Hardcover ISBN:  9781470415648 
Product Code:  GSM/153 
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Electronic ISBN:  9781470418564 
Product Code:  GSM/153.E 
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Book DetailsGraduate Studies in MathematicsVolume: 153; 2014; 338 ppMSC: Primary 22; 11; 20;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, MontgomeryZippin, 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 oneparameter groups and the BakerCampbellHausdorff formula), then presenting a proof of the GleasonYamabe 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 modeltheoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the GleasonYamabe 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.ReadershipGraduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.

Table of Contents

Part 1. Hilbert’s Fifth Problem

Chapter 1. Introduction

Chapter 2. Lie groups, Lie algebras, and the BakerCampbellHausdorff formula

Chapter 3. Building Lie structure from representations and metrics

Chapter 4. Haar measure, the PeterWeyl theorem, and compact or abelian groups

Chapter 5. Building metrics on groups, and the GleasonYamabe theorem

Chapter 6. The structure of locally compact groups

Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis

Chapter 8. Models of ultra approximate groups

Chapter 9. The microscopic structure of approximate groups

Chapter 10. Applications of the structural theory of approximate groups

Part 2. Related articles

Chapter 11. The JordanSchur theorem

Chapter 12. Nilpotent groups and nilprogressions

Chapter 13. Ado’s theorem

Chapter 14. Associativity of the BakerCampbellHausdorffDynkin law

Chapter 15. Local groups

Chapter 16. Central extensions of Lie groups, and cocycle averaging

Chapter 17. The HilbertSmith conjecture

Chapter 18. The PeterWeyl theorem and nonabelian Fourier analysis

Chapter 19. Polynomial bounds via nonstandard analysis

Chapter 20. Loeb measure and the triangle removal lemma

Chapter 21. Two notes on Lie groups


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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, MontgomeryZippin, 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 oneparameter groups and the BakerCampbellHausdorff formula), then presenting a proof of the GleasonYamabe 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 modeltheoretic preliminaries (most notably the theory of ultraproducts), the combinatorial applications of the GleasonYamabe 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.
Graduate students and research mathematicians interested in lie groups, topological groups, geometric group theory, and approximate groups.

Part 1. Hilbert’s Fifth Problem

Chapter 1. Introduction

Chapter 2. Lie groups, Lie algebras, and the BakerCampbellHausdorff formula

Chapter 3. Building Lie structure from representations and metrics

Chapter 4. Haar measure, the PeterWeyl theorem, and compact or abelian groups

Chapter 5. Building metrics on groups, and the GleasonYamabe theorem

Chapter 6. The structure of locally compact groups

Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis

Chapter 8. Models of ultra approximate groups

Chapter 9. The microscopic structure of approximate groups

Chapter 10. Applications of the structural theory of approximate groups

Part 2. Related articles

Chapter 11. The JordanSchur theorem

Chapter 12. Nilpotent groups and nilprogressions

Chapter 13. Ado’s theorem

Chapter 14. Associativity of the BakerCampbellHausdorffDynkin law

Chapter 15. Local groups

Chapter 16. Central extensions of Lie groups, and cocycle averaging

Chapter 17. The HilbertSmith conjecture

Chapter 18. The PeterWeyl theorem and nonabelian Fourier analysis

Chapter 19. Polynomial bounds via nonstandard analysis

Chapter 20. Loeb measure and the triangle removal lemma

Chapter 21. Two notes on Lie groups