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A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat vague and imprecise, and their interpretation depends on the context and also on one’s choice of normalisations, but if one uses a “macroscopic” normalisation, “macroscopic” scales correspond to scales that are comparable to unit size (i.e. bounded above and below by absolute constants), while “microscopic” scales are much smaller, being the minimal scale at which nontrivial behaviour occurs. (Other normalisations are possible, e.g. making the microscopic scale a unit scale, and letting the macroscopic scale go off to infinity; for instance, such a normalisation is often used, at least initially, in the study of groups of polynomial growth. However, for the theory of approximate groups, a macroscopic scale normalisation is more convenient.)
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 :
- Sets such as for some fixed (e.g. ) can be considered to be sets at a global macroscopic scale. Sending to infinity, one enters the large-scale regime.
- Sets such as the sets that appear in the Sanders lemma from the previous set of notes (thus for some fixed , e.g. ) can be considered to be sets at a local macroscopic scale. Sending to infinity, one enters the mesoscopic regime.
- The non-identity element of that is “closest” to the identity in some suitable metric (cf. the proof of Jordan’s theorem from Notes 0) would be an element associated to the microscopic scale. The orbit starts out at microscopic scales, and (assuming some suitable “escape” axioms) will pass through mesoscopic scales and finally entering the macroscopic regime. (Beyond this point, the orbit may exhibit a variety of behaviours, such as periodically returning back to the smaller scales, diverging off to ever larger scales, or filling out a dense subset of some macroscopic set; the escape axioms we will use do not exclude any of these possibilities.)
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.
- Start with an (ultra) strong approximate group .
- From the Gleason lemma, the elements with zero escape norm form a normal subgroup of . Quotient these elements out. Show that all non-identity elements will have positive escape norm.
- Find the non-identity element in (the quotient of) of minimal escape norm. Use the commutator estimate (assuming it is inherited by the quotient) to show that will centralise (most of) this quotient. In particular, the orbit is (essentially) a central subgroup of .
- Quotient this orbit out; then find the next non-identity element in this new quotient of . Again, show that is essentially a central subgroup of this quotient.
- Repeat this process until becomes entirely trivial. Undoing all the quotients, this should demonstrate that is virtually nilpotent, and that is essentially a coset nilprogression.
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.
In the previous notes, we established the Gleason-Yamabe theorem:
Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.
Roughly speaking, this theorem asserts the “mesoscopic” structure of a locally compact group (after restricting to an open subgroup to remove the macroscopic structure, and quotienting out by to remove the microscopic structure) is always of Lie type.
In this post, we combine the Gleason-Yamabe theorem with some additional tools from point-set topology to improve the description of locally compact groups in various situations.
We first record some easy special cases of this. If the locally compact group has the no small subgroups property, then one can take to be trivial; thus is Lie, which implies that is locally Lie and thus Lie as well. Thus the assertion that all locally compact NSS groups are Lie (Theorem 10 from Notes 4) is a special case of the Gleason-Yamabe theorem.
In a similar spirit, if the locally compact group is connected, then the only open subgroup of is the full group ; in particular, by arguing as in the treatment of the compact case (Exercise 19 of Notes 3), we conclude that any connected locally compact Hausdorff group is the inverse limit of Lie groups.
Now we return to the general case, in which need not be connected or NSS. One slight defect of Theorem 1 is that the group can depend on the open neighbourhood . However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make independent of :
Theorem 2 (Gleason-Yamabe theorem, stronger version) Let be a locally compact group. Then there exists an open subgoup of such that, for any open neighbourhood of the identity in , there exists a compact normal subgroup of in such that is isomorphic to a Lie group.
We prove this theorem below the fold. As in previous notes, if is Hausdorff, the group is thus an inverse limit of Lie groups (and if (and hence ) is first countable, it is the inverse limit of a sequence of Lie groups).
It remains to analyse inverse limits of Lie groups. To do this, it helps to have some control on the dimensions of the Lie groups involved. A basic tool for this purpose is the invariance of domain theorem:
Theorem 3 (Brouwer invariance of domain theorem) Let be an open subset of , and let be a continuous injective map. Then is also open.
We prove this theorem below the fold. It has an important corollary:
Corollary 4 (Topological invariance of dimension) If , and is a non-empty open subset of , then there is no continuous injective mapping from to . In particular, and are not homeomorphic.
Exercise 1 (Uniqueness of dimension) Let be a non-empty topological space. If is a manifold of dimension , and also a manifold of dimension , show that . Thus, we may define the dimension of a non-empty manifold in a well-defined manner.
If are non-empty manifolds, and there is a continuous injection from to , show that .
Remark 1 Note that the analogue of the above exercise for surjections is false: the existence of a continuous surjection from one non-empty manifold to another does not imply that , thanks to the existence of space-filling curves. Thus we see that invariance of domain, while intuitively plausible, is not an entirely trivial observation.
As we shall see, we can use Corollary 4 to bound the dimension of the Lie groups in an inverse limit by the “dimension” of the inverse limit . Among other things, this can be used to obtain a positive resolution to Hilbert’s fifth problem:
Theorem 5 (Hilbert’s fifth problem) Every locally Euclidean group is isomorphic to a Lie group.
