14 1. Introduction

to an open subgroup) and also ignoring the very small scales (by allowing

one to quotient out by a small group). In special cases, the conclusion of the

theorem can be simplified. For instance, it is easy to see that an open sub-

group G of a topological group G is also closed (since the complement G\G

is a union of cosets of G ), and so if G is connected, there are no open sub-

groups other than G itself. Thus, in the connected case of Theorem 1.1.13,

one can take G = G . In a similar spirit, if G has the no small subgroups

(NSS) property, that is to say that there exists an open neighbourhood of

the identity that contains no nontrivial subgroups of G, then we can take

K to be trivial. Thus, as a special case of the Gleason-Yamabe theorem, we

see that all connected NSS locally compact groups are Lie; in fact it is not

diﬃcult to then conclude that any locally compact NSS group (regardless of

connectedness) is Lie. Conversely, this claim (which we isolate as Corollary

5.3.3) turns out to be a key step in the proof of Theorem 1.1.13, as we shall

see later. (It is also not diﬃcult to show that all Lie groups are NSS; see

Exercise 5.3.1.)

The proof of the Gleason-Yamabe theorem proceeds in a somewhat

lengthy series of steps in which the initial regularity (local compactness)

on the group G is gradually upgraded to increasingly stronger regularity

(e.g. metrisability, the NSS property, or the locally Euclidean property) un-

til one eventually obtains Lie structure; see Figure 1. A key turning point

in the argument will be the construction of a metric (which we call a Glea-

son metric) on (a large portion of) G which obeys a commutator estimate

similar to (1.2).

While the Gleason-Yamabe theorem does not completely classify all lo-

cally compact groups (as mentioned earlier, it primarily controls the medium-

scale behaviour, and not the very fine-scale or very coarse-scale behaviour,

of such groups), it is still powerful enough for a number of applications, to

which we now turn.

1.2. Approximate groups

We now discuss what appears at first glance to be an unrelated topic, namely

that of additive combinatorics (and its noncommutative counterpart, multi-

plicative combinatorics). One of the main objects of study in either additive

or multiplicative combinatorics are approximate groups — sets A (typically

finite) contained in an additive or multiplicative ambient group G that are

“almost groups” in the sense that they are “almost” closed under either

addition or multiplication. (One can also consider abstract approximate

groups that are not contained in an ambient genuine group, but we will not

do so here.)