Introduction To Lambda Trees

By: Ian Chiswell

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Hardcover | 1 March 2001

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328 Pages

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16.2 x 22.4 x 2.2

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The theory of Lambda-trees has its origin in the work of Lyndon on length functions in groups. The first definition of an "R"-tree was given by Tits in 1977. The importance of Lambda-trees was established by Morgan and Shalen, who showed how to compactify a generalization of Teichmuller space for a finitely generated group using "R"-trees. In that work they were led to define the idea of a Lambda-tree, where Lambda is an arbitrary ordered abelian group. Since then there has been much progress in understanding the structure of groups acting on "R"-trees, notably Rips' theorem on free actions. There has also been some progress for certain other ordered abelian groups Lambda, including some interesting connections with model theory. "Introduction to Lambda-Trees" should prove useful for mathematicians and research students in algebra and topology.

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Preliminaries
Ordered abelian groups	p. 1
Metric spaces	p. 6
Graphs and simplicial trees	p. 14
Valuations	p. 22
[Lambda]-trees and their Construction
Definition and elementary properties	p. 29
Special properties of R-trees	p. 41
Linear subtrees and ends	p. 56
Lyndon length functions	p. 69
Isometries of [Lambda]-trees
Theory of a single isometry	p. 79
Group actions as isometries	p. 88
Pairs of isometries	p. 98
Minimal actions	p. 115
Aspects of Group Actions on [Lambda]-trees
Introduction	p. 127
Actions of special classes of groups	p. 130
The action of the special linear group	p. 144
Measured laminations	p. 150
Hyperbolic surfaces	p. 173
Spaces of actions on R-trees	p. 203
Free Actions
Introduction	p. 215
Harrison's Theorem	p. 219
Some examples	p. 229
Free actions of surface groups	p. 233
Non-standard free groups	p. 237
Rips' Theorem
Systems of isometries	p. 251
Minimal components	p. 265
Independent generators	p. 278
Interval exchanges and conclusion	p. 290
References	p. 297
Index of Notation	p. 307
Index	p. 309
Table of Contents provided by Syndetics. All Rights Reserved.

Introduction To Lambda Trees

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Hardcover

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