 | Mosher M. Quasi-actions on trees II: finite depth Bass-Serre trees / L.Mosher, M.Sageev, K.Whyte. - Providence: American Mathematical Society, 2011. - v, 105 p. - (Memoirs of the American Mathematical Society; vol.214, N 1008). - Bibliogr.: p.101-103. - Ind.: p.105. - ISBN 978-0-8218-4712-1; ISSN 0065-9266
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Chapter 1 Introduction ......................................... 1
1.1 Example applications ....................................... 2
1.2 The methods of proof: a special case ....................... 3
1.3 The general setting ........................................ 4
1.4 Statements of results ...................................... 7
1.5 Structure of the paper ..................................... 9
Chapter 2 Preliminaries ....................................... 11
2.1 Coarse language ........................................... 11
2.2 Coarse properties of subgroups ............................ 14
2.3 Coboundedness principle ................................... 15
2.4 Bass-Serre trees and Bass-Serre complexes ................. 17
2.5 Irreducible graphs of groups .............................. 19
2.6 Coarse PD(n) spaces and groups ............................ 21
2.7 The methods of proof: the general case .................... 26
Chapter 3 Depth Zero Vertex Rigidity .......................... 29
3.1 A sufficient condition for depth zero vertex rigidity ..... 30
3.2 Proof of the Depth Zero Vertex Rigidity Theorem ........... 36
Chapter 4 Finite Depth Graphs of Groups ....................... 49
4.1 Definitions and examples .................................. 49
4.2 Proof of the Vertex-Edge Rigidity Theorem 2.11 ............ 53
4.3 Reduction of finite depth graphs of groups ................ 56
Chapter 5 Tree Rigidity ....................................... 59
5.1 Examples and motivations .................................. 59
5.2 Outline of the Tree Rigidity Theorem ...................... 60
5.3 Special case: isolated edge spaces ........................ 61
5.4 Special case: all edges have depth one .................... 64
5.4.1 Proof of Lemma 5.5: an action on a 2-complex ............ 71
5.4.2 Proof of the Tracks Theorem 5.7 ......................... 74
5.5 Proof of the Tree Rigidity Theorem ........................ 81
Chapter 6 Main Theorems ....................................... 87
Chapter 7 Applications and Examples ........................... 89
7.1 Patterns of edge spaces in a vertex space ................. 89
7.2 Hn vertex groups and Z edge groups ........................ 90
7.3 H3 vertex groups and surface fiber edge groups ............ 91
7.4 Surface vertex groups and cyclic edge groups .............. 92
7.5 Graphs of abelian groups .................................. 94
7.6 Quasi-isometry groups and classification .................. 96
Bibliography .................................................. 101
Index ......................................................... 105
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