Hilbert's fifth problem and related topics
type: Item
instance of: scholarly article
zbMATH Open document ID: 1298.22001
publication date: 2014-08-12T00:00:00Z
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 6329298
zbMATH Keywords: Lie groups
zbMATH Keywords: local Lie groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: locally compact groups
zbMATH Keywords: Haar measure
zbMATH Keywords: Gleason metric
zbMATH Keywords: approximate groups
zbMATH Keywords: Riemannian manifolds
author: Terence Tao
MaRDI profile type: Publication
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title: Hilbert's fifth problem and related topics
published in: Graduate Studies in Mathematics
review text: Hilbert's fifth problem asks about a~topological description of Lie groups without any direct reference to smooth structures. This question can be formalized in a~number of ways but one of a~commonly accepted formulation asks whether any locally Euclidean topological group is necessarily a~Lie group. This question was answered affirmatively by Gleason and by Montgomery and Zippin. The book focuses on three related topics: (a)~Topological description of Lie groups and the classification of locally compact groups, (b)~approximate groups in nonabelian groups and their classification via the Gleason-Yamabe theorem, and (c)~Gromov's theorem on finitely generated groups of polynomial growth and consequences to fundamental groups of Riemannian manifolds.NEWLINENEWLINEThe Gleason-Yamabe Theorem states that if \(G\) is a~locally compact group, then for every open neighbourhood~\(U\) of the identity there exists an open subgroup~\(G'\) of~\(G\) and a~compact normal subgroup \(K\subseteq U\) of~\(G'\) such that \(G'/K\) is isomorphic to a~Lie group. The proof of this theorem is divided into 12 steps which are the content of Chapters 2--5 of the first part of the book. To describe the basic analytic structure theory of Lie groups, they are related to the simpler concept of a~Lie algebra. By fundamental theorems of Lie, the Lie algebra can be used to locally reconstruct the Lie group using the Baker-Campbell-Hausdorff formula. A~Cartan's Theorem says that a~closed subgroup of a~Lie group is a~Lie group and by von Neumann, a~locally compact group with an injective continuous homomorphism into a~Lie group is a~Lie group. The Lie structure of a~locally compact group can be obtained from a~special type of metric called Gleason metric. To build useful representations and metric on locally compact groups Haar measures are exploited and in particular the Peter-Weyl theorem is introduced. The solution of Hilbert's fifth problem is proved in Chapter~6. By combining the Gleason-Yamabe the…
reviewed by: Miroslav Repický
description: scientific article; zbMATH DE number 6329298