.- 1.3 Consequences of the structure theorem.- 1.4 Constructions using amalgams.- 1.5 Examples.- §2 Trees.- 2.1 Graphs.- 2.2 Trees.- 2.3 Subtrees of a graph.- §3 Trees and free groups.- 3.1 Trees of representatives.- 3.2 Graph of a free group.- 3.3 Free actions on a tree.- 3.4 Application: Schreier¿s theorem.- Appendix: Presentation of a group of homeomorphisms.- §4 Trees and amalgams.- 4.1 The case of two factors.- 4.2 Examples of trees associated with amalgams.- 4.3 Applications.- 4.4 Limit of a tree of groups.- 4.5 Amalgams and fundamental domains (general case).- §5 Structure of a group acting on a tree.- 5.1 Fundamental group of a graph of groups.- 5.2 Reduced words.- 5.3 Universal covering relative to a graph of groups . ..- 5.4 Structure theorem.- 5.5 Application: Kurosh¿s theorem.- §6 Amalgams and fixed points.- 6.1 The fixed point property for groups acting on trees.- 6.2 Consequences of property (FA).- 6.3 Examples.- 6.4 Fixed points of an automorphism of a tree.- 6.5 Groups with fixed points (auxiliary results).- 6.6 The case of SL3(Z).- II. SL2.- §1 The tree of SL2 over a local field.- 1.1 The tree.- 1.2 The groups GL(V) and SL(V).- 1.3 Action of GL(V) on the tree of V; stabilizers.- 1.4 Amalgams.- 1.5 Iharäs theorem.- 1.6 Nagao¿s theorem.- 1.7 Connection with Tits systems.- §2 Arithmetic subgroups of the groups GL2 and SL2 over a function field of one variable.- 2.1 Interpretation of the vertices of FX as classes of vector bundles of rank over C 96.- 2.2 Bundles of rank and decomposable bundles 99.- 2.3 Structure of ?X.- 2.4 Examples.- 2.5 Structure of ?.- 2.6 Auxiliary results.- 2.7 Structure of ?: case of a finite field.- 2.8 Homology.- 2.9 Euler-Poincaré characteristic.
Klappentext
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)
J-P. Serre (Fields Medal 1954) is one of the most exceptional mathematicians of our age and also one of the most talented expositors of high-level mathematics
