Preface

Hilbert’s fifth problem, from his famous list of twenty-three problems in

mathematics from 1900, asks for a topological description of Lie groups,

without any direct reference to smooth structure. As with many of Hilbert’s

problems, this question can be formalised in a number of ways, but one com-

monly accepted formulation asks whether any locally Euclidean topological

group is necessarily a Lie group. This question was answered aﬃrmatively by

Montgomery and Zippin [MoZi1952] and Gleason [Gl1952]; see Theorem

1.1.9. As a byproduct of the machinery developed to solve this problem, the

structure of locally compact groups was greatly clarified, leading in particu-

lar to the very useful Gleason-Yamabe theorem (Theorem 1.1.13) describing

such groups. This theorem (and related results) have since had a number

of applications, most strikingly in Gromov’s celebrated theorem [Gr1981]

on groups of polynomial growth (Theorem 1.3.1), and in the classification

of finite approximate groups (Theorem 1.2.12). These results in turn have

applications to the geometry of manifolds, and on related topics in geometric

group theory.

In the fall of 2011, I taught a graduate topics course covering these top-

ics, developed the machinery needed to solve Hilbert’s fifth problem, and

then used it to classify approximate groups and then finally to develop ap-

plications such as Gromov’s theorem. Along the way, one needs to develop

a number of standard mathematical tools, such as the Baker-Campbell-

Hausdorff formula relating the group law of a Lie group to the associated

Lie algebra, the Peter-Weyl theorem concerning the representation-theoretic

structure of a compact group, or the basic facts about ultrafilters and ultra-

products that underlie nonstandard analysis.

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