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Hilbert's Fifth Problem and Related Topics (Hardcover)
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Hilbert's Fifth Problem and Related Topics (Hardcover)
Series: Graduate Studies in Mathematics
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Winner of the 2015 Prose Award for Best Mathematics Book! 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 structural theory 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-Yamabe theorem to approximate groups and groups of
polynomial growth are then given. A large number of relevant
exercises and other supplementary material are also provided.
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