Hilbert著名的23个问题的第5个问题为：是否每个局部Euclid拓扑群实际上都是Lie群。通过Gleason、Montgomery-Zippin、 Yamabe等人的工作，这个问题得到了肯定的回答；更一般地，他们建立了局部紧群令人满意的（介观）结构理论。随后，这种结构理论被用来证明Gromov关于多项式增长群的定理，也用在最近Hrushovski、Breuillard、Green和作者关于近似群结构的工作中。 本书所有材料以统一的方式呈现，从实Lie群和Lie代数的分析结构理论（强调单参数群的作用和Baker-Campbell-Hausdorff公式）开始，然后给出局部紧群的Gleason-Yamabe结构定理的证明（强调Gleason度量的作用），由此得到Hilbert第五问题的解答。在回顾了一些模型论基础知识（特别是超积理论）之后，作者给出了Gleason-Yamabe定理在多项式增长群和近似群中的组合应用。本书还提供了大量相关练习和其他补充材料供读者参考。

前辅文
Part 1. Hilbert's Fifth Problem
  Chapter 1. Introduction
   §1.1. Hilbert's fifth problem
   §1.2. Approximate groups
   §1.3. Gromov's theorem
  Chapter 2. Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula
   §2.1. Local groups
   §2.2. Some differential geometry
   §2.3. The Lie algebra of a Lie group
   §2.4. The exponential map
   §2.5. The Baker-Campbell-Hausdorff formula
  Chapter 3. Building Lie structure from representations and metrics
   §3.1. The theorems of Cartan and von Neumann
   §3.2. Locally compact vector spaces
   §3.3. From Gleason metrics to Lie groups
  Chapter 4. Haar measure, the Peter-Weyl theorem, and compact or abelian groups
   §4.1. Haar measure
   §4.2. The Peter-Weyl theorem
   §4.3. The structure of locally compact abelian groups
  Chapter 5. Building metrics on groups, and the Gleason-Yamabe theorem
   §5.1. Warmup: the Birkhoff-Kakutani theorem
   §5.2. Obtaining the commutator estimate via convolution
   §5.3. Building metrics on NSS groups
   §5.4. NSS from subgroup trapping
   §5.5. The subgroup trapping property
   §5.6. The local group case
  Chapter 6. The structure of locally compact groups
   §6.1. Van Dantzig's theorem
   §6.2. The invariance of domain theorem
   §6.3. Hilbert's fifth problem
   §6.4. Transitive actions
  Chapter 7. Ultraproducts as a bridge between hard analysis and soft analysis
   §7.1. Ultrafilters
   §7.2. Ultrapowers and ultralimits
   §7.3. Nonstandard finite sets and nonstandard finite sums
   §7.4. Asymptotic notation
   §7.5. Ultra approximate groups
  Chapter 8. Models of ultra approximate groups
   §8.1. Ultralimits of metric spaces (Optional)
   §8.2. Sanders-Croot-Sisask theory
   §8.3. Locally compact models of ultra approximate groups
   §8.4. Lie models of ultra approximate groups
  Chapter 9. The microscopic structure of approximate groups
   §9.1. Gleason's lemma
   §9.2. A cheap version of the structure theorem
   §9.3. Local groups
  Chapter 10. Applications of the structural theory of approximate groups
   §10.1. Sets of bounded doubling
   §10.2. Polynomial growth
   §10.3. Fundamental groups of compact manifolds (optional)
   §10.4. A Margulis-type lemma
Part 2. Related Articles
  Chapter 11. The Jordan-Schur theorem
   §11.1. Proofs
  Chapter 12. Nilpotent groups and nilprogressions
   §12.1. Some elementary group theory
   §12.2. Nilprogressions
  Chapter 13. Ado's theorem
   §13.1. The nilpotent case
   §13.2. The solvable case
   §13.3. The general case
  Chapter 14. Associativity of the Baker-Campbell-Hausdorff-Dynkin law
  Chapter 15. Local groups
   §15.1. Lie's third theorem
   §15.2. Globalising a local group
   §15.3. A nonglobalisable group
  Chapter 16. Central extensions of Lie groups, and cocycle averaging
   §16.1. A little group cohomology
   §16.2. Proof of theorem
  Chapter 17. The Hilbert-Smith conjecture
   §17.1. Periodic actions of prime order
   §17.2. Reduction to the p-adic case
  Chapter 18. The Peter-Weyl theorem and nonabelian Fourier analysis
   §18.1. Proof of the Peter-Weyl theorem
   §18.2. Nonabelian Fourier analysis
  Chapter 19. Polynomial bounds via nonstandard analysis
  Chapter 20. Loeb measure and the triangle removal lemma
   §20.1. Loeb measure
   §20.2. The triangle removal lemma
  Chapter 21. Two notes on Lie groups
Bibliography
Index