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