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A Universal Construction for Groups Acting Freely on Real Trees

Book Description

The theory of R-trees is a well-established and important area of geometric group theory and in this book the authors introduce a construction that provides a new perspective on group actions on R-trees. They construct a group RF(G), equipped with an action on an R-tree, whose elements are certain functions from a compact real interval to the group G. They also study the structure of RF(G), including a detailed description of centralizers of elements and an investigation of its subgroups and quotients. Any group acting freely on an R-tree embeds in RF(G) for some choice of G. Much remains to be done to understand RF(G), and the extensive list of open problems included in an appendix could potentially lead to new methods for investigating group actions on R-trees, particularly free actions. This book will interest all geometric group theorists and model theorists whose research involves R-trees.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. 1 - Introduction
    1. 1.1 Finite Words and Free Groups
    2. 1.2 Words Over a Discretely Ordered Abelian Group Λ
    3. 1.3 The Case Where Λ is Densely Ordered
    4. 1.4 The Case Where Λ = ℝ
    5. 1.5 Contents of the Book
  8. 2 - The Group ℛℱ (G)
    1. 2.1 The Monoid (ℱ(G), ∗)
    2. 2.2 Reduced Functions and Reduced Multiplication
    3. 2.3 Cancellation Theory for ℛℱ (G)
    4. 2.4 Proof of Theorem 2.13
    5. 2.5 The Subgroup G0
    6. 2.6 Appendix to Chapter 2
    7. 2.7 Exercises
  9. 3 - The ℝ-Tree XG Associated With ℛℱ (G)
    1. 3.1 Introduction
    2. 3.2 Construction of XG
    3. 3.3 Completeness and Transitivity
    4. 3.4 Cyclic Reduction
    5. 3.5 Classification of Elements
    6. 3.6 Bounded Subgroups
    7. 3.7 Presenting E(G)
    8. 3.8 A Remark Concerning Universality
    9. 3.9 The Degree of Vertices of XG
    10. 3.10 Exercises
  10. 4 - Free ℝ-Tree Actions and Universality
    1. 4.1 Introduction
    2. 4.2 An Embedding Theorem
    3. 4.3 Universality of ℛℱ-Groups and Their Associated ℝ-Trees
    4. 4.4 Exercises
  11. 5 - Exponent Sums
    1. 5.1 Introduction
    2. 5.2 Some Measure Theory
    3. 5.3 The Maps μg
    4. 5.4 The Maps eg
    5. 5.5 The Map eG
  12. 6 - Functoriality
    1. 6.1 Introduction
    2. 6.2 The Functor RF(–)
    3. 6.3 The Functor RF(–)
    4. 6.4 The Functor RF(–)
    5. 6.5 A Remark Concerning the Automorphism Group of ℛℱ(G)/E(G)
    6. 6.6 Exercises
  13. 7 - Conjugacy of Hyperbolic Elements
    1. 7.1 Introduction
    2. 7.2 The Equivalence Relation τG and the Conjugacy Theorem
    3. 7.3 Normalisers of Infinite Cyclic Hyperbolic Subgroups
    4. 7.4 The Main Lemma
    5. 7.5 Proof of Theorem 7.5
    6. 7.6 Exercises
  14. 8 - The Centralisers of Hyperbolic Elements
    1. 8.1 Introduction
    2. 8.2 A Preliminary Lemma
    3. 8.3 The Periods of a Hyperbolic Function
    4. 8.4 The Subset CF of Cf
    5. 8.5 The Subset CF of Cf
    6. 8.6 The Subset Cf of Cf
    7. 8.7 The Main Result
    8. 8.8 The Case when cf is Cyclic
    9. 8.9 An Application: The Non-Existence of Soluble Normal Subgroups
    10. 8.10 More on Centralisers
    11. 8.11 Exercises
  15. 9 - Test Functions: Basic Theory and First Applications
    1. 9.1 Introduction
    2. 9.2 Test Functions: Definition and First Properties
    3. 9.3 Existence of Test Functions
    4. 9.4 The Maps λf
    5. 9.5 Locally Incompatible Test Functions
    6. 9.6 A Subgroup Theorem
    7. 9.7 The Maps λs
    8. 9.8 Exercises
  16. 10 - Test Functions: Existence Theorem and Further Applications
    1. 10.1 Introduction
    2. 10.2 Incompatible Test Functions with Prescribed Centraliser
    3. 10.3 Proof of Theorem 10.1
    4. 10.4 The Cardinality of ℛℱ(G) Revisited
    5. 10.5 An Embedding Theorem
    6. 10.6 The Subgroup Generated by a Set of Incompatible Test Functions
    7. 10.7 A Structure Theorem for ℛℱ (G) and ℛℱ (G)/E(G)
    8. 10.8 Exercises
  17. 11 - A Generalisation to Groupoids
    1. 11.1 Introduction
    2. 11.2 The Construction
    3. 11.3 Cancellation Theory for Aℛℱ (S, G)
    4. 11.4 Proof of Theorem 11.8
    5. 11.5 Cyclic Reduction and Exponent Sums
    6. 11.6 Lyndon Length Functions on Groupoids
    7. 11.7 Functoriality
    8. 11.8 Exercises
  18. Appendix A - The Basics of Λ-Trees
  19. Appendix B - Some Open Problems
  20. References
  21. Index