1.1. Hilbert’s fifth problem 7 multiple of the identity. On the other hand, [g, h] has determinant 1. Given that it is so close to the identity, it must therefore be the identity (if ε is small enough). In other words, g is central in G , and is thus a multiple of the identity. But this contradicts the hypothesis that there are no central elements other than multiples of the identity, and we are done. Commutator estimates such as (1.2) will play a fundamental role in many of the arguments we will see in this text as we saw above, such estimates combine very well with a discreteness hypothesis, but will also be very useful in the continuous setting. Exercise 1.0.3. Generalise Jordan’s theorem to the case when G is a finite subgroup of GLn(C) rather than of Un(C). (Hint: The elements of G are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of Cn. However, if one averages that inner product by the finite group G, one obtains a new inner product on Cn that is preserved by G, which allows one to conjugate G to a subgroup of Un(C). This averaging trick is (a small) part of Weyl’s unitary trick in representation theory.) Remark 1.0.3. We remark that one can strengthen Jordan’s theorem fur- ther by relaxing the finiteness assumption on G to a periodicity assumption see Chapter 11. Exercise 1.0.4 (Inability to discretise nonabelian Lie groups). Show that if n ≥ 3, then the orthogonal group On(R) cannot contain arbitrarily dense finite subgroups, in the sense that there exists an ε = εn 0 depending only on n such that for every finite subgroup G of On(R), there exists a ball of radius ε in On(R) (with, say, the operator norm metric) that is disjoint from G. What happens in the n = 2 case? Remark 1.0.4. More precise classifications of the finite subgroups of Un(C) are known, particularly in low dimensions. For instance, it is a classical result that the only finite subgroups of SO3(R) (which SU2(C) is a double cover of) are isomorphic to either a cyclic group, a dihedral group, or the symmetry group of one of the Platonic solids. 1.1. Hilbert’s fifth problem One of the fundamental categories of objects in modern mathematics is the category of Lie groups, which are rich in both algebraic and analytic structure. Let us now briefly recall the precise definition of what a Lie group is. Definition 1.1.1 (Smooth manifold). Let d ≥ 0 be a natural number. A d- dimensional topological manifold is a Hausdorff topological space M which

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