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In the previous notes, we established the Gleason-Yamabe theorem:
Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that 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 to remove the macroscopic structure, and quotienting out by 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 has the no small subgroups property, then one can take to be trivial; thus is Lie, which implies that 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 is connected, then the only open subgroup of is the full group ; 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 need not be connected or NSS. One slight defect of Theorem 1 is that the group can depend on the open neighbourhood . However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make independent of :
Theorem 2 (Gleason-Yamabe theorem, stronger version) Let be a locally compact group. Then there exists an open subgoup of such that, for any open neighbourhood of the identity in , there exists a compact normal subgroup of in such that is isomorphic to a Lie group.
We prove this theorem below the fold. As in previous notes, if is Hausdorff, the group is thus an inverse limit of Lie groups (and if (and hence ) 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 be an open subset of , and let be a continuous injective map. Then is also open.
We prove this theorem below the fold. It has an important corollary:
Corollary 4 (Topological invariance of dimension) If , and is a non-empty open subset of , then there is no continuous injective mapping from to . In particular, and are not homeomorphic.
Exercise 1 (Uniqueness of dimension) Let be a non-empty topological space. If is a manifold of dimension , and also a manifold of dimension , show that . Thus, we may define the dimension of a non-empty manifold in a well-defined manner.
If are non-empty manifolds, and there is a continuous injection from to , show that .
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 to another does not imply that , 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 in an inverse limit by the “dimension” of the inverse limit . 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 be a locally compact -compact group that acts transitively, faithfully, and continuously on a connected manifold . Then is isomorphic to a Lie group.
Recall that a continuous action of a topological group on a topological space is a continuous map which obeys the associativity law for and , and the identity law for all . The action is transitive if, for every , there is a with , and faithful if, whenever are distinct, one has for at least one .
The -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).
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 is a -adic group . See this previous blog post for further discussion.
In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with the minimal assumption of local compactness on a topological group , and seeing what kind of structures one can build using this assumption. (For simplicity we shall mostly confine our discussion to global groups rather than local groups for now.) In view of the preceding notes, we would like to see two types of structures emerge in particular:
- representations of into some more structured group, such as a matrix group ; and
- metrics on that capture the escape and commutator structure of (i.e. Gleason metrics).
To build either of these structures, a fundamentally useful tool is that of (left-) Haar measure – a left-invariant Radon measure on . (One can of course also consider right-Haar measures; in many cases (such as for compact or abelian groups), the two concepts are the same, but this is not always the case.) This concept generalises the concept of Lebesgue measure on Euclidean spaces , which is of course fundamental in analysis on those spaces.
Haar measures will help us build useful representations and useful metrics on locally compact groups . For instance, a Haar measure gives rise to the regular representation that maps each element of to the unitary translation operator on the Hilbert space of square-integrable measurable functions on with respect to this Haar measure by the formula
(The presence of the inverse is convenient in order to obtain the homomorphism property without a reversal in the group multiplication.) In general, this is an infinite-dimensional representation; but in many cases (and in particular, in the case when is compact) we can decompose this representation into a useful collection of finite-dimensional representations, leading to the Peter-Weyl theorem, which is a fundamental tool for understanding the structure of compact groups. This theorem is particularly simple in the compact abelian case, where it turns out that the representations can be decomposed into one-dimensional representations , better known as characters, leading to the theory of Fourier analysis on general compact abelian groups. With this and some additional (largely combinatorial) arguments, we will also be able to obtain satisfactory structural control on locally compact abelian groups as well.
The link between Haar measure and useful metrics on is a little more complicated. Firstly, once one has the regular representation , and given a suitable “test” function , one can then embed into (or into other function spaces on , such as or ) by mapping a group element to the translate of in that function space. (This map might not actually be an embedding if enjoys a non-trivial translation symmetry , but let us ignore this possibility for now.) One can then pull the metric structure on the function space back to a metric on , for instance defining an -based metric
if is square-integrable, or perhaps a -based metric
if is continuous and compactly supported (with denoting the supremum norm). These metrics tend to have several nice properties (for instance, they are automatically left-invariant), particularly if the test function is chosen to be sufficiently “smooth”. For instance, if we introduce the differentiation (or more precisely, finite difference) operators
(so that ) and use the metric (1), then a short computation (relying on the translation-invariance of the norm) shows that
for all . This suggests that commutator estimates, such as those appearing in the definition of a Gleason metric in Notes 2, might be available if one can control “second derivatives” of ; informally, we would like our test functions to have a “” type regularity.
