Contents
Preface xi
Chapter 1. Introduction 1
1.1. Cocompact is an open condition 1
1.2. Controlled connectivity 1
1.3. The Boundary Criterion 2
1.4. The Geometric Invariants 2
Part 1. Controlled connectivity and openness results 5
Chapter 2. Outline, Main Results and Examples 7
2.1. Non-positively curved spaces 7
2.2. Controlled connectivity: the definition of
CCf'~l
7
2.3. The case of discrete orbits 8
2.4. The Openness Theorem 9
2.5. Connections with Lie groups and local rigidity 10
2.6. The new tool 10
2.7. Summary of the core idea 11
2.8. SL2 examples 11
Chapter 3. Technicalities Concerning the CC"~[ Property 13
3.1. Local and global versions of CCn~~[ 13
3.2. The Invariance Theorem 14
Chapter 4. Finitary Maps and Sheaves of Maps 17
4.1. Sheaves of maps 17
4.2. G-sheaves 18
4.3. Locally finite sheaves 18
4.4. Embedding sheaves into homotopicaUy closed sheaves 19
4.5. Composing sheaves 20
4.6. Homotopy of sheaves 20
4.7. Finitary maps 21
Chapter 5. Sheaves and Finitary Maps Over a Control Space 23
5.1. Displacement function and norm 23
5.2. Shift towards a point of M 24
5.3. Contractions 24
5.4. Guaranteed shift 25
5.5. Defect of a sheaf 26
Chapter 6. Construction of Sheaves with Positive Shift 29
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