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In the previous notes, we established the Gleason-Yamabe theorem:

Theorem 1 (Gleason-Yamabe theorem) Let {G} be a locally compact group. Then, for any open neighbourhood {U} of the identity, there exists an open subgroup {G'} of {G} and a compact normal subgroup {K} of {G'} in {U} such that {G'/K} is isomorphic to a Lie group.

Roughly speaking, this theorem asserts the “mesoscopic” structure of a locally compact group (after restricting to an open subgroup {G'} to remove the macroscopic structure, and quotienting out by {K} to remove the microscopic structure) is always of Lie type.

In this post, we combine the Gleason-Yamabe theorem with some additional tools from point-set topology to improve the description of locally compact groups in various situations.

We first record some easy special cases of this. If the locally compact group {G} has the no small subgroups property, then one can take {K} to be trivial; thus {G'} is Lie, which implies that {G} is locally Lie and thus Lie as well. Thus the assertion that all locally compact NSS groups are Lie (Theorem 10 from Notes 4) is a special case of the Gleason-Yamabe theorem.

In a similar spirit, if the locally compact group {G} is connected, then the only open subgroup {G'} of {G} is the full group {G}; in particular, by arguing as in the treatment of the compact case (Exercise 19 of Notes 3), we conclude that any connected locally compact Hausdorff group is the inverse limit of Lie groups.

Now we return to the general case, in which {G} need not be connected or NSS. One slight defect of Theorem 1 is that the group {G'} can depend on the open neighbourhood {U}. However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make {G'} independent of {U}:

Theorem 2 (Gleason-Yamabe theorem, stronger version) Let {G} be a locally compact group. Then there exists an open subgoup {G'} of {G} such that, for any open neighbourhood {U} of the identity in {G'}, there exists a compact normal subgroup {K} of {G'} in {U} such that {G'/K} is isomorphic to a Lie group.

We prove this theorem below the fold. As in previous notes, if {G} is Hausdorff, the group {G'} is thus an inverse limit of Lie groups (and if {G} (and hence {G'}) is first countable, it is the inverse limit of a sequence of Lie groups).

It remains to analyse inverse limits of Lie groups. To do this, it helps to have some control on the dimensions of the Lie groups involved. A basic tool for this purpose is the invariance of domain theorem:

Theorem 3 (Brouwer invariance of domain theorem) Let {U} be an open subset of {{\bf R}^n}, and let {f: U \rightarrow {\bf R}^n} be a continuous injective map. Then {f(U)} is also open.

We prove this theorem below the fold. It has an important corollary:

Corollary 4 (Topological invariance of dimension) If {n > m}, and {U} is a non-empty open subset of {{\bf R}^n}, then there is no continuous injective mapping from {U} to {{\bf R}^m}. In particular, {{\bf R}^n} and {{\bf R}^m} are not homeomorphic.

Exercise 1 (Uniqueness of dimension) Let {X} be a non-empty topological space. If {X} is a manifold of dimension {d_1}, and also a manifold of dimension {d_2}, show that {d_1=d_2}. Thus, we may define the dimension {\hbox{dim}(X)} of a non-empty manifold in a well-defined manner.

If {X, Y} are non-empty manifolds, and there is a continuous injection from {X} to {Y}, show that {\hbox{dim}(X) \leq \hbox{dim}(Y)}.

Remark 1 Note that the analogue of the above exercise for surjections is false: the existence of a continuous surjection from one non-empty manifold {X} to another {Y} does not imply that {\hbox{dim}(X) \geq \hbox{dim}(Y)}, thanks to the existence of space-filling curves. Thus we see that invariance of domain, while intuitively plausible, is not an entirely trivial observation.

As we shall see, we can use Corollary 4 to bound the dimension of the Lie groups {L_n} in an inverse limit {G = \lim_{n \rightarrow \infty} L_n} by the “dimension” of the inverse limit {G}. Among other things, this can be used to obtain a positive resolution to Hilbert’s fifth problem:

Theorem 5 (Hilbert’s fifth problem) Every locally Euclidean group is isomorphic to a Lie group.

Again, this will be shown below the fold.

Another application of this machinery is the following variant of Hilbert’s fifth problem, which was used in Gromov’s original proof of Gromov’s theorem on groups of polynomial growth, although we will not actually need it this course:

Proposition 6 Let {G} be a locally compact {\sigma}-compact group that acts transitively, faithfully, and continuously on a connected manifold {X}. Then {G} is isomorphic to a Lie group.

Recall that a continuous action of a topological group {G} on a topological space {X} is a continuous map {\cdot: G \times X \rightarrow X} which obeys the associativity law {(gh)x = g(hx)} for {g,h \in G} and {x \in X}, and the identity law {1x = x} for all {x \in X}. The action is transitive if, for every {x,y \in X}, there is a {g \in G} with {gx=y}, and faithful if, whenever {g, h \in G} are distinct, one has {gx \neq hx} for at least one {x}.

