One can also consider “mesoscopic” scales which are intermediate between microscopic and macroscopic scales, or large-scale behaviour at scales that go off to infinity (and in particular are larger than the macroscopic range of scales), although the behaviour of these scales will not be the main focus of this post. Finally, one can divide the macroscopic scales into “local” macroscopic scales (less than for some small but fixed ) and “global” macroscopic scales (scales that are allowed to be larger than a given large absolute constant ). For instance, given a finite approximate group :
For comparison, in the theory of locally compact groups, properties about small neighbourhoods of the identity (e.g. local compactness, or the NSS property) would be properties at the local macroscopic scale, whereas the space of one-parameter subgroups can be interpreted as an object at the microscopic scale. The exponential map then provides a bridge connecting the microscopic and macroscopic scales.
We return now to approximate groups. The macroscopic structure of these objects is well described by the Hrushovski Lie model theorem from the previous set of notes, which informally asserts that the macroscopic structure of an (ultra) approximate group can be modeled by a Lie group. This is already an important piece of information about general approximate groups, but it does not directly reveal the full structure of such approximate groups, because these Lie models are unable to see the microscopic behaviour of these approximate groups.
To illustrate this, let us review one of the examples of a Lie model of an ultra approximate group, namely Exercise 28 from Notes 7. In this example one studied a “nilbox” from a Heisenberg group, which we rewrite here in slightly different notation. Specifically, let be the Heisenberg group
thus is the nonstandard box
where . As the above exercise establishes, is an ultra approximate group with a Lie model given by the formula
for and . Note how the nonabelian nature of (arising from the term in the group law (1)) has been lost in the model , because the effect of that nonabelian term on is only which is infinitesimal and thus does not contribute to the standard part. In particular, if we replace with the abelian group with the additive group law
and let and be defined exactly as with and , but placed inside the group structure of rather than , then and are essentially “indistinguishable” as far as their models by are concerned, even though the latter approximate group is abelian and the former is not. The problem is that the nonabelian-ness in the former example is so microscopic that it falls entirely inside the kernel of and is thus not detected at all by the model.
The problem of not being able to “see” the microscopic structure of a group (or approximate group) also was a key difficulty in the theory surrounding Hilbert’s fifth problem that was discussed in previous notes. A key tool in being able to resolve such structure was to build left-invariant metrics (or equivalently, norms ) on one’s group, which obeyed useful “Gleason axioms” such as the commutator axiom
for sufficiently small , or the escape axiom
when was sufficiently small. Such axioms have important and non-trivial content even in the microscopic regime where or are extremely close to the identity. For instance, in the proof of Jordan’s theorem from Notes 0, which showed that any finite unitary group was boundedly virtually abelian, a key step was to apply the commutator axiom (2) (for the distance to the identity in operator norm) to the most “microscopic” element of , or more precisely a non-identity element of of minimal norm. The key point was that this microscopic element was virtually central in , and as such it restricted much of to a lower-dimensional subgroup of the unitary group, at which point one could argue using an induction-on-dimension argument. As we shall see, a similar argument can be used to place “virtually nilpotent” structure on finite approximate groups. For instance, in the Heisenberg-type approximate groups and discussed earlier, the element will be “closest to the origin” in a suitable sense to be defined later, and is centralised by both approximate groups; quotienting out (the orbit of) that central element and iterating the process two more times, we shall see that one can express both and as a tower of central cyclic extensions, which in particular establishes the nilpotency of both groups.
The escape axiom (3) is a particularly important axiom in connecting the microscopic structure of a group to its macroscopic structure; for instance, as shown in Notes 2, this axiom (in conjunction with the closely related commutator axiom) tends to imply dilation estimates such as that allow one to understand the microscopic geometry of points close to the identity in terms of the (local) macroscopic geometry of points that are significantly further away from the identity.
It is thus of interest to build some notion of a norm (or left-invariant metrics) on an approximate group that obeys the escape and commutator axioms (while being non-degenerate enough to adequately capture the geometry of in some sense), in a fashion analogous to the Gleason metrics that played such a key role in the theory of Hilbert’s fifth problem. It is tempting to use the Lie model theorem to do this, since Lie groups certainly come with Gleason metrics. However, if one does this, one ends up, roughly speaking, with a norm on that only obeys the escape and commutator estimates macroscopically; roughly speaking, this means that one has a macroscopic commutator inequality
and a macroscopic escape property
but such axioms are too weak for analysis at the microscopic scale, and in particular in establishing centrality of the element closest to the identity.
Another way to proceed is to build a norm that is specifically designed to obey the crucial escape property. Given an approximate group in a group , and an element of , we can define the escape norm of by the formula
Thus, equals if lies outside of , equals if lies in but lies outside of , and so forth. Such norms had already appeared in Notes 4, in the context of analysing NSS groups.
As it turns out, this expression will obey an escape axiom, as long as we place some additional hypotheses on which we will present shortly. However, it need not actually be a norm; in particular, the triangle inequality
is not necessarily true. Fortunately, it turns out that by a (slightly more complicated) version of the Gleason machinery from Notes 4 we can establish a usable substitute for this inequality, namely the quasi-triangle inequality
where is a constant independent of . As we shall see, these estimates can then be used to obtain a commutator estimate (2).
However, to do all this, it is not enough for to be an approximate group; it must obey two additional “trapping” axioms that improve the properties of the escape norm. We formalise these axioms (somewhat arbitrarily) as follows:
Definition 1 (Strong approximate group) Let . A strong -approximate group is a finite -approximate group in a group with a symmetric subset obeying the following axioms:
An ultra strong -approximate group is an ultraproduct of strong -approximate groups.
The first trapping condition can be rewritten as
and the second trapping condition can similarly be rewritten as
This makes the escape norms of , and comparable to each other, which will be needed for a number of reasons (and in particular to close a certain bootstrap argument properly). Compare this with equation (12) from Notes 4, which used the NSS hypothesis to obtain similar conclusions. Thus, one can view the strong approximate group axioms as being a sort of proxy for the NSS property.
Example 1 Let be a large natural number. Then the interval in the integers is a -approximate group, which is also a strong -approximate group (setting , for instance). On the other hand, if one places in rather than in the integers, then the first trapping condition is lost and one is no longer a strong -approximate group. Also, if one remains in the integers, but deletes a few elements from , e.g. deleting from ), then one is still a -approximate group, but is no longer a strong -approximate group, again because the first trapping condition is lost.
A key consequence of the Hrushovski Lie model theorem is that it allows one to replace approximate groups by strong approximate groups:
Exercise 1 (Finding strong approximate groups)
- (i) Let be an ultra approximate group with a good Lie model , and let be a symmetric convex body (i.e. a convex open bounded subset) in the Lie algebra . Show that if is a sufficiently small standard number, then there exists a strong ultra approximate group with
and with can be covered by finitely many left translates of . Furthermore, is also a good model for .
- (ii) If is a finite -approximate group, show that there is a strong -approximate group inside with the property that can be covered by left translates of . (Hint: use (i), Hrushovski’s Lie model theorem, and a compactness and contradiction argument.)
The need to compare the strong approximate group to an exponentiated small ball will be convenient later, as it allows one to easily use the geometry of to track various aspects of the strong approximate group.
As mentioned previously, strong approximate groups exhibit some of the features of NSS locally compact groups. In Notes 4, we saw that the escape norm for NSS locally compact groups was comparable to a Gleason metric. The following theorem is an analogue of that result:
Theorem 2 (Gleason lemma) Let be a strong -approximate group in a group .
- (Symmetry) For any , one has .
- (Conjugacy bound) For any , one has .
- (Triangle inequality) For any , one has .
- (Escape property) One has whenever .
- (Commutator inequality) For any , one has .
The proof of this theorem will occupy a large part of the current set of notes. We then aim to use this theorem to classify strong approximate groups. The basic strategy (temporarily ignoring a key technical issue) follows the Bieberbach-Frobenius proof of Jordan’s theorem, as given in Notes 0, is as follows.
There are two main technical issues to resolve to make this strategy work. The first is to show that the iterative step in the argument terminates in finite time. This we do by returning to the Lie model theorem. It turns out that each time one quotients out by an orbit of an element that escapes, the dimension of the Lie model drops by at least one. This will ensure termination of the argument in finite time.
The other technical issue is that while the quotienting out all the elements of zero escape norm eliminates all “torsion” from (in the sense that the quotient of has no non-trivial elements of zero escape norm), further quotienting operations can inadvertently re-introduce such torsion. This torsion can be re-eradicated by further quotienting, but the price one pays for this is that the final structural description of is no longer as strong as “virtually nilpotent”, but is instead a more complicated tower alternating between (ultra) finite extensions and central extensions.
Example 2 Consider the strong -approximate group
in the integers, where is a large natural number not divisible by . As is torsion-free, all non-zero elements of have positive escape norm, and the nonzero element of minimal escape norm here is (or ). But if one quotients by , projects down to , which now has torsion (and all elements in this quotient have zero escape norm). Thus torsion has been re-introduced by the quotienting operation. (A related observation is that the intersection of with is not a simple progression, but is a more complicated object, namely a generalised arithmetic progression of rank two.)
To deal with this issue, we will not quotient out by the entire cyclic group generated by the element of minimal escape norm, but rather by an arithmetic progression , where is a natural number comparable to the reciprocal of the escape norm, as this will be enough to cut the dimension of the Lie model down by one without introducing any further torsion. Of course, this cannot be done in the category of global groups, since the arithmetic progression will not, in general, be a group. However, it is still a local group, and it turns out that there is an analogue of the quotient space construction in local groups. This fixes the problem, but at a cost: in order to make the inductive portion of the argument work smoothly, it is now more natural to place the entire argument inside the category of local groups rather than global groups, even though the primary interest in approximate groups is in the global case when lies inside a global group. This necessitates some technical modification to some of the preceding discussion (for instance, the Gleason-Yamabe theorem must be replaced by the local version of this theorem, due to Goldbring); details can be found in this recent paper of Emmanuel Breuillard, Ben Green, and myself, but will only be sketched here.
— 1. Gleason’s lemma —
Throughout this section, is a strong -approximate group in a global group . We will prove the various estimates in Theorem 2. The arguments will be very close to those in Notes 4; indeed, it is possible to unify the results here with the results in those notes by a suitable modification of the notation, but we will not do so here.
We begin with the easy estimates. The symmetry property is immediate from the symmetry of . Now we turn to the escape property. By symmetry, we may take to be positive (the case is trivial). We may of course assume that is strictly positive, say equal to ; thus and , and . By the first trapping property, this implies that for some .
Let be the first multiple of larger than or equal to , then . Since is less than , we have ; since , we conclude that . In particular this shows that , and the claim follows.
The escape property implies the conjugacy bound:
Exercise 2 Establish the conjugacy bound. (Hint: one can mimic the arguments establishing a nearly identical bound in Section 2 of Notes 4.)
Now we turn to the triangle inequality, which (as in Notes 4) is the most difficult property to establish. Our arguments will closely resemble the proof of Proposition 11 from these notes, with and playing the roles of and from that argument. As in that theorem, we will initially assume that we have an a priori bound of the form
for all , and some (large) independent of , and remove this hypothesis later. We then introduce the norm
then is symmetric, obeys the triangle inequality, and is comparable to in the sense that
We introduce the function by
where . Then takes values between and , equals on , is supported on , and obeys the Lipschitz bound
for all , thanks to the triangle inequality for and (6). We also introduce the function by
then also takes values between and , equals on , is supported on , and obeys the Lipschitz bound
Now we form the convolution by the formula
By construction, is supported on and at least at the identity. As or is supported in , which has cardinality at most , we have the uniform bound
and (7) we have the Lipschitz bound
and restricting to (so that is supported on , which has cardinality at most ) we see from (7), (8) that
for and .
We can use this to improve the bound (10). Indeed, using the telescoping identity
we see that
and thus
whenever . Using the second trapping property, this implies that
In the converse direction, if , then
and thus from the support of , for all . But then by the first trapping property, this implies that for all . We conclude that
The triangle inequality for then implies a triangle inequality for ,
which is (5) with replaced by . If we knew (5) for some large but finite , we could iterate this argument and conclude that (5) held with replaced by , which would give the triangle inequality. Now it is not immediate that (5) holds for any finite , but we can avoid this problem with the usual regularisation trick of replacing with throughout the argument for some small , which makes (5) automatically true with , run the above iteration argument, and then finally send to zero.
Exercise 3 Verify that the modifications to the above argument sketched above actually do establish the triangle inequality.
A final application of the Gleason convolution machinery then gives the final estimate in Gleason’s lemma:
Exercise 4 Use the properties of the escape norm already established (and in particular, the escape property and the triangle inequality) to establish the commutator inequality. (Hint: adapt the argument from Section 2 of Notes 4.)
The proof of Theorem 2 is now complete.
Exercise 5 Generalise Theorem 2 to the setting where is not necessarily finite, but is instead an open precompact subset of a locally compact group . (Hint: replace cardinality by left-invariant Haar measure and follow the arguments in Notes 4 closely.) Note that this already gives most of one of the key results from that set of notes, namely that any NSS group admits a Gleason metric, since it is not difficult to show that NSS groups contain open precompact strong approximate groups.
