Trees Book

	Introduction
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