Contents
Preface xi
Notation xii
Acknowledgments xiii
Part 1. Hilbert’s Fifth Problem
Chapter 1. Introduction 3
§1.1. Hilbert’s fifth problem 7
§1.2. Approximate groups 14
§1.3. Gromov’s theorem 20
Chapter 2. Lie groups, Lie algebras, and the Baker-Campbell-
Hausdorff formula 25
§2.1. Local groups 26
§2.2. Some differential geometry 30
§2.3. The Lie algebra of a Lie group 34
§2.4. The exponential map 35
§2.5. The Baker-Campbell-Hausdorff formula 43
Chapter 3. Building Lie structure from representations and metrics 53
§3.1. The theorems of Cartan and von Neumann 57
§3.2. Locally compact vector spaces 60
§3.3. From Gleason metrics to Lie groups 64
Chapter 4. Haar measure, the Peter-Weyl theorem, and compact or
abelian groups 73
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