— 2. A cheap version of the structure theorem —
In this section we use Theorem 2 to establish a “cheap” version of the structure theorem for ultra approximate groups. We begin by eliminating the elements of zero escape norm. Let us say that an approximate group is NSS if it contains no non-trivial subgroups of the ambient group, or equivalently if every non-identity element of has a positive escape norm. We say that an ultra approximate group is NSS if it is the ultralimit of NSS approximate groups.
Using the Gleason lemma, we can easily reduce to the NSS case:
Exercise 6 (Reduction to the NSS case) Let be a connected Lie group with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let , and let be an ultra strong approximate group which has a good model with
Let be the set of all elements in of zero (nonstandard) escape norm. Show that is a normal nonstandard finite subgroup of that lies in . If is the quotient map, and is the map factored through , show that there exists an ultra strong NSS approximate group in which has as a good model with
and such that is covered by finitely many left-translates of .
Let us now analyse the NSS case. Let be a connected Lie group, with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let be an ultra strong NSS approximate group which has a good model with
If is zero-dimensional, then by connectedness it is trivial, and then (by the properties of a good model) is a nonstandard finite group; since it is NSS, it is also trivial. Now suppose that is not zero-dimensional. Then contains non-identity elements whose image under is arbitrarily close to the identity of ; in particular, does not consist solely of the identity element, and thus contains elements of positive escape norm by the NSS assumption. Let be a non-identity element of with minimal escape norm , then must be the identity (so in particular is infinitesimal). (Note that any non-trivial NSS finite approximate group will contain non-identity elements of minimal escape norm, and the extension of this claim to the ultra approximate group case follows from Los’s theorem.) From Theorem 2 one has
for all . (Here we are now using the nonstandard asymptotic notation, thus means that for some standard .) In particular, from the minimality of , we see that there is a standard such that commutes with all elements with . In particular, if is a sufficiently small standard real number, we can find an ultra approximate subgroup of with
which is centralised by .
Now we show that generates a one-parameter subgroup of the model Lie group .
Exercise 7 (One-parameter subgroups from orbits) Let the notation be as above. Let be such that is infinitesimal but non-zero.
- Show that whenever .
- Show that the map is a one-parameter subgroup in (i.e. a continuous homomorphism from to ).
- Show that there exists an element of such that for all .
Similar statements hold with , replaced by .
We can now quotient out by the centraliser of and reduce the dimension of the Lie model:
Exercise 8 Let be the centraliser of in , and let be the nonstandard cyclic group generated by . (Thus, by the preceding discussion, contains , and is a central subgroup of containing . It will be important for us that and are both nonstandard sets, i.e. ultraproducts of standard sets.)
- (i) Show that is a compact subset of for each standard .
- (ii) Show that is a central subgroup of that contains .
- (iii) Show that is a central subgroup of that is a Lie group of dimension at least one, and so the quotient group is a Lie group of dimension strictly less than the dimension of .
- (iv) Let be the quotient map, and let be the obvious quotient of . Let be a convex symmetric body in the Lie algebra of . Show that for sufficiently small standard , there exists an ultra strong approximate group
with as a good model, with , and with covered by finitely many left-translates of .
Note that the quotient approximate group obtained by the above procedure is not necessarily NSS. However, it can be made NSS by Exercise 6. As such, one can iterate the above exercise until the dimension of the Lie model shrinks all the way to zero, at which point the NSS approximate group one is working with becomes trivial. This leads to a “cheap” structure theorem for approximate groups:
Exercise 9 (Cheap structure theorem) Let be an ultra approximate group in a nonstandard group .
- (i) Show that if has a good model by a connected Lie group , then is nilpotent. (Hint: first use Exercise 1, and then induct on the dimension of .)
- (ii) Show that is covered by finitely many left translates of a nonstandard subgroup of which admits a normal series
for some standard , where for every , is a normal nonstandard subgroup of , and is either a nonstandard finite group or a nonstandard central subgroup of . Furthermore, if is not central, then it is contained in the image of in . (Hint: first use the Lie model theorem and Exercise 1, and then induct on the dimension of .)
Exercise 10 (Cheap structure theorem, finite version) Let be a finite -approximate group in a group . Show that is covered by left-translates of a subgroup of which admits a normal series
for some , where for every , is a normal subgroup of , and is either finite or central in . Furthermore, if is not central, then it is contained in the image of in .
One can push the cheap structure theorem a bit further by controlling the dimension of the nilpotent Lie group in terms of the covering number of the ultra approximate group, as laid out in the following exercise.
Exercise 11 (Nilpotent groups) A Lie algebra is said to be nilpotent if the derived series , , becomes trivial after a finite number of steps.
- (i) Show that a connected Lie group is nilpotent if and only if its Lie algebra is nilpotent.
- (ii) If is a finite-dimensional nilpotent Lie algebra, show that there is a simply connected Lie group with Lie algebra , for which the exponential map is a (global) homeomorphism. Furthermore, any other connected Lie group with Lie algebra is a quotient of by a discrete central subgroup of .
- (iii) If and are as in (ii), show that the pushforward of a Haar measure (or Lebesgue measure) on is a bi-invariant Haar measure on . (Recall from Exercise 6 of Notes 3 that connected nilpotent Lie groups are unimodular.)
- (iv) If and are as in (ii), and is a bi-invariant Haar measure on , show that for all open precompact , where is the dimension of .
- (v) If is a connected (but not necessarily simply connected) nilpotent Lie group, and is the maximal compact normal subgroup of (which exists by Exercise 32 of Notes 7), show that is central, and is simply connected. As a consequence, conclude that if is a left-Haar measure of , then for all open precompact , where is the dimension of .
- (vi) Show that if is an ultra -approximate group which has a Lie model , and is the maximal compact normal subgroup of , then has dimension at most .
- (vii) Show that if is an ultra -approximate group, then there is an ultra -approximate group in that is modeled by a Lie group , such that is covered by finitely many left-translates of . (Hint: has a good model by a locally compact group ; by the Gleason-Yamabe theorem, has an open subgroup and a normal subgroup of inside with a Lie group. Set .)
- (viii) Show that if is an ultra -approximate group, then there is an ultra -approximate group in that is modeled by a nilpotent group of dimension , such that can be covered by finitely many left-translates of .
— 3. Local groups —
The main weakness of the cheap structure theorem in the preceding section is the continual reintroduction of torsion whenever one quotients out by the centraliser , which can destroy the NSS property. We now address the issue of how to fix this, by moving to the context of local groups rather than global groups. We will omit some details, referring to this recent paper for details.
We need to extend many of the notions we have been considering to the local group setting. We begin by generalising the concept of an approximate group.
Definition 3 (Approximate groups) A (local) -approximate group is a subset of a local group which is symmetric and contains the identity, such that is well-defined in , and for which is covered by left translates of (by elements in ). An ultra approximate group is an ultraproduct of -approximate groups.
Note that we make no topological requirements on or in this definition; in particular, we may as well give the local group the discrete topology. There are some minor technical advantages in requiring the local group to be symmetric (so that the inversion map is globally defined) and cancellative (so that or implies ), although these assumptions are essentially automatic in practice.
The exponent here is not terribly important in practice, thanks to the following variant of the Sanders lemma:
Exercise 12 Let be a finite -approximate group in a local group , except with only known to be well-defined rather than . Let . Show that there exists a finite -approximate subgroup in with well-defined and contained in , and with covered by left-translates of (by elements in ). (Hint: adapt the proof of Lemma 1 from Notes 7.)
Just as global approximate groups can be modeled by global locally compact groups (and in particular, global Lie groups), local approximate groups can be modeled by local locally compact groups:
Definition 4 (Good models) Let be a (local) ultra approximate group. A (local) good model for is a homomorphism from to a locally compact Hausdorff local group that obeys the following axioms:
- (Thick image) There exists a neighbourhood of the identity in such that and .
- (Compact image) is precompact.
- (Approximation by nonstandard sets) Suppose that , where is compact and is open. Then there exists a nonstandard finite set such that .
We make the pedantic remark that with our conventions, a global good model of a global approximate group only becomes a local good model of by after restricting the domain of to . It is also convenient for minor technical reasons to assume that the local group is symmetric (i.e. the inversion map is globally defined) but this hypothesis is not of major importance.
The Hrushovski Lie Model theorem can be localised:
Theorem 5 (Local Hrushovski Lie model theorem) Let be a (local) ultra approximate group. Then there is an ultra approximate subgroup of (thus ) with covered by finitely many left-translates of (by elements in ), which has a good model by a connected local Lie group .
The proof of this theorem is basically a localisation of the proof of the global Lie model theorem from Notes 7, and is omitted (see for details). One key replacement is that if is a local approximate group rather than a global one, then the global Gleason-Yamabe theorem (Theorem 1 from Notes 4) must be replaced by the local Gleason-Yamabe theorem of Goldbring, discussed in Section 6 of Notes 4.
One can define the notion of a strong -approximate group and ultra strong approximate group in the local setting without much difficulty, since strong approximate groups only need to work inside , which is well-defined. Using the local Lie model theorem, one can obtain a local version of Exercise 1. The Gleason lemma (Theorem 2) also localises without much difficulty to local strong approximate groups, as does the reduction to the NSS case in Exercise 6.
Now we once again analyse the NSS case. As before, let be a connected (local) Lie group, with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let be a (local) ultra strong NSS approximate group which has a (local) good model with
Again, we assume has dimension at least , since is trivial otherwise. We let be a non-identity element of minimal escape norm. As before, will have an infinitesimal escape norm and lie in the kernel of . If we set , then is an unbounded natural number, and the map will be a local one-parameter subgroup, i.e. a continuous homomorphism from to . This one-parameter subgroup will be non-trivial and centralised by a neighbourhood of the identity in .
In the global setting, we quotiented (the group generated by a large portion of) by the centraliser of . In the local setting, we perform a more “gentle” quotienting, which roughly speaking arises by quotienting by the geometric progression , where is a sufficiently small standard quantity to be chosen later. However, is only a local group rather than a global one, and so we must now digress to introduce the notion of quotients of local groups. It is convenient to restrict attention to symmetric cancellative local groups:
Definition 6 (Cancellative local groups) A local group is symmetric if the inversion operation is globally defined. It is said to be cancellative if the following assertions hold:
- (i) Whenever are such that and are well-defined and equal to each other, then . (Note that this implies in particular that .)
- (ii) Whenever are such that and are well-defined and equal to each other, then .
- (iii) Whenever are such that and are well-defined, then . (In particular, if is symmetric and is well-defined in for some , then is also symmetric.)
Exercise 13 Show that every local group contains an open neighbourhood of the identity which is also a symmetric cancellative local group.
Definition 7 (Sub-local groups) Given two symmetric local groups and , we say that is a sub-local group of if is the restriction of to a symmetric neighbourhood of the identity, and there exists an open neighbourhood of with the property that whenever are such that is defined in , then ; we refer to as an associated neighbourhood for . If is also a global group, we say that is a subgroup of .
If is a sub-local group of , we say that is normal if there exists an associated neighbourhood for with the additional property that whenever are such that is well-defined and lies in , then . We call a normalising neighbourhood of .
Example 3 If are the (additive) local groups and , then is a sub-local group of (with associated neighbourhood ). Note that this is despite not being closed with respect to addition in ; thus we see why it is necessary to allow the associated neighbourhood to be strictly smaller than . In a similar vein, the open interval is a sub-local group of .
The interval is also a sub-local group of ; here, one can take for instance as the associated neighbourhood. As all these examples are abelian, they are clearly normal.
Example 4 Let be a linear transformation on a finite-dimensional vector space , and let be the associated semi-direct product. Let , where is a subspace of that is not preserved by . Then is not a normal subgroup of , but it is a normal sub-local group of , where one can take as a normalising neighbourhood of .
Observe that any sub-local group of a cancellative local group is again a cancellative local group.
One also easily verifies that if is a local homomorphism from to for some open neighbourhood of the identity in , then is a normal sub-local group of , and hence of . Note that the kernel of a local morphism is well-defined up to local identity. If is Hausdorff, then the kernel will also be closed.
Conversely, normal sub-local groups give rise to local homomorphisms into quotient spaces.
Exercise 14 (Quotient spaces) Let be a cancellative local group, and let be a normal sub-local group with normalising neighbourhood . Let be a symmetric open neighbourhood of the identity such that . Show that there exists a cancellative local group and a surjective continuous homomorphism such that, for any , one has if and only if , and for any , one has open if and only if is open.
Example 5 Let be the additive local group , and let be the sub-local group , with normalising neighbourhood . If we then set , then the hypotheses of Exercise 14 are obeyed, and can be identified with , with the projection map .
Example 6 Let be the torus , and let be the sub-local group , where is an irrational number, with normalising neighbourhood . Set . Then the hypotheses of Exercise 14 are again obeyed, and can be identified with the interval , with the projection map for . Note, in contrast, that if one quotiented by the global group generated by , the quotient would be a non-Hausdorff space (and would also contain a dense set of torsion points, in contrast to the interval which is “locally torsion free”). It is because of this pathological behaviour of quotienting by global groups that we need to work with local group quotients instead.
We now return to the analysis of the NSS ultra strong approximate group . We give the ambient local group the discrete topology.
Exercise 15 If is a standard real that is sufficiently small depending on , show that there exists an ultra approximate group with
such that is a sub-local group of with normalising neighbourhood , that is also centralised by .
