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.)

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