The {\sigma}-compact hypothesis is a technical one, and can likely be dropped, but we retain it for this discussion (as in most applications we can reduce to this case).

Exercise 2 Show that Proposition 6 implies Theorem 5.

Remark 2 It is conjectured that the transitivity hypothesis in Proposition 6 can be dropped; this is known as the Hilbert-Smith conjecture. It remains open; the key difficulty is to figure out a way to eliminate the possibility that {G} is a {p}-adic group {{\bf Z}_p}. See this previous blog post for further discussion.

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This is yet another post in a series on basic ingredients in the structural theory of locally compact groups, which is closely related to Hilbert’s fifth problem.

In order to understand the structure of a topological group {G}, a basic strategy is to try to split {G} into two smaller factor groups {H, K} by exhibiting a short exact sequence

\displaystyle  0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0.

If one has such a sequence, then {G} is an extension of {H} by {K} (which includes direct products {H \times K} and semidirect products {H \ltimes K} as examples, but can be more general than these situations, as discussed in this previous blog post). In principle, the problem of understanding the structure of {G} then splits into three simpler problems:

  1. (Horizontal structure) Understanding the structure of the “horizontal” group {H}.
  2. (Vertical structure) Understanding the structure of the “vertical” group {K}.
  3. (Cohomology) Understanding the ways in which one can extend {H} by {K}.

The “cohomological” aspect to this program can be nontrivial. However, in principle at least, this strategy reduces the study of the large group {G} to the study of the smaller groups {H, K}. (This type of splitting strategy is not restricted to topological groups, but can also be adapted to many other categories, particularly those of groups or group-like objects.) Typically, splitting alone does not fully kill off a structural classification problem, but it can reduce matters to studying those objects which are somehow “simple” or “irreducible”. For instance, this strategy can often be used to reduce questions about arbitrary finite groups to finite simple groups.

A simple example of splitting is as follows. Given any topological group {G}, one can form the connected component {G^\circ} of the identity – the maximal connected set containing the identity. It is not difficult to show that {G^\circ} is a closed (and thus also locally compact) normal subgroup of {G}, whose quotient {G/G^\circ} is another locally compact group. Furthermore, due to the maximal connected nature of {G^\circ}, {G/G^\circ} is totally disconnected – the only connected sets are the singletons. In particular, {G/G^\circ} is Hausdorff (the identity element is closed). Thus we have obtained a splitting

\displaystyle  0 \rightarrow G^\circ \rightarrow G \rightarrow G/G^\circ \rightarrow 0

of an arbitrary locally compact group into a connected locally compact group {G^\circ}, and a totally disconnected locally compact group {G/G^\circ}. In principle at least, the study of locally compact groups thus splits into the study of connected locally compact groups, and the study of totally disconnected locally compact groups (though the cohomological issues are not always trivial).

In the structural theory of totally disconnected locally compact groups, the first basic theorem in the subject is van Dantzig’s theorem (which we prove below the fold):

Theorem 1 (Van Danztig’s theorem) Every totally disconnected locally compact group {G} contains a compact open subgroup {H} (which will of course still be totally disconnected).

Example 1 Let {p} be a prime. Then the {p}-adic field {{\bf Q}_p} (with the usual {p}-adic valuation) is totally disconnected locally compact, and the {p}-adic integers {{\bf Z}_p} are a compact open subgroup.

Of course, this situation is the polar opposite of what occurs in the connected case, in which the only open subgroup is the whole group.

In view of van Dantzig’s theorem, we see that the “local” behaviour of totally disconnected locally compact groups can be modeled by the compact totally disconnected groups, which are better understood (for instance, one can start analysing them using the Peter-Weyl theorem, as discussed in this previous post). The global behaviour however remains more complicated, in part because the compact open subgroup given by van Dantzig’s theorem need not be normal, and so does not necessarily induce a splitting of {G} into compact and discrete factors.

Example 2 Let {p} be a prime, and let {G} be the semi-direct product {{\bf Z} \ltimes {\bf Q}_p}, where the integers {{\bf Z}} act on {{\bf Q}_p} by the map {m: x \mapsto p^m x}, and we give {G} the product of the discrete topology of {{\bf Z}} and the {p}-adic topology on {{\bf Q}_p}. One easily verifies that {G} is a totally disconnected locally compact group. It certainly has compact open subgroups, such as {\{0\} \times {\bf Z}_p}. However, it is easy to show that {G} has no non-trivial compact normal subgroups (the problem is that the conjugation action of {{\bf Z}} on {{\bf Q}_p} has all non-trivial orbits unbounded).

Returning to more general locally compact groups, we obtain an immediate corollary:

Corollary 2 Every locally compact group {G} contains an open subgroup {H} which is “compact-by-connected” in the sense that {H/H^\circ} is compact.