By Exercise 14, we may now form the quotient set . Show that this is an ultra approximate group that is modeled by , where is an open neighbourhood of the identity in and is the local one-parameter subgroup of introduced earlier. In particular, is modeled by a local Lie group of dimension one less than the dimension of .
Now we come to a key observation, which is the main reason why we work in the local groups category in the first place:
We will in fact prove a stronger claim:
Lemma 9 (Lifting lemma) If , then there exists such that and , where is the projection map.
Since is NSS, all non-identity elements of have non-zero escape norm, and so by the lifting lemma, all non-identity elements of also have non-zero escape norm, giving Lemma 8.
Proof: (Proof of Lemma 9) We choose to be a lift of (i.e. an element of in ) that minimises the escape norm . (Such a minimum exists since is nonstandard finite, thanks to Los’s theorem.) If is trivial, then so is and there is nothing to prove. Therefore we may assume that is not the identity and hence, since is NSS, that it has positive escape norm. Suppose, by way of contradiction, that . Our goal will be to reach a contradiction by finding another lift of with strictly smaller escape norm than . We will do this by setting for some suitably chosen .
We may assume that is infinitesimal, since otherwise there is nothing to prove; in particular lies in the kernel of the local model . We may thus find a lift of in the kernel of . In particular, we may assume that has infinitesimal escape norm.
Set , then is unbounded. By hypothesis, ; thus whenever . In particular, for every (standard) integer , . This implies that the group generated by lies in . In particular, lies in the kernel of , and hence lies in for all .
By (an appropriate local version of) Exercise 7, we can find such that
whenever ; since lies in for , we conclude that must be parallel to the generator of . Similarly, we have
whenever (say) for some that is also parallel to . In particular, for some
Since is the minimal escape norm of non-identity elements of , we have , and thus for some ; in particular, . Comparing this with (11) we see that
and thus
and hence
By the Euclidean algorithm, we can thus find a nonstandard integer number such that the quantity
lies in the interval . In particular
If we set then (as commutes with ) we see for all that
for all . In particular, . Since is also a lift of , this contradicts the minimality of , and the claim follows.
Because the NSS property is preserved, it is possible to improve upon Exercise 9:
Exercise 16 Strengthen Exercise 9 by ensuring the final quotient is nonstandard finite, and all the other quotients are central in .
As a consequence, one obtains a stronger structure theorem than Exercise 9. Call a symmetric subset containing the identity in a local group nilpotent of step at most if every iterated commutator in of length is well-defined and trivial.
Exercise 17 (Helfgott-Lindenstrauss conjecture)
- (i) Let be a (local) NSS ultra strong approximate group. Show that there is a symmetric subset of containing the identity which is nilpotent of some finite step, such that is covered by a finite number of left translates of .
- (ii) Let be a global NSS ultra strong approximate group with ambient group . Show that there is a nonstandard nilpotent subgroup of such that is covered by a finite number of left translates of .
- (iii) Let be an NSS strong -approximate group in a global group . Show that there is a nilpotent subgroup of of step such that can be covered by a finite number of left translates of .
- (iv) Let be a -approximate group in a global group . Show that there exists a subgroup of and a normal subgroup of contained in , such that is covered by left-translates of , and is nilpotent of step .
In fact, a stronger statement is true, involving the nilprogressions defined in Notes 6:
- (i) If is an NSS ultra strong approximate group, then there is an ultra nilprogression in such that contains , and can be covered by finitely many left-translates of .
- (ii) If is an ultra approximate group, then there is an ultra coset nilprogression in such that contains , and can be covered by finitely many left-translates of .
- (iii) For all , there exists such that, given a finite -approximate group in a group , one can find a coset nilprogression in of rank at most and step at most such that contains , and can be covered by at most left-translates of .
This proposition is established in this paper. The key point is to use the lifting lemma to observe that if (with the notation of the preceding discussion) contains a large nilprogression, then also contains a large nilprogression. One consequence of this proposition is that there is essentially no difference between local and global approximate groups, at the qualitative level at least:
Corollary 11 Let be a local -approximate group. Then there exists a -approximate subgroup of , with covered by left-translates of , such that is isomorphic to a global -approximate subgroup.
This is because coset nilprogressions (or large fractions thereof) can be embedded into global groups; again, see this paper for details.
For most applications, one does not need the full strength of Proposition 10; Exercise 17 will suffice. We will give some examples of this in the next set of notes.
]]>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.
— 1. Van Dantzig’s theorem —
Recall that a (non-empty) topological space is connected if the only clopen (i.e. closed and open) subsets of are the whole space and the empty set ; a non-empty topological space is disconnected if it is not connected. (By convention, the empty set is considered to be neither connected nor disconnected, somewhat analogously to how the natural number is neither considered prime nor composite.)
At the opposite extreme to connectedness is the property of being a totally disconnected space. This is a space whose only connected subsets are the singleton sets. Typical examples of totally disconnected spaces include discrete spaces (e.g. the integers with the discrete topology) and Cantor spaces (such as the standard Cantor set).
Most topological spaces are neither connected nor totally disconnected, but some intermediate combination of both. In the case of topological groups , this rather vague assertion can be formalised as follows.
Exercise 3
- Define a connected component of a topological space to be a maxmial connected set. Show that the connected components of form a partition of , thus every point in belongs to exactly one connected component.
- Let be a topological group, and let be the connected component of the identity. Show that is a closed normal subgroup of , and that is a totally disconnected subgroup of . Thus, one has a short exact sequence
of topological groups that describes as an extension of a totally disconnected group by a connected group.
- Conversely, if one has a short exact sequence
of topological groups, with connected and totally disconnected, show that is isomorphic to , and is isomorphic to .
- If is locally compact, show that and are also locally compact.
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. (Note however that even if one has a complete understanding of the factors of a short exact sequence , it may still be a non-trivial issue to fully understand the combined group , due to the possible presence of non-trivial group cohomology. See for instance this previous blog post for more discussion.)
For totally disconnected locally compact groups, one has the following fundamental theorem of van Dantzig’s theorem:
Theorem 7 (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.
To prove van Dantzig’s theorem, we first need a lemma from point set topology, which shows that totally disconnected spaces contain enough clopen sets to separate points:
Lemma 8 Let be a totally disconnected compact Hausdorff space, and let be distinct points in . Then there exists a clopen set that contains but not .
Proof: Let be the intersection of all the clopen sets that contain (note that is obviously clopen). Clearly is closed and contains . Our objective is to show that consists solely of . As is totally disconnected, it will suffice to show that is connected.
Suppose this is not the case, then we can split where are disjoint non-empty closed sets; without loss of generality, we may assume that lies in . As all compact Hausdorff spaces are normal, we can thus enclose in disjoint open subsets of . In particular, the topological boundary is compact and lies outside of . By definition of , we thus see that for every , we can find a clopen neighbourhood of that avoids ; by compactness of (and the fact that finite intersections of clopen sets are clopen), we can thus find a clopen neighbourhood of that is disjoint from . One then verifies that is a clopen neighbourhood of that is disjoint from , contradicting the definition of , and the claim follows.
Now we can prove van Dantzig’s theorem. We will use an argument from the book of Hewitt and Ross. Let be totally disconnected locally compact (and thus Hausdorff). Then we can find a compact neighbourhood of the identity. By Lemma 8, for every , we can find a clopen neighbourhood of the identity that avoids ; by compactness of , we may thus find a clopen neighbourhood of the identity that avoids . By intersecting this neighbourhood with , we may thus find a compact clopen neighbourhood of the identity. As is both compact and open, we may then the continuity of the group operations find a symmetric neighbourhood of the identity such that . In particular, if we let be the group generated by , then is an open subgroup of contained in and is thus compact as required.
Remark 3 The same argument shows that a totally disconnected locally compact group contains arbitrarily small compact open subgroups, or in other words the compact open subgroups form a neighbourhood base for the identity.
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. Thanks to the Gleason-Yamabe theorem for compact groups, such groups are the inverse limits of compact totally disconnected Lie groups. But it is easy to see that a compact totally disconnected Lie group must be finite, and so compact totally disconnected groups are necessarily profinite. 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).
We can pull van Dantzig’s theorem back to more general locally compact groups:
Exercise 4 Let be a locally compact group.
- Show that contains an open subgroup which is “compact-by-connected” in the sense that is compact. (Hint: apply van Dantzig’s theorem to .)
- If is compact-by-connected, and is an open neighbourhood of the identity, show that there exists a compact subgroup of in such that is isomorphic to a Lie group. (Hint: use Theorem 1, and observe that any open subgroup of the compact-by-connected group has finite index and thus has only finitely many conjugates.) Conclude Theorem 2.
- Show that any locally compact Hausdorff group contains an open subgroup that is isomorphic to an inverse limit of Lie groups , which each Lie group has at most finitely many connected components. Furthermore, each is isomorphic to for some compact normal subgroup of , with for . If is first countable, show that this inverse limit can be taken to be a sequence (so that the index set is simply the natural numbers with the usual ordering), and the then shrink to zero in the sense that they lie inside any given open neighbourhood of the identity for large enough.
Exercise 5 Let be a totally disconnected locally compact group. Show that every compact subgroup of is contained in a compact open subgroup. (Hint: van Dantzig’s theorem provides a compact open subgroup, but it need not contain . But is there a way to modify it so that it is normalised by ? Why would being normalised by be useful?)
— 2. The invariance of domain theorem —
In this section we give a proof of the invariance of domain theorem. The main topological tool for this is Brouwer’s famous fixed point theorem:
Theorem 9 (Brouwer fixed point theorem) Let be a continuous function on the unit ball in a Euclidean space . Then has at least one fixed point, thus there exists with .
This theorem has many proofs. We quickly sketch one of these proofs as follows:
Exercise 6 For this exercise, suppose for sake of contradiction that Theorem 9 is false, thus there is a continuous map from to with no fixed point.
- Show that there exists a smooth map from to with no fixed point.
- Show that there exists a smooth map from to the unit sphere , which equals the identity function on .
- Show that there exists a smooth map from to the unit sphere , which equals the map on a neighbourhood of .
- By computing the integral in two different ways (one by using Stokes’ theorem, and the other by using the -dimensional nature of the sphere ), establish a contradiction.
Now we prove Theorem 3. By rescaling and translation invariance, it will suffice to show the following claim:
Theorem 10 (Invariance of domain, again) Let be an continuous injective map. Then lies in the interior of .
Let be as in Theorem 10. The map is a continuous bijection between compact Hausdorff spaces and is thus a homeomorphism. In particular, the inverse map is continuous. Using the Tietze extension theorem, we can find a continuous function that extends .
The function has a zero on , namely at . We can use the Brouwer fixed point theorem to show that this zero is stable:
Lemma 11 (Stability of zero) Let be a continuous function such that for all . Then has at least one zero (i.e. there is a such that ).
Proof: Apply Theorem 9 to the function
Now suppose that Theorem 10 failed, so that is not an interior point of . We will use this to locate a small perturbation of that no longer has a zero on , contradicting Lemma 11.
We turn to the details. Let be a small number. By continuity of , we see (if is chosen small enough) that we have whenever and .
On the other hand, since is not an interior point of , there exists a point with that lies outside . By translating if necessary, we may take ; thus avoids zero, , and we have
Let denote the set , where
and
By construction, is compact but does not contain . Crucially, there is a continuous map defined by setting
Note that is continuous and well-defined since avoids zero. Informally, is a perturbation of caused by pushing out a small distance away from the origin (and hence also away from ), with being the “pushing” map.
By construction, is non-zero on ; since is compact, is bounded from below on by some . By shrinking if necessary we may assume that .
By the Weierstrass approximation theorem, we can find a polynomial such that
for all ; in particular, does not vanish on . At present, it is possible that vanishes on . But as is smooth and has measure zero, also has measure zero; so by shifting by a small generic constant we may assume without loss of generality that also does not vanish on . (If one wishes, one can use an algebraic geometry argument here instead of a measure-theoretic one, noting that lies in an algebraic hypersurface and can thus be generically avoided by perturbation. A purely topological way to avoid zeroes in is also given in Kulpa’s paper.)
Now consider the function defined by
This is a continuous function that is never zero. From (3), (2) we have
whenever is such that . On the other hand, if , then from (2), (1) we have
and hence by (3) and the triangle inequality
Thus in all cases we have
for all . But this, combined with the non-vanishing nature of , contradicts Lemma 11.
— 3. Hilbert’s fifth problem —
We now establish Theorem 5. Let be a locally Euclidean group. By Exercise 5 of Notes 0, is Hausdorff; it is also locally compact and first countable. Thus, by Exercise 4, such a group contains an open subgroup which is isomorphic to the inverse limit of Lie groups , each of which has only finitely many components. Clearly, is also locally Euclidean. If it is Lie, then is locally Lie and thus Lie. Thus, by replacing with if necessary, we may assume without loss of generality that is the inverse limit , each of which has only finitely many components.
By Exercise 4, each is isomorphic to the quotient of by some compact normal subgroup with . In particular, is isomorphic to the quotient of by a compact normal subgroup . By Cartan’s theorem (Theorem 2 of Notes 2), is also a Lie group. Among other things, this implies that the quotient homomorphism from the Lie algebra of to the Lie algebra of is surjective; indeed, it is the quotient map by the Lie algebra of . This implies that there is a continuous map from to that inverts the quotient map; in other words, we have a continuous map from the one-parameter subgroups of to the one-parameter subgroups of , such that for all .
