**Mathematical Surveys and Monographs**

Volume: 75;
2000;
195 pp;
Hardcover

MSC: Primary 03;
Secondary 22; 54

**Print ISBN: 978-0-8218-2002-5
Product Code: SURV/75**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

**Electronic ISBN: 978-1-4704-1302-6
Product Code: SURV/75.E**

List Price: $68.00

AMS Member Price: $54.40

MAA Member Price: $61.20

# Classification and Orbit Equivalence Relations

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*Greg Hjorth*

Actions of Polish groups are ubiquitous in mathematics. In
certain branches of ergodic theory and functional analysis, one finds
a systematic study of the group of measure-preserving transformations
and the unitary group. In logic, the analysis of countable models
intertwines with results concerning the actions of the infinite
symmetric group.

This text develops the theory of Polish group actions entirely from
scratch, ultimately presenting a coherent theory of the resulting
orbit equivalence classes that may allow complete classification by
invariants of an indicated form. The book concludes with a criterion
for an orbit equivalence relation classifiable by countable structures
considered up to isomorphism.

This self-contained volume offers a complete treatment of this
active area of current research and develops a difficult general
theory classifying a class of mathematical objects up to some relevant
notion of isomorphism or equivalence.

Greg Hjorth received the Carol Karp Prize for outstanding work on turbulence and countable Borel equivalence relations from the Association of Symbolic Logic.

#### Readership

Graduate students and research mathematicians interested in set theory, and topological groups and Lie groups.

#### Reviews & Endorsements

Presents a beautiful example of a theory whose creation and consequences, in one way or another, involve most fields of mathematics.

-- Bulletin of the LMS

This monograph is an important contribution to the ongoing study … of definable equivalence relations on Polish spaces … essential reading for anyone interested in the subject of definable equivalence relations, but is also important for mathematicians working in areas lying outside the usual boundaries of set theory.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Classification and Orbit Equivalence Relations

- Contents ix10 free
- Preface xi12 free
- Chapter 1. An outline 120 free
- Chapter 2. Definitions and technicalities 524
- Chapter 3. Turbulence 3756
- Chapter 4. Classifying homeomorphisms 5978
- Chapter 5. Infinite dimensional group representations 7594
- Chapter 6. A generalized Scott analysis 81100
- Chapter 7. GE groups 115134
- Chapter 8. The dark side 149168
- Chapter 9. Beyond Borel 155174
- Chapter 10. Looking ahead 177196
- Appendix A. Ordinals 181200
- Appendix B. Notation 185204
- Bibliography 189208
- Index 193212