Clara L¨oh
Geometric group theory, an introduction
July 13, 2011 – 13:56 Preliminary version Please send corrections and suggestions to
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Clara L¨oh
[email protected] http://www.mathematik.uni-regensburg.de/loeh/ NWF I – Mathematik Universit¨at Regensburg 93040 Regensburg Germany
Contents 1 Introduction
1
2 Generating groups
7
2.1
Review of the category of groups 2.1.1 2.1.2 2.1.3
2.2
2.3
Axiomatic description of groups Concrete groups – automorphism groups Normal subgroups and quotients
8 10 13
Groups via generators and relations
15
2.2.1 2.2.2 2.2.3
16 17 22
Generating sets of groups Free groups Generators and relations
New groups out of old
28
2.3.1 2.3.2
28 31
Products and extensions Free products and free amalgamated products
3 Groups → geometry, I: Cayley graphs 3.1 3.2 3.3
8
35
Review of graph notation Cayley graphs Cayley graphs of free groups
36 39 42
3.3.1 3.3.2 3.3.3
43 45 47
Free groups and reduced words Free groups → trees Trees → free groups
iv
Contents
4 Groups → geometry, II: Group actions 4.1
4.2
4.3 4.4 4.5
Review of group actions
50
4.1.1 4.1.2 4.1.3
51 54 58
Free actions Orbits Application: Counting via group actions
Free groups and actions on trees
60
4.2.1 4.2.2
61 62
Spanning trees Completing the proof
Application: Subgroups of free groups are free The ping-pong lemma Application: Free subgroups of matrix groups
5 Groups → geometry, III: Quasi-isometry 5.1 5.2
Quasi-isometry types of metric spaces Quasi-isometry types of groups 5.2.1
5.3
5.4 5.5
First examples
6.3
66 69 72
75 76 83 86
ˇ The Svarc-Milnor lemma
88
5.3.1 5.3.2 5.3.3
89 91
Quasi-geodesics and quasi-geodesic spaces ˇ The Svarc-Milnor lemma ˇ Applications of the Svarc-Milnor lemma to group theory, geometry and topology
96
The dynamic criterion for quasi-isometry
101
5.4.1
107
Applications of the dynamic criterion
Preview: Quasi-isometry invariants and geometric properties
109
5.5.1 5.5.2
109 110
Quasi-isometry invariants Geometric properties of groups and rigidity
6 Growth types of groups 6.1 6.2
49
113
Growth functions of finitely generated groups Growth types of groups
114 117
6.2.1 6.2.2 6.2.3
117 119 123
Growth types Growth types and quasi-isometry Application: Volume growth of manifolds
Groups of polynomial growth
127
6.3.1 6.3.2
127 129
Nilpotent groups Growth of nilpotent groups
Contents
v 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7
Groups of polynomial growth are virtually nilpotent Application: Being virtually nilpotent is a geometric property Application: More on polynomial growth Application: Quasi-isometry rigidity of free Abelian groups Application: Expanding maps of manifolds
7 Hyperbolic groups 7.1
7.2
7.3 7.4 7.5
133 133 134 135
137
Classical curvature, intuitively
138
7.1.1 7.1.2
138