Exercise 7 By iterating these maps and passing to the inverse limit, conclude that for each , there is a continuous map such that for all .
Because is a Lie group, the exponential map is a homeomorphism from a neighbourhood of the origin in to a neighbourhood of the identity in . We can thus obtain a continuous map from a neighbourhood of the identity in to . Since , this map is injective.
Now we use the hypothesis that is locally Euclidean (and in particular, has a well-defined dimension ). By Exercise 1, we have
for all . On the other hand, since each is a quotient of the next Lie group , one has
Since there are only finitely many possible values for the (necessarily integral) dimension between and , we conclude that the dimension must eventually stabilise, i.e. one has
for all sufficiently large . By discarding the first few terms in the sequence and relabeling, we may thus assume that the dimension is constant for all . Since , this implies that the Lie groups have dimension zero for all . As the are also compact, they are thus finite. Thus each is a finite extension of . As is the inverse limit of the as , we conclude that is a profinite group, i.e. the inverse limit of finite groups. In particular, is totally disconnected.
We now study the short exact squence
playing off the locally connected nature of the Lie group against the totally disconnected nature of .
As discussed earlier, we have a continuous injective map from a neighbourhood of the identity in to that partially inverts the quotient map. By translation, we may normalise . As is locally connected, we can find a connected neighborhood of the identity in such that .
Now consider the set . On the one hand, this set is contained in and contains ; on the other hand, it is connected. As is totally disconnected, this set must equal , thus for all . A similar argument based on consideration of the set shows that for all . Thus is a homomorphism from the local group to .
Finally, for any , a consideration of the set reveals that commutes with . As a consequence, we see that the preimage of under the quotient map is isomorphic as a local group to , after identifying with for any and .
On the other hand, is locally Euclidean, and hence is locally Euclidean also, and in particular locally connected. This implies that is locally connected; but as is also totally disconnected, it must be discrete. This is now locally isomorphic to and hence to , and is thus locally Lie and hence Lie as required. (Here, we say that two groups are locally isomorphic if they have neighbourhoods of the identity which are isomorphic to each other as local groups.)
Exercise 8 Let be a locally compact Hausdorff first-countable group which is “finite-dimensional” in the sense that it does not contain continuous injective images of non-trivial open sets of Euclidean spaces of arbitrarily large dimension. Show that is locally isomorphic to the direct product of a Lie group and a totally disconnected compact group . (Note that this local isomorphism does not necessarily extend to a global isomorphism, as the example of the solenoid group shows.)
Remark 4 Of course, it is possible for locally compact groups to be infinite-dimensional; a simple example is the infinite-dimensional torus , which is compact, abelian, metrisable, and locally connected, but infinite dimensional. (It will still be an inverse limit of Lie groups, though.)
Exercise 9 Show that a topological group is Lie if and only if it is locally compact, Hausdorff, first-countable, locally connected, and finite-dimensional.
Remark 5 It is interesting to note that this characterisation barely uses the real numbers , which are of course fundamental in defining the smooth structure of a Lie group; the only remaining reference to comes through the notion of finite dimensionality. It is also possible, using dimension theory, to obtain alternate characterisations of finite dimensionality (e.g. finite Lebesgue covering dimension) that avoid explicit mention of the real line, thus capturing the concept of a Lie group using only the concepts of point-set topology (and the concept of a group, of course).
— 4. Transitive actions —
We now prove Proposition 6. As this is a stronger statement than Theorem 5, it will not be surprising that we will be using a very similar argument to prove the result.
Let be a locally compact -compact group that acts transitively, faithfully, and continuously on a connected manifold . The advantage of transitivity is that one can now view as a homogeneous space of , where is the stabiliser of a point (and is thus a closed subgroup of ). Note that a priori, we only know that and are identifiable as sets, with the identification map defined by setting being continuous; but thanks to the -compact hypothesis, we can upgrade to a homeomorphism. Indeed, as is -compact, is also; and so given any compact neighbourhood of the identity in , can be covered by countably many translates of . By the Baire category theorem, one of these translates has an image in with non-empty interior, which implies that has as an interior point. From this it is not hard to see that the map is open; as it is also a continuous bijection, it is therefore a homeomorphism.
By the Gleason-Yamabe theorem (Theorem 2), has an open subgroup that is the inverse limit of Lie groups. (Note that is Hausdorff because it acts faithfully on the Hausdorff space .) acts transitively on , which is an open subset of and thus also a manifold. Thus, we may assume without loss of generality that is itself the inverse limit of Lie groups.
As is -compact, the manifold is also. As acts faithfully on , this makes first countable; and so (by Exercise 4) is the inverse limit of a sequence of Lie groups , with each projecting surjectively onto , and with the shrinking to the identity.
Let be the projection of onto ; this is a closed subgroup of the Lie group , and each projects surjectively onto . Then are manifolds, and is the inverse limit of the .
Exercise 10 Show that the dimensions of the are be non-decreasing, and bounded above by the dimension of . (Hint: repeat the arguments of the previous section. The need no longer be compact, but they are still closed, and this still suffices to make the preceding arguments go through.)
Thus, for large enough, the dimensions of must be constant; by renumbering, we may assume that all the have the same dimension. As each is a cover of with structure group , we conclude that the are zero-dimensional and compact, and thus finite. On the other hand, is locally connected, which implies that the are eventually trivial. Indeed, if we pick a simply connected neighbbourhood of the identity in , then by local connectedness of , there exists a connected neighbourhood of the identity in whose projection to is contained in . Being open, must contain one of the . If is non-trivial for any , then the projection of to will then be disconnected (as this projection will be contained in a neighbourhood with the topological structure of , and its intersection with the latter fibre is at least as large as . We conclude that is trivial for large enough, and so is a Lie group as required.
]]>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.
In the next set of notes, we will combine the Gleason-Yamabe theorem with some topological analysis (and in particular, using the invariance of domain theorem) to establish some further control on locally compact groups, and in particular obtaining a solution to Hilbert’s fifth problem.
To prove the Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a Gleason metric:
Definition 2 Let be a topological group. A Gleason metric on is a left-invariant metric which generates the topology on and obeys the following properties for some constant , writing for :
- (Escape property) If and is such that , then .
- (Commutator estimate) If are such that , then
where is the commutator of and .
Theorem 3 (Building Lie structure from Gleason metrics) Let be a locally compact group that has a Gleason metric. Then is isomorphic to a Lie group.
The second tool is the existence of a left-invariant Haar measure on any locally compact group; see Theorem 3 from Notes 3. Finally, we will also need the compact case of the Gleason-Yamabe theorem (Theorem 8 from Notes 3), which was proven via the Peter-Weyl theorem:
Theorem 4 (Gleason-Yamabe theorem for compact groups) Let be a compact Hausdorff group, and let be a neighbourhood of the identity. Then there exists a compact normal subgroup of contained in such that is isomorphic to a linear group (i.e. a closed subgroup of a general linear group ).
To finish the proof of the Gleason-Yamabe theorem, we have to somehow use the available structures on locally compact groups (such as Haar measure) to build good metrics on those groups (or on suitable subgroups or quotient groups). The basic construction is as follows:
Definition 5 (Building metrics out of test functions) Let be a topological group, and let be a bounded non-negative function. Then we define the pseudometric by the formula
and the semi-norm by the formula
Note that one can also write
where is the “derivative” of in the direction .
Exercise 1 Let the notation and assumptions be as in the above definition. For any , establish the metric-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
- (Left-invariance) . In particular, .
In particular, we have the norm-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
We remark that the first three properties of in the above exercise ensure that is indeed a pseudometric.
To get good metrics (such as Gleason metrics) on groups , it thus suffices to obtain test functions that obey suitably good “regularity” properties. We will achieve this primarily by means of two tricks. The first trick is to obtain high-regularity test functions by convolving together two low-regularity test functions, taking advantage of the existence of a left-invariant Haar measure on . The second trick is to obtain low-regularity test functions by means of a metric-like object on . This latter trick may seem circular, as our whole objective is to get a metric on in the first place, but the key point is that the metric one starts with does not need to have as many “good properties” as the metric one ends up with, thanks to the regularity-improving properties of convolution. As such, one can use a “bootstrap argument” (or induction argument) to create a good metric out of almost nothing. It is this bootstrap miracle which is at the heart of the proof of the Gleason-Yamabe theorem (and hence to the solution of Hilbert’s fifth problem).
The arguments here are based on the nonstandard analysis arguments used to establish Hilbert’s fifth problem by Hirschfeld and by Goldbring (and also some unpublished lecture notes of Goldbring and van den Dries). However, we will not explicitly use any nonstandard analysis in this post.
— 1. Warmup: the Birkhoff-Kakutani theorem —
To illustrate the basic idea of using test functions to build metrics, let us first establish a classical theorem on topological groups, which gives a necessary and sufficient condition for metrisability. Recall that a topological space is metrisable if there is a metric on that space that generates the topology.
Theorem 6 (Birkhoff-Kakutani theorem) A topology group is metrisable if and only if it is Hausdorff and first countable.
Remark 1 The group structure is crucial; for instance, the long line is Hausdorff and first countable, but not metrisable.
We now prove this theorem (following the arguments in this book of Montgomery and Zippin). The “only if” direction is easy, so it suffices to establish the “if” direction. The key lemma is
Lemma 7 (Urysohn-type lemma) Let be a Hausdorff first countable group. Then there exists a bounded continuous function with the following properties:
- (Unique maximum) , and for all .
- (Neighbourhood base) The sets for form a neighbourhood base at the identity.
- (Uniform continuity) For every , there exists an open neighbourhood of the identity such that for all and .
Note that if had a left-invariant metric, then the function would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.
Exercise 2 Let be a Hausdorff first countable group, and let be as in Lemma 7. Show that is a metric on (so in particular, only vanishes when ) and that generates the topology of (thus every set which is open with respect to is open in , and vice versa).
In view of the above exercise, we see that to prove the Birkhoff-Kakutani theorem, it suffices to prove Lemma 7, which we now do. By first countability, we can find a countable neighbourhood base
of the identity. As is Hausdorff, we must have
Using the continuity of the group operations, we can recursively find a sequence of nested open neighbourhoods of the identity
such that each is symmetric (i.e. if and only if ), is contained in , and is such that for each . In particular the are also a neighbourhood base of the identity with
For every dyadic rational in , we can now define the open sets by setting
where is the binary expansion of with . By repeated use of the hypothesis we see that the are increasing in ; indeed, we have the inclusion
We now set
with the understanding that if the supremum is over the empty set. One easily verifies using (4) that is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 7, and the Birkhoff-Kakutani theorem follows.
Exercise 3 Let be a topological group. Show that is completely regular, that is to say for every closed subset in and every , there exists a continuous function that equals on and vanishes on .
Exercise 4 (Reduction to the metrisable case) Let be a locally compact group, let be an open neighbourhood of the identity, and let be the group generated by .
- (i) Construct a sequence of open neighbourhoods of the identity
with the property that and for all , where and .
- (ii) If we set , show that is a closed normal subgroup in , and the quotient group is Hausdorff and first countable (and thus metrisable, by the Birkhoff-Kakutani theorem).
- (iii) Conclude that to prove the Gleason-Yamabe theorem (Theorem 1), it suffices to do so under the assumption that is metrisable.
The above arguments are essentially in this paper of Gleason.
Exercise 5 (Birkhoff-Kakutani theorem for local groups) Let be a local group which is Hausdorff and first countable. Show that there exists an open neighbourhood of the identity which is metrisable.
— 2. Obtaining the commutator estimate via convolution —
We now return to the main task of constructing Gleason metrics. The first thing we will do is dispense with the commutator property (1). Thus, define a weak Gleason metric on a topological group to be a left-invariant metric which generates the topology on and obeys the escape property for some constant , thus one has
Theorem 8 Every weak Gleason metric is a Gleason metric (possibly after adjusting the constant ).
We now prove this theorem. The key idea here is to involve a bump function formed by convolving together two Lipschitz functions. The escape property (5) will be crucial in obtaining quantitative control of the metric geometry at very small scales, as one can study the size of a group element very close to the origin through its powers , which are further away from the origin.
Specifically, let be a small quantity to be chosen later, and let be a non-negative Lipschitz function supported on the ball which is not identically zero. For instance, one could use the explicit function
where , although the exact form of will not be important for our argument. Being Lipschitz, we see that
for all (where we allow implied constants to depend on , , and ), where denotes the sup norm.
Let be a left-invariant Haar measure on , the existence of which was established in Theorem 3 from Notes 3. We then form the convolution , with convolution defined using the formula
This is a continuous function supported in , and gives a metric and a norm as usual.
We now prove a variant of the commutator estimate (1), namely that
whenever . To see this, we first use the left-invariance of Haar measure to write
We would like to similarly move the operator over to the second factor, but we run into a difficulty due to the non-abelian nature of . Nevertheless, we can still do this provided that we twist that operator by a conjugation. More precisely, we have
where is conjugated by . If , the integrand is only non-zero when . Applying (6), we obtain the bound
To finish the proof of (8), it suffices to show that
whenever and .
