1.1. Hilbert’s fifth problem 13 This conjecture is true in several cases for instance, one can show using the Peter-Weyl theorem (which we will discuss in Chapter 4) that it is true for compact groups, and we will later see that it is also true for connected locally compact groups (see Theorem 6.0.11). However, it is not quite true in general, as the following example shows. Exercise 1.1.4. Let p be a prime, and let T : Qp Qp be the automor- phism Tx := px. Let G := Qp T Z be the semidirect product of Qp and Z twisted by T more precisely, G is the Cartesian product Qp × Z with the product topology and the group law (x, n)(y, m) := (x + T n y, n + m). Show that G is a locally compact group which is not isomorphic to a Lie group, and that Zp × {0} is an open neighbourhood of the identity that contains no nontrivial normal subgroups of G. Conclude that the conjecture stated above is false. The difficulty in the above example was that it was not easy to keep a subgroup normal with respect to the entire group Qp T Z. Note however that G contains a “large” (and more precisely, open) subgroup Qp × {0} which is the projective limit of Lie groups. So the above examples do not rule out that the conjecture can still be salvaged if one passes from a group G to an open subgroup G . This is indeed the case: Theorem 1.1.13 (Gleason-Yamabe theorem [Gl1951, Ya1953b]). Let G obey the following hypotheses: (1) (Group-like object) G is a topological group. (2) (Weak regularity) G is locally compact. Then for every open neighbourhood U of the identity, there exists a sub- group G of G and a compact normal subgroup K of G with the following properties: (i) (G /K is close to G) G is an open subgroup of G, and K is con- tained in U. (ii) (Lie-type structure) G /K is isomorphic to a Lie group. The proof of this theorem will occupy the next few sections of this text, being finally proven in Chapter 5. As stated, G may depend on U, but one can in fact take the open subgroup G to be uniform in the choice of U we will show this in later sections. Theorem 1.1.9 can in fact be deduced from Theorem 1.1.13 and some topological arguments involving the invariance of domain theorem this will be shown in Chapter 6. The Gleason-Yamabe theorem asserts that locally compact groups are “essentially” Lie groups, after ignoring the very large scales (by restricting
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