Again, this will be shown below the fold.
Another application of this machinery is the following variant of Hilbert’s fifth problem, which was used in Gromov’s original proof of Gromov’s theorem on groups of polynomial growth, although we will not actually need it this course:
Proposition 6 Let be a locally compact -compact group that acts transitively, faithfully, and continuously on a connected manifold . Then is isomorphic to a Lie group.
Recall that a continuous action of a topological group on a topological space is a continuous map which obeys the associativity law for and , and the identity law for all . The action is transitive if, for every , there is a with , and faithful if, whenever are distinct, one has for at least one .
The -compact hypothesis is a technical one, and can likely be dropped, but we retain it for this discussion (as in most applications we can reduce to this case).
Remark 2 It is conjectured that the transitivity hypothesis in Proposition 6 can be dropped; this is known as the Hilbert-Smith conjecture. It remains open; the key difficulty is to figure out a way to eliminate the possibility that is a -adic group . See this previous blog post for further discussion.
In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:
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.
In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with the minimal assumption of local compactness on a topological group , and seeing what kind of structures one can build using this assumption. (For simplicity we shall mostly confine our discussion to global groups rather than local groups for now.) In view of the preceding notes, we would like to see two types of structures emerge in particular:
- representations of into some more structured group, such as a matrix group ; and
- metrics on that capture the escape and commutator structure of (i.e. Gleason metrics).
To build either of these structures, a fundamentally useful tool is that of (left-) Haar measure – a left-invariant Radon measure on . (One can of course also consider right-Haar measures; in many cases (such as for compact or abelian groups), the two concepts are the same, but this is not always the case.) This concept generalises the concept of Lebesgue measure on Euclidean spaces , which is of course fundamental in analysis on those spaces.
Haar measures will help us build useful representations and useful metrics on locally compact groups . For instance, a Haar measure gives rise to the regular representation that maps each element of to the unitary translation operator on the Hilbert space of square-integrable measurable functions on with respect to this Haar measure by the formula
(The presence of the inverse is convenient in order to obtain the homomorphism property without a reversal in the group multiplication.) In general, this is an infinite-dimensional representation; but in many cases (and in particular, in the case when is compact) we can decompose this representation into a useful collection of finite-dimensional representations, leading to the Peter-Weyl theorem, which is a fundamental tool for understanding the structure of compact groups. This theorem is particularly simple in the compact abelian case, where it turns out that the representations can be decomposed into one-dimensional representations , better known as characters, leading to the theory of Fourier analysis on general compact abelian groups. With this and some additional (largely combinatorial) arguments, we will also be able to obtain satisfactory structural control on locally compact abelian groups as well.
The link between Haar measure and useful metrics on is a little more complicated. Firstly, once one has the regular representation , and given a suitable “test” function , one can then embed into (or into other function spaces on , such as or ) by mapping a group element to the translate of in that function space. (This map might not actually be an embedding if enjoys a non-trivial translation symmetry , but let us ignore this possibility for now.) One can then pull the metric structure on the function space back to a metric on , for instance defining an -based metric
if is square-integrable, or perhaps a -based metric
if is continuous and compactly supported (with denoting the supremum norm). These metrics tend to have several nice properties (for instance, they are automatically left-invariant), particularly if the test function is chosen to be sufficiently “smooth”. For instance, if we introduce the differentiation (or more precisely, finite difference) operators
(so that ) and use the metric (1), then a short computation (relying on the translation-invariance of the norm) shows that
for all . This suggests that commutator estimates, such as those appearing in the definition of a Gleason metric in Notes 2, might be available if one can control “second derivatives” of ; informally, we would like our test functions to have a “” type regularity.
If was already a Lie group (or something similar, such as a local group) then it would not be too difficult to concoct such a function by using local coordinates. But of course the whole point of Hilbert’s fifth problem is to do without such regularity hypotheses, and so we need to build test functions by other means. And here is where the Haar measure comes in: it provides the fundamental tool of convolution
between two suitable functions , which can be used to build smoother functions out of rougher ones. For instance:
Exercise 1 Let be continuous, compactly supported functions which are Lipschitz continuous. Show that the convolution using Lebesgue measure on obeys the -type commutator estimate
for all and some finite quantity depending only on .
This exercise suggests a strategy to build Gleason metrics by convolving together some “Lipschitz” test functions and then using the resulting convolution as a test function to define a metric. This strategy may seem somewhat circular because one needs a notion of metric in order to define Lipschitz continuity in the first place, but it turns out that the properties required on that metric are weaker than those that the Gleason metric will satisfy, and so one will be able to break the circularity by using a “bootstrap” or “induction” argument.
We will discuss this strategy – which is due to Gleason, and is fundamental to all currently known solutions to Hilbert’s fifth problem – in later posts. In this post, we will construct Haar measure on general locally compact groups, and then establish the Peter-Weyl theorem, which in turn can be used to obtain a reasonably satisfactory structural classification of both compact groups and locally compact abelian groups.
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