4.1. Topological and locally compact groups
Definition 4.1.1. A group is a topological group if it is a topological space and the multiplication and the inversion maps are continuous. A topological group is a locally compact group if it is a Hausdorff space with a relative open, compact set.
The Hausdorffness condition can be removed from the definition of a locally compact group. Analysis on Hausdorff spaces factors then through a quotient group, which is Hausdorff.
Proposition 4.1.2 ([27, Proposition 1.1.6]). Let G be a topological group. A continuous map fromGto aT1-space factors through the quotient groupG/cl{1}, wherecl{1}is the closure of the set, which contains only the identity element and is a closed, normal subgroup. The quotient groupG/cl{1} is a Hausdorff topological group.
The assumption thatGis locally compact is crucial. This property is essentially equivalent to the existence of quasi-invariant Radon measures on a group. A Radon measure is a functional on the space Cc(G) of continuous, compactly supported functionsG→C. In a case where the group does not admit a compact, relatively open set, the spaceCc(G)contains no non-zero element.
4.2. The identity component
Definition-Theorem 4.2.1 ([27, Proposition 4.1.2]). The path-connected compo-nent of the unit element in a topological group is a closed, normal subgroup. IfG is a locally compact group, we denote this component byG0.
SinceG0is normal, we can consider the group extension 1→G0→G→G/G0→1.
This extension provides us with a good starting point for understanding the local structure of a locally compact group. It also allows us to introduce the following standard definitions:
Definition 4.2.2. LetGbe a locally compact group.
• The groupGis (path-)connected ifG=G0.
• The groupGis almost connected if the quotient groupG/G0 is compact.
• The groupGis locally pro-finite if the groupG0={1}.
It follows that for each topological group G, the quotient group G/G0 is a totally disconnected group. We want to have a sufficiently flexible definition of a Lie group, to the extent that discrete groups are Lie group. We also want to remove
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the condition of being para-compact. We consider a singleton, i.e., a point with the discrete topology, as the zero-dimensional Euclidean space.
Definition 4.2.3. A smooth manifold is a topological space which is locally home-omorphic to the Euclidean space, possibly of dimension zero, and whose transition maps are smooth. A Lie group is a smooth manifold whose group operations are smooth.
We will see shortly that almost connected groups are projective limits of such Lie groups.
4.3. Van Dantzig’s Theorem and its consequences Theorem 4.3.1 (Van Dantzig’s Theorem).
• Van Dantzig’s Theorem: Every locally profinite group contains a neighbor-hood base at the identity of open, compact subgroups.
• In a locally pro-finite group, every compact subgroup is contained in an open, compact subgroup.
Proof. Van Dantzig’s Theorem is well-known. See e.g. [93, Theorem 2.3, page 54] or [27, Theorem 4.1.6, page 95] for a proof. The second point is a conclusion of van Dantzig’s Theorem; letGbe a totally disconnected group, and letK be a compact subgroup. Pick an open compact subgroupO ofG. Consider
O0= \
k∈K
k−1Ok.
There exists a finite set F ⊂K such that O0 = \
k∈F
k−1Ok,
henceO0 is a compact, open subgroup as a closed subset ofK·O·K and commutes with any element ofK. The group generated byO0 andK equalsO0·Kand is an
open compact group, which containsK.
Corollary 4.3.2 (Every compact subgroup is contained in an almost connected, open subgroup). LetGbe a locally compact group with a compact subgroupK, then it contains a closed subgroupH
(1) which is open, (2) almost connected, (3) and which containsK.
Proof. Consider the surjectionq:GG/G0. Consider an open, compact group O ⊂G/G0 withO ⊃q(K), which exists by Theorem 4.3.1. The pullback q−1(O)is an open, closed and almost connected subgroup, which containsK.
4.4. Approximation by Lie groups
The following results are often referred to as the solution of Hilbert’s fifth problem.
Theorem 4.4.1 (Gleason-Yamabe Theorem). Let Gbe a locally compact, almost connected group, then it admits a net1 of normal compact subgroups N = {N}, which is partially ordered by inclusion, and satisfies
1It is sufficient to consider sequences if and only ifGis metrizable.
4.4. APPROXIMATION BY LIE GROUPS 61
(1) G= lim
← G/N, or equivalently T
N∈NN ={1}, (2) G/N is isomorphic to a Lie group.
Proofs can be found in [131, Theorem 5, page 364] and [93, Theorem 4.6, page 175]. We have introduced Lie groups in a way that they include discrete groups. We give two special cases as examples:
Example 4.4.2.
• LetGbe a compact group, thus every irreducible representation ofGis unitarizable and finite-dimensional by the Peter-Weyl Theorem. LetSbe the set of finite subsets of the irreducible representations, which is partially ordered by inclusion, and forS ∈SdefineρS =⊕ρ∈Sρ. It is true thatG is the projective limit of the images ofρS:
G∼= lim
← S⊂Sfinite
imρS.
The image of everyρS is certainly closed asGis compact and is a subset of the compact Lie group U(dim(ρS)). Every closed subgroup of a Lie group is a Lie group. So every compact group is homeomorphic to a projective limit of compact Lie groups. In particular, every compact, totally disconnected group is a projective limit of finite groups, i.e., a pro-finite group.
• Let A be a locally compact abelian group. Every neighborhood of the identity contains a compact open normal subgroupK such thatA/K∼= Rn×Tm×D for some discrete abelian groupD, so it is a Lie group by definition. See [58, Corollary 7.54].
Corollary 4.4.3. LetGbe a locally compact group, letK be a compact subgroup.
Then there exists an open, closed subgroupG⊂Gand a net of compact subgroups N, which is partially ordered by inclusion, and satisfies
(1) GcontainsK, (2) G= lim
← G/N, or equivalently T
N∈NN ={1}, (3) G/N is isomorphic to a Lie group.
Example 4.4.4 (Lie group and locally pro-finite groups).
• If Gis a Lie group, then we can pickG=G and the family of normal, compact subgroupsN ={{1}}.
• IfGis a locally pro-finite group, then every almost connected subgroupG is compact, i.e., pro-finite. Let N be the family of normal, finite-index subgroups, which becomes a net under the partial ordering of inclusion.
CHAPTER 5