Theorem 1 is deep and difficult result, but the discussion in the previous posts has reduced the proof of this Theorem to that of establishing two simpler results, involving the concepts of a no small subgroups (NSS) subgroup, and that of a Gleason metric. We briefly recall the relevant definitions:
Definition 2 (NSS) A topological group is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood of the identity in that contains no subgroups of other than the trivial subgroup .
Definition 3 (Gleason metric) 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 .
The remaining steps in the resolution of Hilbert’s fifth problem are then as follows:
Theorem 4 (Reduction to the NSS case) Let be a locally compact group, 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 NSS and locally compact.
The purpose of this post is to establish these two results, using arguments that are originally due to Gleason. We will split this task into several subtasks, each of which improves the structure on the group by some amount:
Proposition 6 (From locally compact to metrisable) Let be a locally compact group, 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 metrisable.
For any open neighbourhood of the identity in , let be the union of all the subgroups of that are contained in . (Thus, for instance, is NSS if and only if is trivial for all sufficiently small .)
Proposition 7 (From metrisable to subgroup trapping) Let be a locally compact metrisable group. Then has the subgroup trapping property: for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that generates a subgroup contained in .
Proposition 8 (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.
Proposition 9 (From NSS to the escape property) Let be a locally compact NSS group. Then there exists a left-invariant metric on generating the topology on which obeys the escape property (1) for some constant .
Propositions 6–10 are all proven separately, but their proofs share some common strategies and ideas. The first main idea is to construct metrics on a locally compact group by starting with a suitable “bump function” (i.e. a continuous, compactly supported function from to ) and pulling back the metric structure on by using the translation action , thus creating a (semi-)metric
One easily verifies that this is indeed a (semi-)metric (in that it is non-negative, symmetric, and obeys the triangle inequality); it is also left-invariant, and so we have , where
where is the difference operator ,
This construction was already seen in the proof of the Birkhoff-Kakutani theorem, which is the main tool used to establish Proposition 6. For the other propositions, the idea is to choose a bump function that is “smooth” enough that it creates a metric with good properties such as the commutator estimate (2). Roughly speaking, to get a bound of the form (2), one needs to have “ regularity” with respect to the “right” smooth structure on By regularity, we mean here something like a bound of the form
for all . Here we use the usual asymptotic notation, writing or if for some constant (which can vary from line to line).
The following lemma illustrates how regularity can be used to build Gleason metrics.
Proof: We begin with the commutator property (2). Observe the identity
But from (4) (and the triangle inequality) we have
and thus we have the “Taylor expansion”
which gives (1).
It remains to obtain that have the desired regularity property. In order to get such regular bump functions, we will use the trick of convolving together two lower regularity bump functions (such as two functions with “ regularity” in some sense to be determined later). In order to perform this convolution, we will use the fundamental tool of (left-invariant) Haar measure on the locally compact group . Here we exploit the basic fact that the convolution
of two functions tends to be smoother than either of the two factors . This is easiest to see in the abelian case, since in this case we can distribute derivatives according to the law
which suggests that the order of “differentiability” of should be the sum of the orders of and separately.
These ideas are already sufficient to establish Proposition 10 directly, and also Proposition 9 when comined with an additional bootstrap argument. The proofs of Proposition 7 and Proposition 8 use similar techniques, but is more difficult due to the potential presence of small subgroups, which require an application of the Peter-Weyl theorem to properly control. Both of these theorems will be proven below the fold, thus (when combined with the preceding posts) completing the proof of Theorem 1.
The presentation here is based on some unpublished notes of van den Dries and Goldbring on Hilbert’s fifth problem. I am indebted to Emmanuel Breuillard, Ben Green, and Tom Sanders for many discussions related to these arguments.
— 1. From escape to the commutator estimate —
The general strategy here is to keep using the Gleason strategy of using the regularity one already has on the group to build good bump functions to create metrics that give even more regularity on . As with many such “bootstrap” arguments, the deepest and most difficult steps are the earliest ones, in which one has very little regularity to begin with; conversely, the easiest and most straightforward steps tend to be the final ones, when one already has most of the regularity that one needs, thus having plenty of structure and tools available to climb the next rung of the regularity ladder. (For instance, to get from regularity of a topological group to or real analytic regularity is relatively routine, with two different such approaches indicated in the preceding blog posts.) In particular, the easiest task to accomplish will be that of Proposition 10, which establishes the commutator estimate (2) once the rest of the structural control on the group is in place.
We now prove this proposition. As indicated in the introduction, the key idea here is to involve a bump function formed by convolving together two Lipschitz functions. The escape property (1) 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
for all (where we allow implied constants to depend on , , and ).
Let be a non-trivial left-invariant Haar measure on (see for instance this previous blog post for a construction of Haar measure on locally compact groups). We then form the convolution , with convolution defined using (6); this is a continuous function supported in , and gives a metric and a norm .
We now prove a variant of (4), namely that
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 (7), 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 (1). Let be a natural number such that , then and so . Conjugating by , this implies that , and so by (1), 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:
- If with sufficiently small, then .
- If with sufficiently small, then .
Then by the triangle inequality
This implies that and have overlapping support, and hence lies in . By the escape property (1), this implies (if is small enough) that , and the claim follows.
