**Graduate Studies in Mathematics**

Volume: 153;
2014;
338 pp;
Hardcover

MSC: Primary 22; 11; 20;

Print ISBN: 978-1-4704-1564-8

Product Code: GSM/153

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

**Electronic ISBN: 978-1-4704-1856-4
Product Code: GSM/153.E**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

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#### Supplemental Materials

# Hilbert’s Fifth Problem and Related Topics

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*Terence Tao*

Winner of the 2015 Prose Award for Best Mathematics Book!

In the fifth of his famous list of 23
problems, Hilbert asked if every topological group which was locally
Euclidean was in fact a Lie group. Through the work of Gleason,
Montgomery-Zippin, Yamabe, and others, this question was solved
affirmatively; more generally, a satisfactory description of the
(mesoscopic) structure of locally compact groups was
established. Subsequently, this structure theory was used to prove
Gromov's theorem on groups of polynomial growth, and more recently in
the work of Hrushovski, Breuillard, Green, and the author on the
structure of approximate groups.

In this graduate text, all of this material is presented in a
unified manner, starting with the analytic structural theory of real
Lie groups and Lie algebras (emphasising the role of one-parameter
groups and the Baker-Campbell-Hausdorff formula), then presenting a
proof of the Gleason-Yamabe structure theorem for locally compact
groups (emphasising the role of Gleason metrics), from which the
solution to Hilbert's fifth problem follows as a corollary. After
reviewing some model-theoretic preliminaries (most notably the theory
of ultraproducts), the combinatorial applications of the
Gleason-Yamabe theorem to approximate groups and groups of polynomial
growth are then given. A large number of relevant exercises and other
supplementary material are also provided.

#### Readership

Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.

#### Table of Contents

# Table of Contents

## Hilbert's Fifth Problem and Related Topics

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Part 1: Hilbert’s Fifth Problem 116 free
- Chapter 1. Introduction 318
- Chapter 2. Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula 2540
- Chapter 3. Building Lie structure from representations and metrics 5368
- Chapter 4. Haar measure, the Peter-Weyl theorem, and compact or abelian groups 7388
- Chapter 5. Building metrics on groups, and the Gleason-Yamabe theorem 99114
- Chapter 6. The structure of locally compact groups 125140
- Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis 139154
- Chapter 8. Models of ultra approximate groups 167182
- Chapter 9. The microscopic structure of approximate groups 197212
- Chapter 10. Applications of the structural theory of approximate groups 219234
- Part 2: Related articles 231246
- Chapter 11. The Jordan-Schur theorem 233248
- Chapter 12. Nilpotent groups and nilprogressions 237252
- Chapter 13. Ado’s theorem 249264
- Chapter 14. Associativity of the Baker-Campbell-Hausdorff-Dynkin law 259274
- Chapter 15. Local groups 265280
- Chapter 16. Central extensions of Lie groups, and cocycle averaging 277292
- Chapter 17. The Hilbert-Smith conjecture 289304
- Chapter 18. The Peter-Weyl theorem and nonabelian Fourier analysis 297312
- Chapter 19. Polynomial bounds via nonstandard analysis 307322
- Chapter 20. Loeb measure and the triangle removal lemma 313328
- Chapter 21. Two notes on Lie groups 325340
- Bibliography 329344
- Index 335350 free
- Back Cover Back Cover1354