If was already a Lie group (or something similar, such as a local group) then it would not be too difficult to concoct such a function by using local coordinates. But of course the whole point of Hilbert’s fifth problem is to do without such regularity hypotheses, and so we need to build test functions by other means. And here is where the Haar measure comes in: it provides the fundamental tool of convolution
between two suitable functions , which can be used to build smoother functions out of rougher ones. For instance:
Exercise 1 Let be continuous, compactly supported functions which are Lipschitz continuous. Show that the convolution using Lebesgue measure on obeys the -type commutator estimate
for all and some finite quantity depending only on .
This exercise suggests a strategy to build Gleason metrics by convolving together some “Lipschitz” test functions and then using the resulting convolution as a test function to define a metric. This strategy may seem somewhat circular because one needs a notion of metric in order to define Lipschitz continuity in the first place, but it turns out that the properties required on that metric are weaker than those that the Gleason metric will satisfy, and so one will be able to break the circularity by using a “bootstrap” or “induction” argument.
We will discuss this strategy – which is due to Gleason, and is fundamental to all currently known solutions to Hilbert’s fifth problem – in later posts. In this post, we will construct Haar measure on general locally compact groups, and then establish the Peter-Weyl theorem, which in turn can be used to obtain a reasonably satisfactory structural classification of both compact groups and locally compact abelian groups.
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 , a basic strategy is to try to split into two smaller factor groups by exhibiting a short exact sequence
If one has such a sequence, then is an extension of by (which includes direct products and semidirect products 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 then splits into three simpler problems:
- (Horizontal structure) Understanding the structure of the “horizontal” group .
- (Vertical structure) Understanding the structure of the “vertical” group .
- (Cohomology) Understanding the ways in which one can extend by .
The “cohomological” aspect to this program can be nontrivial. However, in principle at least, this strategy reduces the study of the large group to the study of the smaller groups . (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 , one can form the connected component of the identity – the maximal connected set containing the identity. It is not difficult to show that is a closed (and thus also locally compact) normal subgroup of , whose quotient is another locally compact group. Furthermore, due to the maximal connected nature of , is totally disconnected – the only connected sets are the singletons. In particular, is Hausdorff (the identity element is closed). Thus we have obtained a splitting
of an arbitrary locally compact group into a connected locally compact group , and a totally disconnected locally compact group . 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 contains a compact open subgroup (which will of course still be totally disconnected).
Example 1 Let be a prime. Then the -adic field (with the usual -adic valuation) is totally disconnected locally compact, and the -adic integers 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 into compact and discrete factors.
Example 2 Let be a prime, and let be the semi-direct product , where the integers act on by the map , and we give the product of the discrete topology of and the -adic topology on . One easily verifies that is a totally disconnected locally compact group. It certainly has compact open subgroups, such as . However, it is easy to show that has no non-trivial compact normal subgroups (the problem is that the conjugation action of on has all non-trivial orbits unbounded).
Returning to more general locally compact groups, we obtain an immediate corollary:
Corollary 2 Every locally compact group contains an open subgroup which is “compact-by-connected” in the sense that is compact.
Indeed, one applies van Dantzig’s theorem to the totally disconnected group , 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 with the property that given any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that 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 from Example 2. This is of course not a Lie group. However, any open neighbourhood of the identity in will contain the compact subgroup for some integer . The open subgroup then has isomorphic to the discrete finite group , which is certainly a Lie group. Thus 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 of Lie groups indexed by partially ordered set which is directed in the sense that every finite subset of has an upper bound, together with continuous homomorphisms for all which form a category in the sense that for all . Then we can form the inverse limit
which is the subgroup of consisting of all tuples which are compatible with the in the sense that for all . If we endow with the product topology, then is a closed subgroup of , and thus has the structure of a topological group, with continuous homomorphisms which are compatible with the in the sense that for all . Such an inverse limit need not be locally compact; for instance, the inverse limit
of Euclidean spaces with the usual coordinate projection maps is isomorphic to the infinite product space with the product topology, which is not locally compact. However, if an inverse limit
of Lie groups is locally compact, it can be easily seen to be a generalised Lie group. Indeed, by local compactness, any open neighbourhood of the identity will contain an open precompact neighbourhood of the identity; by construction of the product topology (and the directed nature of ), this smaller neighbourhood will in turn will contain the kernel of one of the , which will be compact since the preceding neighbourhood was precompact. Quotienting out by this we obtain a locally compact subgroup of the Lie group , 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 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, contains the open subgroup , which is the inverse limit of the discrete (and thus Lie) groups for (where we give the usual ordering, and use the obvious projection maps).