We can achieve this by the escape property (5). Let be a natural number such that , then and so . Conjugating by , this implies that , and so by (5), we have (if is small enough), and the claim follows.
Next, we claim that the norm is locally comparable to the original norm . More precisely, we claim:
Claim 2 follows easily from (9) and (6), so we turn to Claim 1. Let , and let be a natural number such that
Then by the triangle inequality
This implies that and have overlapping support, and hence lies in . By the escape property (5), this implies (if is small enough) that , and the claim follows.
Combining Claim 2 with (8) we see that
whenever are small enough. Now we use the identity
and the triangle inequality to conclude that
whenever are small enough. Theorem 8 then follows from Claim 1 and Claim 2.
— 3. Building metrics on NSS groups —
We will now be able to build metrics on groups using a set of hypotheses that do not explicitly involve any metric at all. The key hypothesis will be the no small subgroups (NSS) property:
Definition 9 (No small subgroups) A topological group has the no small subgroups (or NSS) property if there exists an open neighbourhood of the identity which does not contain any subgroup of other than the trivial group.
Exercise 7 Show that any group with a weak Gleason metric is NSS.
For an example of a group which is not NSS, consider the infinite-dimensional torus . From the definition of the product topology, we see that any neighbourhood of the identity in this torus contains an infinite-dimensional subtorus, and so this group is not NSS.
Exercise 8 Show that for any prime , the -adic groups and are not NSS. What about the solenoid group ?
Exercise 9 Show that an NSS group is automatically Hausdorff. (Hint: use Exercise 3 from Notes 3.)
Exercise 10 Show that an NSS locally compact group is automatically metrisable. (Hint: use Exercise 4.)
Exercise 11 (NSS implies escape property) Let be a locally compact NSS group. Show that if is a sufficiently small neighbourhood of the identity, then for every , there exists a positive integer such that . Furthermore, for any other neighbourhood of the identity, there exists a positive integer such that if , then .
We can now prove the following theorem (first proven in full generality by Yamabe), which is a key component in the proof of the Gleason-Yamabe theorem and in the wider theory of Hilbert’s fifth problem.
Theorem 10 Every NSS locally compact group admits a weak Gleason metric. In particular, by Theorem 8 and Theorem 3, every NSS locally compact group is isomorphic to a Lie group.
In view of this theorem and Exercise 6, we see that for locally compact groups, the property of being a Lie group is equivalent to the property of being an NSS group. This is a major advance towards both the Gleason-Yamabe theorem and Hilbert’s fifth problem, as it has reduced the property of being a Lie group into a condition that is almost purely algebraic in nature.
We now prove Theorem 10. An important concept will be that of an escape norm associated to an open neighbourhood of a group , defined by the formula
for any , where ranges over the natural numbers (thus, for instance , with equality iff ). Thus, the longer it takes for the orbit to escape , the smaller the escape norm.
Strictly speaking, the escape norm is not necessarily a norm, as it need not obey the symmetry, non-degeneracy, or triangle inequalities; however, we shall see that in many situations, the escape norm behaves similarly to a norm, even if it does not exactly obey the norm axioms. Also, as the name suggests, the escape norm will be well suited for establishing the escape property (5).
It is possible for the escape norm of a non-identity element to be zero, if contains the group generated by . But if the group has the NSS property, then we see that this cannot occur for all sufficiently small (where “sufficiently small” means “contained in a suitably chosen open neighbourhood of the identity”). In fact, more is true: if are two sufficiently small open neighbourhoods of the identity in a locally compact NSS group , then the two escape norms are comparable, thus we have
for all (where the implied constants can depend on ).
By symmetry, it suffices to prove the second inequality in (12). By (11), it suffices to find an integer such that whenever is such that , then . But this follows from Exercise 11. This concludes the proof of (12).
Exercise 12 Let be a locally compact group. Show that if is a left-invariant metric on obeying the escape property (5) that generates the topology, then is NSS, and is comparable to for all sufficiently small and for all sufficiently small . (In particular, any two left-invariant metrics obeying the escape property and generating the topology are locally comparable to each other.)
Henceforth is a locally compact NSS group. We now establish a metric-like property on the escape norm .
Proposition 11 (Approximate triangle inequality) Let be a sufficiently small open neighbourhood of the identity. Then for any and any , one has
(where the implied constant can depend on ).
Of course, in view of (12), the exact choice of is irrelevant, so long as it is small. It is slightly convenient to take to be symmetric (thus ), so that for all .
Proof: We will use a bootstrap argument. Assume to start with that we somehow already have a weaker form of the conclusion, namely
for all and some huge constant ; we will then deduce the same estimate with a smaller value of . Afterwards we will show how to remove the hypothesis (13).
Now suppose we have (13) for some . Motivated by the argument in the previous section, we now try to convolve together two “Lipschitz” functions. For this, we will need some metric-like functions. Define the modified escape norm by the formula
where the infimum is over all possible ways to split as a finite product of group elements. From (13), we have
and we have the triangle inequality
for any . We also have the symmetry property . Thus gives a left-invariant semi-metric on by defining
We can now define a “Lipschitz” function by setting
On the one hand, we see from (14) that this function takes values in obeys the Lipschitz bound
for any . On the other hand, it is supported in the region where , which by (14) (and (11)) is contained in .
We could convolve with itself in analogy to the preceding section, but in doing so, we will eventually end up establishing a much worse estimate than (13) (in which the constant is replaced with something like ). Instead, we will need to convolve with another function , that we define as follows. We will need a large natural number (independent of ) to be chosen later, then a small open neighbourhood of the identity (depending on ) to be chosen later. We then let be the function
Similarly to , we see that takes values in and obeys the Lipschitz-type bound
for all and . Also, is supported in , and hence (if is sufficiently small depending on ) is supported in , just as is.
The functions need not be continuous, but they are compactly supported, bounded, and Borel measurable, and so one can still form their convolution , which will then be continuous and compactly supported; indeed, is supported in .
We have a lower bound on how big is, since
(where we allow implied constants to depend on , but remain independent of , , or ). This gives us a way to compare with . Indeed, if , then (as in the proof of Claim 1 in the previous section) we have ; this implies that
for all , and hence by (12) we have
also. In the converse direction, we have
thanks to (15). But we can do better than this, as follows. For any , we have the analogue of (10), namely
If , then the integrand vanishes unless . By continuity, we can find a small open neighbourhood of the identity such that for all and ; we conclude from (15), (16) that
whenever and . To use this, we observe the telescoping identity
for any and natural number , and thus by the triangle inequality
whenever and . Using the trivial bound , we then have
optimising in we obtain
and hence by (12)
where the implied constant in can depend on , but is crucially independent of . Note the essential gain of here compared with (18). We also have the norm inequality
Combining these inequalities with (17) we see that
Thus we have improved the constant in the hypothesis (13) to . Choosing large enough and iterating, we conclude that we can bootstrap any finite constant in (13) to .
Of course, there is no reason why there has to be a finite for which (13) holds in the first place. However, one can rectify this by the usual trick of creating an epsilon of room. Namely, one replaces the escape norm by, say, for some small in the definition of and in the hypothesis (13). Then the bound (13) will be automatic with a finite (of size about ). One can then run the above argument with the requisite changes and conclude a bound of the form
uniformly in ; we omit the details. Sending , we have thus shown Proposition 11.
Now we can finish the proof of Theorem 10. Let be a locally compact NSS group, and let be a sufficiently small neighbourhood of the identity. From Proposition 11, we see that the escape norm and the modified escape norm are comparable. We have seen is a left-invariant pseudometric. As is NSS and is small, there are no non-identity elements with zero escape norm, and hence no non-identity elements with zero modified escape norm either; thus is a genuine metric.
We now claim that generates the topology of . Given the left-invariance of , it suffices to establish two things: firstly, that any open neighbourhood of the identity contains a ball around the identity in the metric; and conversely, any such ball contains an open neighbourhood around the identity.
To prove the first claim, let be an open neighbourhood around the identity, and let be a smaller neighbourhood of the identity. From (12) we see (if is small enough) that is comparable to , and contains a small ball around the origin in the metric, giving the claim. To prove the second claim, consider a ball in the metric. For any positive integer , we can find an open neighbourhood of the identity such that , and hence for all . For large enough, this implies that , and the claim follows.
To finish the proof of Theorem 10, we need to verify the escape property (5). Thus, we need to show that if , are such that is sufficiently small, then we have . We may of course assume that is not the identity, as the claim is trivial otherwise. As is comparable to , we know that there exists a natural number such that . Let be a neighbourhood of the identity small enough that . We have for all , so and hence . Let be the first multiple of larger than , then and so . Since , this implies . Since is divisible by , we conclude that , and the claim follows from (12).
— 4. NSS from subgroup trapping —
In view of Theorem 10, the only remaining task in the proof of the Gleason-Yamabe theorem is to locate “big” subquotients of a locally compact group with the NSS property. We will need some further notation. Given a neighbourhood of the identity in a topological group , let denote the union of all the subgroups of that are contained in . Thus, a group is NSS if is trivial for all sufficiently small .
We will need a property that is weaker than NSS:
Definition 12 (Subgroup trapping) A topological group has the subgroup trapping property if, for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that generates a subgroup contained in .
Clearly, every NSS group has the subgroup trapping property. Informally, groups with the latter property do have small subgroups, but one cannot get very far away from the origin just by combining together such subgroups.
Example 1 The infinite-dimensional torus does not have the NSS property, but it does have the subgroup trapping property.
It is difficult to produce an example of a group that does not have the subgroup trapping property; the reason for this will be made clear in the next section. For now, we establish the following key result.
Proposition 13 (From subgroup trapping to NSS) Let be a locally compact group with the subgroup trapping property, and let be an open neighbourhood of the identity in . Then there exists an open subgroup of , and a compact subgroup of contained in , such that is locally compact and NSS. In particular, by Theorem 10, is isomorphic to a Lie group.
Intuitively, the idea is to use the subgroup trapping property to find a small compact normal subgroup that contains for some small , and then quotient this group out to get an NSS group. Unfortunately, because is not necessarily contained in , this quotienting operation may create some additional small subgroups. To fix this, we need to pass from the compact subgroup to a smaller one. In order to understand the subgroups of compact groups, the main tool will be Gleason-Yamabe theorem for compact groups (Theorem 4).
For us, the main reason why we need the compact case of the Gleason-Yamabe theorem is that Lie groups automatically have the NSS property, even though need not. Thus, one can view Theorem 4 as giving the compact case of Proposition 13.
We now prove Proposition 13, using an argument of Yamabe. Let be a locally compact group with the subgroup trapping property, and let be an open neighbourhood of the identity. We may find a smaller neighbourhood of the identity with , which in particular implies that ; by shrinking if necessary, we may assume that is compact. By the subgroup trapping property, one can find an open neighbourhood of the identity such that is contained in , and thus is a compact subgroup of contained in . By shrinking if necessary we may assume .
Ideally, if were normal and contained in , then the quotient group would have the NSS property. Unfortunately need not be normal, and need not be contained in , but we can fix this as follows. Applying Theorem 4, we can find a compact normal subgroup of contained in such that is isomorphic to a Lie group, and in particular is NSS. In particular, we can find an open symmetric neighbourhood of the identity in such that and that the quotient space has no non-trivial subgroups in , where is the quotient map.
We now claim that is normalised by . Indeed, if , then the conjugate of is contained in and hence in . As is a group, it must thus be contained in and hence in . But then is a subgroup of that is contained in , and is hence trivial by construction. Thus , and so is normalised by . If we then let be the subgroup of generated by and , we see that is an open subgroup of , with a compact normal subgroup of .
To finish the job, we need to show that has the NSS property. It suffices to show that has no nontrivial subgroups. But any subgroup in pulls back to a subgroup in , hence in , hence in , hence in ; since has no nontrivial subgroups, the claim follows. This concludes the proof of Proposition 13.
— 5. The subgroup trapping property —
In view of Theorem 10, Proposition 13, and Exercise 4, we see that the Gleason-Yamabe theorem (Theorem 1) now reduces to the following claim.
Proposition 14 Every locally compact metrisable group has the subgroup trapping property.
We now prove this proposition, which is the hardest step of the entire proof and uses almost all the tools already developed. In particular, it requires both Theorem 4 and Gleason’s convolution trick, as well as some of the basic theory of Hausdorff distance; as such, this is perhaps the most “infinitary” of all the steps in the argument.
The Gleason-type arguments can be encapsulated in the following proposition, which is a weak version of the subgroup trapping property:
Proposition 15 (Finite trapping) Let be a locally compact group, let be an open precompact neighbourhood of the identity, and let be an integer. Then there exists an open neighbourhood of the identity with the following property: if is a symmetric set containing the identity, and is such that , then .
Informally, Proposition 15 asserts that subsets of grow much more slowly than “large” sets such as . We remark that if one could replace in the conclusion here by , then a simple induction on (after first shrinking to lie in ) would give Proposition 14. It is the loss of in the exponent that necessitates some non-trivial additional arguments.
