Description
Springer Trees by Jean-Pierre Serre, J. Stilwell
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._x000D_ Table of contents :- _x000D_
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 F\X 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-Poincare characteristic._x000D_