Indeed, one applies van Dantzig’s theorem to the totally disconnected group {G/G^\circ}, and then pulls back the resulting compact open subgroup.

Now we mention another application of van Dantzig’s theorem, of more direct relevance to Hilbert’s fifth problem. Define a generalised Lie group to be a topological group {G} with the property that given any open neighbourhood {U} of the identity, there exists an open subgroup {G'} of {G} and a compact normal subgroup {N} of {G'} in {U} such that {G'/N} is isomorphic to a Lie group. It is easy to see that such groups are locally compact. The deep Gleason-Yamabe theorem, which among other things establishes a satisfactory solution to Hilbert’s fifth problem (and which we will not prove here), asserts the converse:

Theorem 3 (Gleason-Yamabe theorem) Every locally compact group is a generalised Lie group.

Example 3 We consider the locally compact group {G = {\bf Z} \ltimes {\bf Q}_p} from Example 2. This is of course not a Lie group. However, any open neighbourhood {U} of the identity in {G} will contain the compact subgroup {N := \{0\} \times p^j {\bf Z}_p} for some integer {j}. The open subgroup {G' := \{0\} \times {\bf Z}_p} then has {G'/N} isomorphic to the discrete finite group {{\bf Z}/p^j{\bf Z}}, which is certainly a Lie group. Thus {G} is a generalised Lie group.

One important example of generalised Lie groups are those locally compact groups which are an inverse limit (or projective limit) of Lie groups. Indeed, suppose we have a family {(G_i)_{i\in I}} of Lie groups {G_i} indexed by partially ordered set {I} which is directed in the sense that every finite subset of {I} has an upper bound, together with continuous homomorphisms {\pi_{i \rightarrow j}: G_i \rightarrow G_j} for all {i > j} which form a category in the sense that {\pi_{j \rightarrow k} \circ \pi_{i \rightarrow j} = \pi_{i \rightarrow k}} for all {i>j>k}. Then we can form the inverse limit

\displaystyle G := \lim_{\stackrel{\leftarrow}{i \in I}} G_i,

which is the subgroup of {\prod_{i \in I} G_i} consisting of all tuples {(g_i)_{i \in I} \in \prod_{i \in I} G_i} which are compatible with the {\pi_{i \rightarrow j}} in the sense that {\pi_{i \rightarrow j}(g_i) = g_j} for all {i>j}. If we endow {\prod_{i \in I} G_i} with the product topology, then {G} is a closed subgroup of {\prod_{i \in I} G_i}, and thus has the structure of a topological group, with continuous homomorphisms {\pi_i: G \rightarrow G_i} which are compatible with the {\pi_{i \rightarrow j}} in the sense that {\pi_{i \rightarrow j} \circ \pi_i = \pi_j} for all {i>j}. Such an inverse limit need not be locally compact; for instance, the inverse limit

\displaystyle \lim_{\stackrel{\leftarrow}{n \in {\bf N}}} {\bf R}^n

of Euclidean spaces with the usual coordinate projection maps is isomorphic to the infinite product space {{\bf R}^{\bf N}} with the product topology, which is not locally compact. However, if an inverse limit

\displaystyle G = \lim_{\stackrel{\leftarrow}{i \in I}} G_i

of Lie groups is locally compact, it can be easily seen to be a generalised Lie group. Indeed, by local compactness, any open neighbourhood {G} of the identity will contain an open precompact neighbourhood of the identity; by construction of the product topology (and the directed nature of {I}), this smaller neighbourhood will in turn will contain the kernel of one of the {\pi_i}, which will be compact since the preceding neighbourhood was precompact. Quotienting out by this {\pi_i} we obtain a locally compact subgroup of the Lie group {G_i}, which is necessarily again a Lie group by Cartan’s theorem, and the claim follows.

In the converse direction, it is possible to use Corollary 2 to obtain the following observation of Gleason:

Theorem 4 Every Hausdorff generalised Lie group contains an open subgroup that is an inverse limit of Lie groups.

We show Theorem 4 below the fold. Combining this with the (substantially more difficult) Gleason-Yamabe theorem, we obtain quite a satisfactory description of the local structure of locally compact groups. (The situation is particularly simple for connected groups, which have no non-trivial open subgroups; we then conclude that every connected locally compact Hausdorff group is the inverse limit of Lie groups.)

Example 4 The locally compact group {G := {\bf Z} \ltimes {\bf Q}_p} is not an inverse limit of Lie groups because (as noted earlier) it has no non-trivial compact normal subgroups, which would contradict the preceding analysis that showed that all locally compact inverse limits of Lie groups were generalised Lie groups. On the other hand, {G} contains the open subgroup {\{0\} \times {\bf Q}_p}, which is the inverse limit of the discrete (and thus Lie) groups {\{0\} \times {\bf Q}_p/p^j {\bf Z}_p} for {j \in {\bf Z}} (where we give {{\bf Z}} the usual ordering, and use the obvious projection maps).

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