Proof: } Let be small enough to be chosen later, and let be as in the proposition. Once again we will convolve together two “Lipschitz” functions to obtain a good bump function which generates a useful metric for analysing the situation. The first bump function will be defined by the formula
Then takes values in , equals on , is supported in , and obeys the Lipschitz type property
for all . The second bump function is similarly defined by the formula
where , where is a quantity depending on and to be chosen later. If is small enough depending on and , then , and so also takes values in , equals on , is supported in , and obeys the Lipschitz type property
Now let . Then is supported on and (where implied constants can depend on , ). As before, we conclude that whenever is sufficiently small.
Now suppose that ; we will estimate . From (19) one has
(note that and commute). For the first term, we can compute
and
Since , , so by (21) we conclude that
For the second term, we similarly expand
Using (21), (20) we conclude that
Putting this together we see that
for all , which in particular implies that
for all . For sufficiently large, this gives as required.
We will also need the following compactness result in the Hausdorff distance
between two non-empty closed subsets of a metric space .
Example 2 In with the usual metric, the finite sets converge in Hausdorff distance to the closed interval .
Exercise 13 Show that the space of non-empty closed subsets of a compact metric space is itself a compact metric space (with the Hausdorff distance as the metric). (Hint: use the Heine-Borel theorem.)
Now we can prove Proposition 14. Let be a locally compact group endowed with some metric , and let be an open neighbourhood of the identity; by shrinking we may assume that is precompact. Let be a sequence of balls around the identity with radius going to zero, then is a symmetric set in that contains the identity. If, for some , for every , then and we are done. Thus, we may assume for sake of contradiction that there exists such that and ; since the go to zero, we have . By Proposition 15, we can also find such that .
The sets are closed subsets of ; by Exercise 13, we may pass to a subsequence and assume that they converge to some closed subset of . Since the are symmetric and contain the identity, is also symmetric and contains the identity. For any fixed , we have for all sufficiently large , which on taking Hausdorff limits implies that . In particular, the group is a compact subgroup of contained in .
Let be a small neighbourhood of the identity in to be chosen later. By Theorem 4, we can find a normal subgroup of contained in such that is NSS. Let be a neigbourhood of the identity in so small that has no small subgroups. A compactness argument then shows that there exists a natural number such that for any that is not in , at least one of must lie outside of .
Now let be a small parameter. Since , we see that does not lie in the -neighbourhood of if is small enough, where is the projection map. Let be the first integer for which does not lie in , then and as (for fixed ). On the other hand, as , we see from another application of Proposition 15 that if is sufficiently large depending on .
On the other hand, since converges to a subset of in the Hausdorff distance, we know that for large enough, and hence is contained in the -neighbourhood of . Thus we can find an element of that lies within of a group element of , but does not lie in ; thus lies inside . By construction of , we can find such that lies in . But also lies within of , which lies in and hence in , where denotes a quantity depending on that goes to zero as . We conclude that and are separated by , which leads to a contradiction if is sufficiently small (note that and are compact and disjoint, and hence separated by a positive distance), and the claim follows.
Exercise 14 Let be a compact metric space, denote the space of non-empty closed and connected subsets of . Show that with the Hausdorff metric is also a compact metric space.
— 6. The local group case —
In the thesis of Goldbring (and also the later paper of Goldbring and van den Dries), the above theory was extended to the setting of local groups. In fact, there is relatively little difficulty (other than some notational difficulties) in doing so, because the analysis in the previous sections can be made to take place on a small neighbourhood of the origin. This extension to local groups is not simply a generalisation for its own sake; it will turn out that it will be natural to work with local groups when we classify approximate groups in later notes.
One technical issue that comes up in the theory of local groups is that basic cancellation laws such as , which are easily verified for groups, are not always true for local groups. However, this is a minor issue as one can always recover the cancellation laws by passing to a slightly smaller local group, as follows.
Definition 16 (Cancellative local group) A local group is said to be symmetric if the inverse operation is always well-defined. It is said to be cancellative if it is symmetric, and the following axioms hold:
- (i) Whenever are such that and are well-defined and equal to each other, then . (Note that this implies in particular that .)
- (ii) Whenever are such that and are well-defined and equal to each other, then .
- (iii) Whenever are such that and are well-defined, then . (In particular, if is symmetric and is well-defined in for some , then is also symmetric.)
Clearly, all global groups are cancellative, and more generally the restriction of a global group to a symmetric neighbourhood of the identity s cancellative. While not all local groups are cancellative, we have the following substitute:
Exercise 15 Let be a local group. Show that there is a neighbourhood of the identity which is cancellative (thus, the restriction of to is cancellative).
Note that any symmetric neighbourhood of the identity in a cancellative local group is again a cancellative local group. Because of this, it turns out in practice that we may restrict to the cancellative setting without much loss of generality.
Next, we need to localise the notion of a quotient of a global group by a normal subgroup . Recall that in order for a subset og a global group to be a normal subgroup, it has to be symmetric, contain the identity, be closed under multiplication (thus whenever , and closed under conjugation (thus whenever and ). We now localise this concept as follows:
Definition 17 (Normal sublocal group) Let be a cancellative local group. A subset of is said to be a normal sublocal group if there is an open neighbourhood of (called a normalising neighbourhood of ) obeying the following axioms:
- (Identity and inverse) is symmetric and contains the identity.
- (Local closure) If and is well-defined in , then .
- (Normality) If are such that is well-defined in , then .
(Strictly speaking, one should refer to the pair as the normal sublocal group, rather than just , but by abuse of notation we shall omit the normalising neighbourhood when referring to the normal sublocal group.)
It is easy to see that if is a normal sublocal group of , then is itself a cancellative local group, using the topology and group structure formed by restriction from . (Note how the open neighbourhood is needed to ensure that the domain of the multiplication map in remains open.)
Example 3 In the global group , the open interval is a normal sub-local subgroup if one takes (say) as the normalising neighbourhood.
Example 4 Let be the shift map , and let be the semidirect product of and . Then if is any (global) subgroup of , the set is a normal sub-local subgroup of (with normalising neighbourhood ). This is despite the fact that will, in general, not be normal in in the classical (global) sense.
As observed by Goldbring, one can define the operation of quotienting a local group by a normal sub-local group, provided that one restricts to a sufficiently small neighbourhood of the origin:
Exercise 16 (Quotient spaces) Let be a cancellative local group, and let be a normal sub-local group with normalising neighbourhood . Let be a symmetric open neighbourhood of the identity such that is well-defined and contained in . Show that there exists a cancellative local group and a surjective continuous homomorphism such that, for any , one has if and only if , and for any , one has open if and only if is open.
It is not difficult to show that the quotient defined by the above exercise is unique up to local isomorphism, so we will abuse notation and talk about “the” quotient space given by the above construction.
We can now state the local version of the Gleason-Yamabe theorem, first proven by Goldbring in his thesis, and later reproven by Goldbring and van den Dries by a slightly different method:
Theorem 18 (Local Gleason-Yamabe theorem) Let be a locally compact local group. Then there exists an open symmetric neighbourhood of the identity, and a compact global group in that is normalised by , such that is well-defined and isomorphic to a local Lie group.
The proofs of this theorem by Goldbring and Goldbring-van den Dries were phrased in the language of nonstandard analysis. However, it is possible to translate those arguments to standard analysis arguments, which closely follow the arguments given in previous sections and notes. (Actually, our arguments are not a verbatim translation of those in Goldbring and Goldbring-van den Dries, as we have made a few simplifications in which the role of Gleason metrics is much more strongly emphasised.) We briefly sketch the main points here.
As in the global case, the route to obtaining (local) Lie structure is via Gleason metrics. On a local group , we define a local Gleason metric to be a metric defined on some symmetric open neighbourhood of the identity with (say) well-defined (to avoid technical issues), which generates the topology of , and which obeys the following version of the left-invariance, escape and commutator properties:
One can then verify (by localisation of the arguments in Notes 2) that any locally compact local Lie group with a local Gleason metric is locally Lie (i.e. some neighbourhood of the identity is isomorphic to a local Lie group); see Exercise 10 from Notes 2. Next, one can define the notion of a weak local Gleason metric by dropping the commutator estimate, and one can verify an analogue of Theorem 8, namely that any weak local Gleason metric is automatically a local Gleason metric, after possibly shrinking the neighbourhood and adjusting the constant as necessary. The proof of this statement is essentially the same as that in Theorem 8 (which is already localised to small neighbourhoods of the identity), but uses a local Haar measure instead of a global Haar measure, and requires some preliminary shrinking of the neighbourhood to ensure that all group-theoretic operations (and convolutions) are well-defined. We omit the (rather tedious) details.
Now we define the concept of an NSS local group as a local group which has an open neighbourhood of the identity that contains no non-trivial global subgroups. The proof of Theorem 10 is already localised to small neighbourhoods of the identity, and it is possible (after being sufficiently careful with the notation) to translate that argument to the local setting, and conclude that any NSS local group admits a weak Gleason metric on some open neighbourhood of the identity, and is hence locally Lie. (A typical example of being “sufficiently careful with the notation”: to define the escape norm (11), one adopts the convention that a statement such as is automatically false if are not all well-defined. The induction hypothesis (13) will play a key role in ensuring that all expressions involved are well-defined and localised to a suitably small neighbourhood of the identity.) Again, we omit the details.
The next step is to obtain a local version of Proposition 13. Here we encounter a slight difficulty because in a general local group , we do not have a good notion of the group generated by a set of generators in . As such, the subgroup trapping property does not automatically translate to the local group setting as defined in Definition 19. However, this difficulty can be easily avoided by rewording the definition:
Definition 19 (Subgroup trapping) A local group has the subgroup trapping property if, for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that is contained in a global subgroup that is in turn contained in . (Here, is, as before, the union of all the global subgroups contained in .)
Because is now contained in a global group , the group generated by is well-defined. As is in the open neighbourhood , one can then also form the closure ; if we choose small enough to be precompact, then this is a compact global group (and thus describable by the Gleason-Yamabe theorem for such groups, Theorem 4). Because of this, it is possible to adapt Proposition 13 without much difficulty to the local setting to conclude that given any locally compact local group with the subgroup trapping property, there exists an open symmetric neighbourhood of the identity, and a compact global group in that is normalised by , such that is well-defined and NSS (and thus locally isomorphic to a local Lie group).
Finally, to finish the proof of Theorem 18, one has to establish the analogue of Proposition 14, namely that one has to show that every locally compact metrisable local group has the subgroup trapping property. (It is not difficult to adapt Exercise 4 to the local group setting to reduce to the metrisable case.) The first step is to prove the local group analogue of Proposition 15 (again adopting the obvious convention that a statement such as is only considered true if is well-defined, and adding the additional hypothesis that is sufficiently small in order to ensure that all manipulations are justified). This can be done by a routine modification of the proof. But then one can modify the rest of the argument in Proposition 14 to hold in the local setting as well (note, as in the proof of Proposition 13, that the compact set generated in the course of this argument remains a global group rather than a local one, and so one can again use Theorem 4 without difficulty). Again, we omit the details.
]]>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.
— 1. Haar measure —
For technical reasons, it is convenient to not work with an absolutely general locally compact group, but to restrict attention to those groups that are both -compact and Hausdorff, in order to access measure-theoretic tools such as the Fubini-Tonelli theorem and the Riesz representation theorem without bumping into unwanted technical difficulties. Intuitively, -compact groups are those groups that do not have enormously “large” scales – scales are too coarse to be “seen” by any compact set. Similarly, Hausdorff groups are those groups that do not have enormously “small” scales – scales that are too small to be “seen” by any open set. A simple example of a locally compact group that fails to be -compact is the real line with the discrete topology; conversely, a simple example of a locally compact group that fails to be Hausdorff is the real line with the trivial topology.
As the two exercises below show, one can reduce to the -compact Hausdorff case without much difficulty, either by restricting to an open subgroup to eliminate the largest scales and recover -compactness, or to quotient out by a compact normal subgroup to eliminate the smallest scales and recover the Hausdorff property.
Exercise 2 Let be a locally compact group. Show that there exists an open subgroup which is locally compact and -compact. (Hint: take the group generated by a compact neighbourhood of the identity.)
Exercise 3 Let be a locally compact group. Let be the topological closure of the identity element.
- (i) Show that given any open neighbourhood of a point in , there exists a neighbourhood of whose closure lies in . (Hint: translate to the identity and select so that .) In other words, is a regular space.
- (ii) Show that for any group element , that the sets and are either equal or disjoint.
- (iii) Show that is a compact normal subgroup of .
- (iv) Show that the quotient group (equipped with the quotient topology) is a locally compact Hausdorff group.
- (v) Show that a subset of is open if and only if it is the preimage of an open set in .
Now that we have restricted attention to the -compact Hausdorff case, we can now define the notion of a Haar measure.
Definition 1 (Radon measure) Let be a -compact locally compact Hausdorff topological space. The Borel -algebra on is the -algebra generated by the open subsets of . A Borel measure is a countably additive non-negative measure on the Borel -algebra. A Radon measure is a Borel measure obeying three additional axioms:
- (i) (Local finiteness) One has for every compact set .
- (ii) (Inner regularity) One has for every Borel measurable set .
- (iii) (Outer regularity) One has for every Borel measurable set .
Definition 2 (Haar measure) Let be a -compact locally compact Hausdorff group. A Radon measure is left-invariant (resp. right-invariant) if one has (resp. ) for all and Borel measurable sets . A left-invariant Haar measure is a non-zero Radon measure which is left-invariant; a right-invariant Haar measure is defined similarly. A bi-invariant Haar measure is a Haar measure which is both left-invariant and right-invariant.
