Chapter 1

Introduction

This text focuses on three related topics:

• Hilbert’s fifth problem on the topological description of Lie groups,

as well as the closely related (local) classification of locally compact

groups (the Gleason-Yamabe theorem, see Theorem 1.1.13);

• Approximate groups in nonabelian groups, and their classification

[Hr2012], [BrGrTa2011] via the Gleason-Yamabe theorem; and

• Gromov’s theorem [Gr1981] on groups of polynomial growth, as

proven via the classification of approximate groups (as well as some

consequences to fundamental groups of Riemannian manifolds).

These three families of results exemplify two broad principles (part of

what I like to call the the dichotomy between structure and randomness

[Ta2008]):

• (Rigidity) If a group-like object exhibits a weak amount of regular-

ity, then it (or a large portion thereof) often automatically exhibits

a strong amount of regularity as well.

• (Structure) Furthermore, this strong regularity manifests itself ei-

ther as Lie type structure (in continuous settings) or nilpotent type

structure (in discrete settings). (In some cases, “nilpotent” should

be replaced by sister properties such as “abelian”, “solvable”, or

“polycyclic”.)

Let us illustrate these two principles with two simple examples, one in

the continuous setting and one in the discrete setting. We begin with a

continuous example. Given an n×n complex matrix A ∈ Mn(C), define the

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