Combining Claim 2 with (8) we see that
whenever are small enough; arguing as in the proof of Lemma 11 we conclude that
whenever are small enough. Proposition 10 then follows from Claim 1 and Claim 2.
— 2. From NSS to the escape property —
Now we turn to establishing Proposition 9. An important concept will be that of an escape norm associated to an open neighbourhood of a group , defined by the formula
for any . 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 (1).
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 . Equivalently: for every , one has for some . If is small enough, then by the NSS property, we know that for each , we have for some . As is locally compact, we can make and hence compact, and so we can make uniformly bounded in by a compactness argument, and the claim follows.
Exercise 1 Let be a locally compact group. Show that if is a left-invariant metric on obeying the escape property (1) that generates the topology, then is NSS, and is comparable to for all sufficiently small . (In particular, any two left-invariant metrics obeying the escape property and generating the topology are comparable to each other.)
Henceforth is a locally compact NSS group.
(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 .
for all and some huge constant , and 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
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
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
whenever and . To use this, we apply (5) and conclude that
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
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 12.
Now we can finish the proof of Proposition 9. Let be a locally compact NSS group, and let be a sufficiently small neighbourhood of the identity. From Proposition 12, we see that the escape norm and the modified escape norm are comparable. We have seen is a left-invariant semi-metric. 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 Proposition 9, we need to verify the escape property (1). 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).
— 3. From subgroup trapping to NSS —
We now turn to the task of proving Proposition 8. 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 the Peter-Weyl theorem. Actually, we will just need the following weak version of that theorem:
Theorem 13 (Weak Peter-Weyl theorem) Let be a compact group, and let be a neighbourhood of the identity in . Then there exists a finite-dimensional real linear representation of (i.e. a continuous homomorphism from to the general linear group of a finite-dimensional real vector space ) whose kernel lies in . Equivalently, there exists a compact normal subgroup of contained in such that is isomorphic to a compact subgroup of .
Proof: As is compact, it has a Haar probability measure . Let be a symmetric open neighbourhood of the identity such that . The convolution operator given by is a self-adjoint integral operator on a probability space with bounded measurable kernel and is thus compact (indeed, it is a Hilbert-Schmidt integral operator). By the spectral theorem, then decomposes as the orthogonal sum of the eigenspaces of , with all the eigenspaces corresponding to non-zero eigenvalues being finite-dimensional.
Note that commutes with the left translation operators for every , so all of the eigenspaces are invariant with respect to this action, and so we have finite-dimensional linear represenations for each non-zero eigenvalue .
Let , then (the supports are disjoint). The function lies in the direct sum of the with non-zero, and so there must exist at least one such that the projections of and to are distinct. We conclude that is non-trivial for this and ; by continuity, the same is true for all in an open neighbourhood of . By compactness of , we may thus find a finite number of non-zero eigenvalues such that for each , is non-trivial for at least one . The representation can then be seen to have all the required properties.
For us, the main reason why we need the Peter-Weyl theorem is that the linear spaces automatically have the NSS property, even though need not. Thus, one can view Theorem 13 as giving the compact case of Theorem 4.
We now prove Proposition 8, 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 13, we can find a compact normal subgroup of contained in such that is isomorphic to a linear 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.
— 4. From metrisable to subgroup trapping —
We now perform the most difficult step, which is to establish Proposition 7. This step will require both the weak Peter-Weyl theorem (Theorem 13) and the Gleason technology, 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 14 (Finite trapping) Let be a locally compact group, let be an open 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 14 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 7. 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
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
for all .
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 (5) one has
(note that and commute). For the first term, we can compute
Since , , so by (20) we conclude that
For the second term, we similarly expand
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 1 In with the usual metric, the finite sets converge in Hausdorff distance to the closed interval .
Proof: It is easy to see that the Hausdorff distance is indeed a metric on , and that this metric is complete. The total boundedness of easily implies the total boundedness of (indeed, once one can cover by the -neighbourhood of a finite set , one can cover by the -neighbourhood of , by “rounding” off any closed subset of to the nearest subset of ). The claim then follows from the Heine-Borel theorem.
Now we can prove Proposition 7. 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 14, we can also find such that .
The sets are closed subsets of ; by Lemma 15, 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 13, 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 14 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.
— 5. From locally compact to metrisable —
We finally establish Proposition 6, which is actually one of the easier steps of the argument (because the conclusion is so weak). This argument is also due to Gleason. Let be a locally compact group, and let be an open neighbourhood of the identity. Let be a symmetric precompact neighbourhood of the identity in . We can then recursively construct a sequence
of symmetric precompact neighbourhoods such that for each . In particular
If we then form
then is compact, symmetric, contains the origin, and ; thus is normal. Also, since , we have , thus is normalised by . Thus if is the group generated by , then is an open subgroup of and is a normal subgroup of .
Let be the quotient map, then we see that are nested open sets with compact and whose intersection is the identity. From this one easily verifies that they form a neighbourhood base for . Thus is first countable and Hausdorff, and thus metrisable by the Birkhoff-Kakutani theorem. As is locally compact, and are also locally compact, and the claim follows.