Introduction to Lambda Trees.
Title:
Introduction to Lambda Trees.
Author:
Chiswell, Ian.
ISBN:
9789812810533
Personal Author:
Physical Description:
1 online resource (327 pages)
Contents:
Contents -- Chapter 1. Preliminaries -- 1. Ordered abelian groups -- 2. Metric spaces -- 3. Graphs and simplicial trees -- 4. Valuations -- Chapter 2. Λ-trees and their Construction -- 1. Definition and elementary properties -- 2. Special properties of R-trees -- 3. Linear subtrees and ends -- 4. Lyndon length functions -- Chapter 3. Isometries of Λ-trees -- 1. Theory of a single isometry -- 2. Group actions as isometries -- 3. Pairs of isometries -- 4. Minimal actions -- Chapter 4. Aspects of Group Actions on Λ-trees -- 1. Introduction -- 2. Actions of special classes of groups -- 3. The action of the special linear group -- 4. Measured laminations -- 5. Hyperbolic surfaces -- 6. Spaces of actions on R-trees -- Chapter 5. Free Actions -- 1. Introduction -- 2. Harrison's Theorem -- 3. Some examples -- 4. Free actions of surface groups -- 5. Non-standard free groups -- Chapter 6. Rips' Theorem -- 1. Systems of isometries -- 2. Minimal components -- 3. Independent generators -- 4. Interval exchanges and conclusion -- References -- Index of Notation -- Index.
Abstract:
The theory of Λ-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 Λ-trees was established by Morgan and Shalen, who showed how to compactify a generalisation of Teichmüller space for a finitely generated group using R -trees. In that work they were led to define the idea of a Λ-tree, where Λ 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 Λ, including some interesting connections with model theory. Introduction to Λ-Trees will prove to be useful for mathematicians and research students in algebra and topology. Contents: Λ-Trees and Their Construction; Isometries of Λ-Trees; Aspects of Group Actions on Λ-Trees; Free Actions; Rips' Theorem. Readership: Mathematicians and research students in algebra and topology.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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