Trees by Jean-Pierre Serre (PDF)

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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.

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