Hilbert's Fifth Problem and Related Topics
Description:... 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,
Montgomery-Zippin, 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 structuraltheory of real Lie groups and Lie algebras (emphasising the role of
one-parameter groups and the Baker-Campbell-Hausdorff formula), then
presenting a proof of the Gleason-Yamabe 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 model-theoretic preliminaries (most notably the theory
of ultraproducts), the combinatorial applications of the Gleason-Yamabetheorem to approximate groups and groups of polynomial growth are thengiven. A large number of relevant exercises and other supplementary
material are also provided.
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