Note that we do not consider the zero measure to be a Haar measure.
Example 1 A large part of the foundations of Lebesgue measure theory (e.g. most of these lecture notes of mine) can be summed up in the single statement that Lebesgue measure is a (bi-invariant) Haar measure on Euclidean spaces .
Example 2 If is a countable discrete group, then counting measure is a bi-invariant Haar measure.
Example 3 If is a left-invariant Haar measure on a -compact locally compact Hausdorff group , then the reflection defined by is a right-invariant Haar measure on , and the scalar multiple is a left-invariant Haar measure on for any .
Exercise 4 If is a left-invariant Haar measure on a -compact locally compact Hausdorff group , show that for any non-empty open set .
Let be a left-invariant Haar measure on a -compact locally compact Hausdorff group. Let be the space of all continuous, compactly supported complex-valued functions ; then is absolutely integrable with respect to (thanks to local finiteness), and one has
for all (thanks to left-invariance). Similarly for right-invariant Haar measures (but now replacing by ).
The fundamental theorem regarding Haar measures is:
Theorem 3 (Existence and uniqueness of Haar measure) Let be a -compact locally compact Hausdorff group. Then there exists a left-invariant Haar measure on . Furthermore, this measure is unique up to scalars: if are two left-invariant Haar measures on , then for some scalar .
Similarly if “left-invariant” is replaced by “right-invariant” throughout. (However, we do not claim that every left-invariant Haar measure is automatically right-invariant, or vice versa.)
To prove this theorem, we will rely on the Riesz representation theorem:
Theorem 4 (Riesz representation theorem) Let be a -compact locally compact Hausdorff space. Then to every linear functional which is non-negative (thus whenever ), one can associate a unique Radon measure such that for all . Conversely, for each Radon measure , the functional is a non-negative linear functional on .
We now establish the uniqueness component of Theorem 3. We shall just prove the uniqueness of left-invariant Haar measure, as the right-invariant case is similar (and also follows from the left-invariant case by Example 3). Let be two left-invariant Haar measures on . We need to prove that is a scalar multiple of . From the Riesz representation theorem, it suffices to show that is a scalar multiple of . Equivalently, it suffices to show that
for all .
To show this, the idea is to approximate both and by superpositions of translates of the same function . More precisely, fix , and let . As the functions and are continuous and compactly supported, they are uniformly continuous, in the sense that we can find an open neighbourhood of the identity such that and for all and ; we may also assume that the are contained in a compact set that is uniform in . By Exercise 4 and Urysohn’s lemma, we can then find an “approximation to the identity” supported in such that . Since
for all in the support of , we conclude that
uniformly in ; also, the left-hand side has uniformly compact support in . If we integrate against , we conclude that
where the implied constant in the notation can depend on but not on . But by the left-invariance of , the left-hand side is also
which by the Fubini-Tonelli theorem is
which by the left-invariance of is
which simplifies to . We conclude that
and similarly
which implies that
Sending we obtain the claim.
Exercise 5 Obtain another proof of uniqueness of Haar measure by investigating the translation-invariance properties of the Radon-Nikodym derivative of with respect to .
Now we show existence of Haar measure. Again, we restrict attention to the left-invariant case (using Example 3 if desired). By the Riesz representation theorem, it suffices to find a functional from the space of non-negative continuous compactly supported functions to the non-negative reals obeying the following axioms:
Here, is the translation operation as discussed in the introduction.
We will construct this functional by an approximation argument. Specifically, we fix a non-zero . We will show that given any finite number of functions and any , one can find a functional that obeys the following axioms:
Once one has established the existence of these approximately additive functionals , one can then construct the genuinely additive functional (and thus a left-invariant Haar measure) by a number of standard compactness arguments. For instance:
These approaches are more or less equivalent to each other, and the choice of which approach to use is largely a matter of personal taste.
It remains to obtain the approximate functionals for a given and . As with the uniqueness claim, the basic idea is to approximate all the functions by translates of a given function . More precisely, let be a small quantity (depending on and ) to be chosen later. By uniform continuity, we may find a neighbourhood of the identity such that for all and . Let be a function, not identically zero, which is supported in .
To motivate the argument that follows, pretend temporarily that we have a left-invariant Haar measure available, and let be the integral of with respect to this measure. Then , and by left-invariance one has
and thus
for any scalars and . In particular, if we introduce the covering number
of a given function by , we have
This suggests using a scalar multiple of as the approximate linear functional (noting that can be defined without reference to any existing Haar measure); in view of the normalisation , it is then natural to introduce the functional
(This functional is analogous in some ways to the concept of outer measure or the upper Darboux integral in measure theory.) Note from compactness that is finite for every , and from the non-triviality of we see that , so is well-defined as a map from to . It is also easy to verify that obeys the homogeneity, left-invariance, and normalisation axioms. From the easy inequality
we also obtain the uniform bound axiom, and from the infimal nature of we also easily obtain the subadditivity property
To finish the construction, it thus suffices to show that
for each , if is chosen sufficiently small depending on .
Fix . By definition, we have the pointwise bound
and
if is small enough. Indeed, we have
If is non-zero, then by the construction of and , one has and , which implies that
Using (3) we thus have
which gives (5); a similar argument gives (6). From the subadditivity (and monotonicity) of , we conclude that
and
where equals on the support of . Summing and using (4), we conclude that
and the claim follows by taking small enough. This concludes the proof of Theorem 3.
Exercise 6 State and prove a generalisation of Theorem 3 in which the hypothesis that is Hausdorff and -compact are dropped. (This requires extending concepts such as “Borel -algebra”, “Radon measure”, and “Haar measure” to the non-Hausdorff or non--compact setting. Note that different texts sometimes have inequivalent definitions of these concepts in such settings; because of this (and also because of the potential breakdown of some basic measure-theoretic tools such as the Fubini-Tonelli theorem), it is usually best to avoid working with Haar measure in the non-Hausdorff or non--compact case unless one is very careful.)
Remark 1 An important special case of the Haar measure construction arises for compact groups . Here, we can normalise the Haar measure by requiring that (i.e. is a probability measure), and so there is now a unique (left-invariant) Haar probability measure on such a group. In Exercise 7 we will see that this measure is in fact bi-invariant.
Remark 2 The above construction, based on the Riesz representation theorem, is not the only way to construct Haar measure. Another approach that is common in the literature is to first build a left-invariant outer measure and then use the Carathéodory extension theorem. Roughly speaking, the main difference between that approach and the one given here is that it is based on covering compact or open sets by other compact or open sets, rather than covering continuous, compactly supported functions by other continuous, compactly supported functions. In the compact case, one can also construct Haar probability measure by defining to be the mean of , or more precisely the unique constant function that is an average of translates of . See Exercise 6 of these notes for further discussion (the post there focuses on the abelian case, but the argument extends to the nonabelian setting).
The following exercise explores the distinction between left-invariance and right-invariance.
Exercise 7 Let be a -compact locally compact Hausdorff group, and let be a left-invariant Haar measure on .
- (i) Show that for each , there exists a unique positive real (independent of the choice of ) such that for all Borel measurable sets and for all absolutely integrable . In particular, a left-invariant Haar measure is right-invariant if and only if for all .
- (ii) Show that the map is a continuous homomorphism from to the multiplicative group . (This homomorphism is known as the modular function, and is said to be unimodular if is identically equal to .)
- Show that for any , one has . (Hint: take another function and evaluate in two different ways, one of which involves replacing by .) In particular, in a unimodular group one has and for any Borel set and any .
- (iii) Show that is unimodular if it is compact.
- (iv) If is a Lie group with Lie algebra , show that , where is the adjoint representation of , defined by requiring for all (cf. Lemma 13 of Notes 1).
- (v) If is a connected Lie group with Lie algebra , show that is unimodular if and only if for all , where is the adjoint representation of .
- (vi) Show that is unimodular if it is a connected nilpotent Lie group.
- (vii) Let be a connected Lie group whose Lie algebra is such that (where is the linear span of the commutators with ). (This condition is in particular obeyed when the Lie algebra is semisimple.) Show that is unimodular.
- (viii) Let be the group of pairs with the composition law . (One can interpret as the group of orientation-preserving affine transformations on the real line.) Show that is a connected Lie group that is not unimodular.
In the case of a Lie group, one can also build Haar measures by starting with a non-invariant smooth measure, and then correcting it. Given a smooth manifold , define a smooth measure on to be a Radon measure which is a smooth multiple of Lebesgue measure when viewed in coordinates, thus for any smooth coordinate chart , the pushforward measure takes the form for some smooth function , thus
for all . We say that the smooth measure is nonvanishing if is non-zero on for every coordinate chart .
Exercise 8 Let be a Lie group, and let be a nonvanishing smooth measure on .
- Show that for every , there exists a unique smooth function such that
- Verify the cocycle equation for all .
- Show that the measure defined by
is a left-invariant Haar measure on .
There are a number of ways to generalise the Haar measure construction. For instance, one can define a local Haar measure on a local group . If is a neighbourhood of the identity in a -compact locally compact Hausdorff local group , we define a local left-invariant Haar measure on to be a non-zero Radon measure on with the property that whenever and is a Borel set such that is well-defined and also in .
Exercise 9 (Local Haar measure) Let be a -compact locally compact Hausdorff local group, and let be an open neighbourhood of the identity in such that is symmetric (i.e. is well-defined and equal to ) and is well-defined in . By adapting the arguments above, show that there is a local left-invariant Haar measure on , and that it is unique up to scalar multiplication. (Hint: a new technical difficulty is that there are now multiple covering numbers of interest, namely the covering numbers associated to various small powers of . However, as long as one keeps track of which covering number to use at various junctures, this will not cause difficulty.)
One can also sometimes generalise the Haar measure construction from groups to spaces that acts transitively on.
Definition 5 (Group actions) Given a topological group and a topological space , define a (left) continuous action of on to be a continuous map from to such that and for all and .
This action is said to be transitive if for any , there exists such that , and in this case is called a homogeneous space with structure group , or homogenous -space for short.
For any , we call the stabiliser of ; this is a closed subgroup of .
If are smooth manifolds (so that is a Lie group) and the action is a smooth map, then we say that we have a smooth action of on .
Exercise 10 If acts transitively on a space , show that all the stabilisers are conjugate to each other, and is homeomorphic to the quotient spaces after weakening the topology of the quotient space (or strengthening the topology of the space .
If and are -compact, locally compact, and Hausdorff, a (left) Haar measure is a non-zero Radon measure on such that for all Borel and .
Exercise 11 Let be a -compact, locally compact, and Hausdorff group (left) acting continuously and transitively on a -compact, locally compact, and Hausdorff space .
- (i) (Uniqueness up to scalars) Show that if are (left) Haar measures on , then for some .
- (ii) (Compact case) Show that if is compact, then is compact too, and a Haar measure on exists.
- (iii) (Smooth unipotent case) Suppose that the action is smooth (so that is a Lie group and is a smooth manifold). Let be a point of . Suppose that for each , the derivative map of the map at is unimodular (i.e. it has determinant ). Show that a Haar measure on exists.
- (iv) (Smooth case) Suppose that the action is smooth. Show that any Haar measure on is necessarily smooth. Conclude that a Haar measure exists if and only if the derivative maps are unimodular.
- (v) (Counterexample) Let be the group from Example 7(viii), acting on by the action . Show that there is no Haar measure on . (This can be done either through (iv), or by an elementary direct argument.)
— 2. The Peter-Weyl theorem —
We now restrict attention to compact groups , which we will take to be Hausdorff for simplicity (although the results in this section will easily extend to the non-Hausdorff case using Exercise 3). By the previous discussion, there is a unique bi-invariant Haar probability measure on , which gives rise in particular to the Hilbert space of square-integrable functions on (quotiented out by almost everywhere equivalence, as usual), with norm
and inner product
For every group element , the translation operator is defined by
One easily verifies that is both the inverse and the adjoint of , and so is a unitary operator. The map is then a continuous homomorphism from to the unitary group of (where we give the latter group the strong operator topology), and is known as the regular representation of .
For our purposes, the regular representation is too “big” of a representation to work with because the underlying Hilbert space is usually infinite-dimensional. However, we can find smaller representations by locating left-invariant closed subspaces of , i.e. closed linear subspaces of with the property that for all . Then the restriction of to becomes a representation to the unitary group of . In particular, if has some finite dimension , this gives a representation of by a unitary group after expressing in coordinates.
We can build invariant subspaces from applying spectral theory to an invariant operator, and more specifically to a convolution operator. If , we define the convolution by the formula
Exercise 12 Show that if , then is well-defined and lies in , and in particular also lies in .
For , let denote the right-convolution operator . This is easily seen to be a bounded linear operator on . Using the properties of Haar measure, we also observe that will be self-adjoint if obeys the condition
and it also commutes with left-translations:
In particular, for any , the eigenspace
will be a closed invariant subspace of . Thus we see that we can generate a large number of representations of by using the eigenspace of a convolution operator.
