Hilbert's fifth problem and related topics
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 theorem with additional tools of point set topology the description of locally compact groups is improved in various situations. In the remaining Chapters 7--10 of Part~I the ``soft analysis'' of locally compact groups is replaced by the ``hard analysis'' of approximate groups. For the classification of approximate groups ultraproducts and approximate Lie structures are used. This classification has applications to geometric group theory and the geometry of manifolds and it includes also the Gromov's theorem.NEWLINENEWLINEPart~II with 11~chapters contains results that are at least in some aspects related to results of Part~I. They include the Jordan-Schur theorem and results on nilpotent groups, the associativity of the Baker-Campbell-Hausdorff-Dynkin law, local groups, local extensions of Lie groups, nonabelian Fourier analysis, and Loeb measure. It is also shown that the open Hilbert-Smith conjecture (which is a~generalization of Hilbert's fifth problem) follows from its \(p\)-adic reduction.
scientific article; zbMATH DE number 6329298