This is another post in a series on various components to the solution of Hilbert’s fifth problem. One interpretation of this problem is to ask for a purely topological classification of the topological groups which are isomorphic to Lie groups. (Here we require Lie groups to be finite-dimensional, but allow them to be disconnected.)
There are some obvious necessary conditions on a topological group in order for it to be isomorphic to a Lie group; for instance, it must be Hausdorff and locally compact. These two conditions, by themselves, are not quite enough to force a Lie group structure; consider for instance a -adic field for some prime , which is a locally compact Hausdorff topological group which is not a Lie group (the topology is locally that of a Cantor set). Nevertheless, it turns out that by adding some key additional assumptions on the topological group, one can recover Lie structure. One such result, which is a key component of the full solution to Hilbert’s fifth problem, is the following result of von Neumann:
Theorem 1 Let be a locally compact Hausdorff topological group that has a faithful finite-dimensional linear representation, i.e. an injective continuous homomorphism into some linear group. Then can be given the structure of a Lie group. Furthermore, after giving this Lie structure, becomes smooth (and even analytic) and non-degenerate (the Jacobian always has full rank).
This result is closely related to a theorem of Cartan:
Theorem 2 (Cartan’s theorem) Any closed subgroup of a Lie group , is again a Lie group (in particular, is an analytic submanifold of , with the induced analytic structure).
Indeed, Theorem 1 immediately implies Theorem 2 in the important special case when the ambient Lie group is a linear group, and in any event it is not difficult to modify the proof of Theorem 1 to give a proof of Theorem 2. However, Theorem 1 is more general than Theorem 2 in some ways. For instance, let be the real line , which we faithfully represent in the -torus using an irrational embedding for some fixed irrational . The -torus can in turn be embedded in a linear group (e.g. by identifying it with , or ), thus giving a faithful linear representation of . However, the image is not closed (it is a dense subgroup of a -torus), and so Cartan’s theorem does not directly apply ( fails to be a Lie group). Nevertheless, Theorem 1 still applies and guarantees that the original group is a Lie group.
(On the other hand, the image of any compact subset of under a faithful representation must be closed, and so Theorem 1 is very close to the version of Theorem 2 for local groups.)
The key to building the Lie group structure on a topological group is to first build the associated Lie algebra structure, by means of one-parameter subgroups.
Definition 3 A one-parameter subgroup of a topological group is a continuous homomorphism from the real line (with the additive group structure) to .
Remark 1 Technically, is a parameterisation of a subgroup , rather than a subgroup itself, but we will abuse notation and refer to as the subgroup.
In a Lie group , the one-parameter subgroups are in one-to-one correspondence with the Lie algebra , with each element giving rise to a one-parameter subgroup , and conversely each one-parameter subgroup giving rise to an element of the Lie algebra; we will establish these basic facts in the special case of linear groups below the fold. On the other hand, the notion of a one-parameter subgroup can be defined in an arbitrary topological group. So this suggests the following strategy if one is to try to represent a topological group as a Lie group:
- First, form the space of one-parameter subgroups of .
- Show that has the structure of a (finite-dimensional) Lie algebra.
- Show that “behaves like” the tangent space of at the identity (in particular, the one-parameter subgroups in should cover a neighbourhood of the identity in ).
- Conclude that has the structure of a Lie group.
It turns out that this strategy indeed works to give Theorem 1 (and variants of this strategy are ubiquitious in the rest of the theory surrounding Hilbert’s fifth problem).
Below the fold, I record the proof of Theorem 1 (based on the exposition of Montgomery and Zippin). I plan to organise these disparate posts surrounding Hilbert’s fifth problem (and its application to related topics, such as Gromov’s theorem or to the classification of approximate groups) at a later date.
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