Another important fact about these operators, is that the are compact, i.e. they map bounded sets to precompact sets. This is a consequence of the following more general fact:
Exercise 13 (Compactness of integral operators) Let and be -finite measure spaces, and let . Define an integral operator by the formula
- Show that is a bounded linear operator, with operator norm bounded by . (Hint: use duality.)
- Show that is a compact linear operator. (Hint: approximate by a linear combination of functions of the form for and , plus an error which is small in norm, so that becomes approximated by the sum of a finite rank operator and an operator of small operator norm.)
Note that is an integral operator with kernel ; from the invariance properties of Haar measure we see that if (note here that we crucially use the fact that is compact, so that ). Thus we conclude that the convolution operator is compact when is compact.
Exercise 14 Show that if is non-zero, then is not compact on . This example demonstrates that compactness of is needed in order to ensure compactness of .
We can describe self-adjoint compact operators in terms of their eigenspaces:
Theorem 6 (Spectral theorem) Let be a compact self-adjoint operator on a complex Hilbert space . Then there exists an at most countable sequence of non-zero reals that converge to zero and an orthogonal decomposition
of into the eigenspace (or kernel) of , and the -eigenspaces , which are all finite-dimensional.
Proof: From self-adjointness we see that all the eigenspaces are orthogonal to each other, and only non-trivial for real. If , then has an orthonormal basis of eigenfunctions , each of which is enlarged by a factor of at least by . In particular, this basis cannot be infinite, because otherwise the image of this basis by would have no convergent subsequence, contradicting compactness. Thus is finite-dimensional for any , which implies that is finite-dimensional for every non-zero , and those non-zero with non-trivial can be enumerated to either be finite, or countable and go to zero.
Let be the orthogonal complement of . If is trivial, then we are done, so suppose for sake of contradiction that is non-trivial. As all of the are invariant, and is self-adjoint, is also invariant, with being self-adjoint on . As is orthogonal to the kernel of , has trivial kernel in . More generally, has no eigenvectors in .
Let be the unit ball in . As has trivial kernel and is non-trivial, . Using the identity
valid for all self-adjoint operators (see Exercise 15 below). Thus, we may find a sequence of vectors of norm at most such that
for some . Since , we conclude that
By compactness of , we may pass to a subsequence so that converges to a limit , and thus . As has no eigenvectors, must be trivial; but then converges to zero, a contradiction.
Exercise 15 Establish (9) whenever is a bounded self-adjoint operator on . (Hint: Bound by the right-hand side of (8) whenever are vectors of norm at most , by playing with for various choices of scalars , in the spirit of the proof of the Cauchy-Schwarz inequality.)
This leads to the consequence that we can find non-trivial finite-dimensional representations on at least a single non-identity element:
Theorem 7 (Baby Peter-Weyl theorem) Let be a compact Hausdorff group with Haar measure , and let be a non-identity element of . Then there exists a finite-dimensional invariant subspace of on which is not the identity.
Proof: Suppose for contradiction that is the identity on every finite-dimensional invariant subspace of , thus annihilates every such subspace. By Theorem 6, we conclude that has range in the kernel of every convolution operator with , thus for any with obeying (7), i.e.
for any such . But one may easily construct such that is non-zero at the identity and vanishing at (e.g. one can set where is an open symmetric neighbourhood of the identity, small enough that lies outside ). This gives the desired contradiction.
Remark 3 The full Peter-Weyl theorem describes rather precisely all the invariant subspaces of . Roughly speaking, the theorem asserts that for each irreducible finite-dimensional representation of , different copies of (viewed as an invariant -space) appear in , and that they are all orthogonal and make up all of ; thus, one has an orthogonal decomposition
of -spaces. Actually, this is not the sharpest form of the theorem, as it only describes the left -action and not the right -action; see this previous blog post for a precise statement and proof of the Peter-Weyl theorem in its strongest form. This form is of importance in Fourier analysis and representation theory, but in this course we will only need the baby form of the theorem (Theorem 7), which is an easy consequence of the full Peter-Weyl theorem (since, if is not the identity, then is clearly non-trivial on and hence on at least one of the factors).
The Peter-Weyl theorem leads to the following structural theorem for compact groups:
Theorem 8 (Gleason-Yamabe theorem for compact groups) Let be a compact Hausdorff group, and let be a neighbourhood of the identity. Then there exists a compact normal subgroup of contained in such that is isomorphic to a linear group (i.e. a closed subgroup of a general linear group ).
Note from Cartan’s theorem (Theorem 2 from Notes 2) that every linear group is Lie; thus, compact Hausdorff groups are “almost Lie” in some sense.
Proof: Let be an element of . By the baby Peter-Weyl theorem, we can find a finite-dimensional invariant subspace of on which is non-trivial. Identifying such a subspace with for some finite , we thus have a continuous homomorphism such that is non-trivial. By continuity, will also be non-trivial for some open neighbourhood of . Using the compactness of , one can then find a finite number of such continuous homomorphisms such that for each , at least one of is non-trivial. If we then form the direct sum
then is still a continuous homomorphism, which is now non-trivial for any ; thus the kernel of is a compact normal subgroup of contained in . There is thus a continuous bijection from the compact space to the Hausdorff space , and so the two spaces are homeomorphic. As is a compact (hence closed) subgroup of , the claim follows.
Exercise 16 Show that the hypothesis that is Hausdorff can be omitted from Theorem 8. (Hint: use Exercise 3.)
Exercise 17 Show that any compact Lie group is isomorphic to a linear group. (Hint: first find a neighbourhood of the identity that is so small that it does not contain any non-trivial subgroups.) The property of having no small subgroups will be an important one in later notes.
One can rephrase the Gleason-Yamabe theorem for compact groups in terms of the machinery of inverse limits (also known as projective limits).
Definition 9 (Inverse limits of groups) Let be a family of groups indexed by a partially ordered set . Suppose that for each in , there is a surjective homomorphism which obeys the composition law for all . (If one wishes, one can take a category-theoretic perspective and view these surjections as describing a functor from the partially ordered set to the category of groups.) We then define the inverse limit to be the set of all tuples in the product set such that for all ; one easily verifies that this is also a group. We let denote the coordinate projection maps .
If the are topological groups and the are continuous, we can give the topology induced from ; one easily verifies that this makes a topological group, and that the are continuous homomorphisms.
Exercise 18 (Universal description of inverse limit) Let be a family of groups with the surjective homomorphisms as in Definition 9. Let be the inverse limit, and let be another group. Suppose that one has homomorphisms for each such that for all . Show that there exists a unique homomorphism such that for all .
Establish the same claim with “group” and “homomorphism” replaced by “topological group” and “continuous homomorphism” throughout.
Exercise 19 Let be a prime. Show that is isomorphic to the inverse limit of the cyclic groups with (with the usual ordering), using the obvious projection homomorphisms from to for .
Exercise 20 Show that every compact Hausdorff group is isomorphic (as a topological group) to an inverse limit of linear groups. (Hint: take the index set to be the set of all non-empty finite collections of open neighbourhoods of the identity, indexed by inclusion.) If the compact Hausdorff group is metrisable, show that one can take the inverse limit to be indexed instead by the natural numbers with the usual ordering.
Exercise 21 Let be an abelian group with a homomorphism into the unitary group of a finite-dimensional space . Show that can be decomposed as the vector space sum of one-dimensional -invariant spaces. (Hint: By the spectral theorem for unitary matrices, any unitary operator on decomposes into eigenspaces, and any operator commuting with must preserve each of these eigenspaces. Now induct on the dimension of .)
Exercise 22 (Fourier analysis on compact abelian groups) Let be a compact abelian Hausdorff group with Haar probability measure . Define a character to be a continuous homomorphism to the unit circle , and let be the collection of all such characters.
- (i) Show that for every not equal to the identity, there exists a character such that . (Hint: combine the baby Peter-Weyl theorem with the preceding exercise.)
- (ii) Show that every function in is the limit in the uniform topology of finite linear combinations of characters. (Hint: use the Stone-Weierstrass theorem.)
- (iii) Show that the characters for form an orthonormal basis of .
— 3. The structure of locally compact abelian groups —
We now use the above machinery to analyse locally compact abelian groups. We follow some combinatorial arguments of Pontryagin, as presented in the text of Montgomery and Zippin.
We first make a general observation that locally compact groups contain open subgroups that are “finitely generated modulo a compact set”. Call a subgroup of a topological group cocompact if the quotient space is compact.
Lemma 10 Let be a locally compact group. Then there exists an open subgroup of which has a cocompact finitely generated subgroup .
Proof: Let be a compact neighbourhood of the identity. Then is also compact and can thus be covered by finitely many copies of , thus
for some finite set , which we may assume without loss of generality to be contained in . In particular, if is the group generated by , then
Multiplying this on the left by powers of and inducting, we conclude that
for all . If we then let be the group generated by , then lies in and . Thus is the image of the compact set under the quotient map, and the claim follows.
In the abelian case, we can improve this lemma by combining it with the following proposition:
Proposition 11 Let be a locally compact Hausdorff abelian group with a cocompact finitely generated subgroup. Then has a cocompact discrete finitely generated subgroup.
To prove this proposition, we need the following lemma.
Lemma 12 Let be a locally compact Hausdorff group, and let . Then the group generated by is either precompact or discrete (or both).
Proof: By replacing with the closed subgroup we may assume without loss of generality that is dense in .
We may assume of course that is not discrete. This implies that the identity element is not an isolated point in , and thus for any neighbourhood of the identity , there exist arbitrarily large such that ; since we may take these to be large and positive rather than large and negative.
Let be a precompact symmetric neighbourhood of the identity, then (say) is covered by a finite number of left-translates of . As is dense, we conclude that is covered by a finite number of translates of left-translates of by powers of . Using the fact that there are arbitrarily large with , we may thus cover by a finite number of translates of with . In particular, if , then there exists an such that . Iterating this, we see that the set is left-syndetic, in that it has bounded gaps as one goes to . Similarly one can argue that this set is right-syndentic and thus syndetic. This implies that the entire group is covered by a bounded number of translates of and is thus precompact as required.
Now we can prove Proposition 11.
Proof: Let us say that a locally compact Hausdorff abelian group has rank at most if it has a cocompact subgroup generated by at most generators. We will induct on the rank . If has rank , then the cocompact subgroup is trivial, and the claim is obvious; so suppose that has some rank , and the claim has already been proven for all smaller ranks.
By hypothesis, has a cocompact subgroup generated by generators . By Lemma 12, the group is either precompact or discrete. If it is discrete, then we can quotient out by that group to obtain a locally compact Hausdorff abelian group of rank at most ; by induction hypothesis, has a cocompact discrete subgroup, and so does also. Hence we may assume that is precompact, and more generally that is precompact for each . But as we are in an abelian group, is the product of all the , and is thus also precompact, so is compact. But is a quotient of and is also compact, and so itself is compact, and the claim follows in this case.
We can then combine this with the Gleason-Yamabe theorem for compact groups to obtain
Theorem 13 (Gleason-Yamabe theorem for abelian groups) Let be a locally compact abelian Hausdorff group, and let be a neighbourhood of the identity. Then there exists a compact normal subgroup of contained in such that is isomorphic to a Lie group.
Proof: By Lemma 10 and Proposition 11, we can find an open subgroup of and discrete cocompact subgroup of . By shrinking as necessary, we may assume that is symmetric and only intersects at the identity. Let be the projection to the compact abelian group , then is a neighbourhood of the identity in . By Theorem 8, one can find a compact normal subgroup of in such that is isomorphic to a linear group, and thus to a Lie group. If we set , it is not difficult to verify that is also a compact normal subgroup of . If is the quotient map, then is a discrete subgroup of and from abstract nonsense one sees that is isomorphic to the Lie group . Thus is locally Lie. Since is an open subgroup of the abelian group , is locally Lie also, and is thus is isomorphic to a Lie group by Exercise 15 of Notes 1.
Exercise 23 Show that the Hausdorff hypothesis can be dropped from the above theorem.
Exercise 24 (Characters separate points) Let be a locally compact Hausdorff abelian group, and let be not equal to the identity. Show that there exists a character (see Exercise 22) such that . This result can be used as the foundation of the theory of Pontryagin duality in abstract harmonic analysis, but we will not pursue this here; see for instance this text of Rudin.
Exercise 25 Show that every locally compact abelian Hausdorff group is isomorphic to the inverse limit of abelian Lie groups.
Thus, in principle at least, the study of locally compact abelian group is reduced to that of abelian Lie groups, which are more or less easy to classify:
Exercise 26
- Show that every discrete subgroup of is isomorphic to for some .
- Show that every connected abelian Lie group is isomorphic to for some natural numbers . (Hint: first show that the kernel of the exponential map is a discrete subgroup of the Lie algebra.) Conclude in particular the divisibility property that if and then there exists with .
- Show that every compact abelian Lie group is isomorphic to for some natural number and a which is a finite product of finite cyclic groups. (You may need the classification of finitely generated abelian groups, and will also need the divisibility property to lift a certain finite group from a certain quotient space back to .)
- Show that every abelian Lie group contains an open subgroup that is isomorphic to for some natural numbers and a finite product of finite cyclic groups.
]]>Remark 4 Despite the quite explicit description of (most) abelian Lie groups, some interesting behaviour can still occur in locally compact abelian groups after taking inverse limits; consider for instance the solenoid example (Exercise 6 from Notes 0).