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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