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 2Let be a topological group. AGleason metricon 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 formulaand the semi-norm by the formula

Note that one can also write

where is the “derivative” of in the direction .

Exercise 1Let 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 1The 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 2Let 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 3Let 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 8Every 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:

- If with sufficiently small, then .
- If with sufficiently small, then .

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 theno 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 7Show 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 8Show that for any prime , the -adic groups and are not NSS. What about the solenoid group ?

Exercise 9Show that an NSS group is automatically Hausdorff. (Hint:use Exercise 3 from Notes 3.)

Exercise 10Show 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 10Every 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 12Let 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 thesubgroup trapping propertyif, 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 1The 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 14Every 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 2In with the usual metric, the finite sets converge in Hausdorff distance to the closed interval .

Exercise 13Show 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 14Let be a compact metric space, denote the space of non-empty closed andconnectedsubsets 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 besymmetricif the inverse operation is always well-defined. It is said to becancellativeif 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 15Let 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 bea normal sublocal groupif there is an open neighbourhood of (called anormalising neighbourhoodof ) 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 3In the global group , the open interval is a normal sub-local subgroup if one takes (say) as the normalising neighbourhood.

Example 4Let 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:

- (Left-invariance) If are such that , then .
- (Escape property) If and , then are well-defined in and .
- (Commutator estimate) If are such that , then is well-defined in and (1) holds.

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 thesubgroup trapping propertyif, 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.

Theorem 1 (Hilbert’s fifth problem)Let be a topological group which is locally Euclidean. Then is isomorphic to a Lie group.

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 haveno small subgroups, or isNSSfor 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. AGleason metricon 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.

Theorem 5 (Gleason’s lemma)Let be a locally compact NSS group. Then has a Gleason metric.

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 thesubgroup 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 .

Proposition 10 (From escape to the commutator estimate)Let be a locally compact group with a left-invariant metric that obeys the escape property (1). Then also obeys the commutator property (2).

It is clear that Propositions 6, 7, and 8 combine to give Theorem 4, and Propositions 9, 10 combine to give Theorem 5.

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.

Lemma 11Suppose that obeys (4). Then the (semi-)metric (and associated (semi-)norm ) obey the escape property (1) and the commutator property (2).

*Proof:* We begin with the commutator property (2). Observe the identity

whence

From the triangle inequality (and translation-invariance of the norm) we thus see that (2) follows from (4). Similarly, to obtain the escape property (1), observe the telescoping identity

for any and natural number , and thus by the triangle inequality

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

where . Being Lipschitz, we see that

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

whenever . 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 (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 .

Claim 2 follows easily from (9) and (7), 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 (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 1Let 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.

Proposition 12 (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 , 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

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 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

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 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

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 (5) one has

(note that and commute). For the first term, we can compute

and

Since , , so by (20) we conclude that

For the second term, we similarly expand

Using (20), (19) 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 1In with the usual metric, the finite sets converge in Hausdorff distance to the closed interval .

Lemma 15The space of non-empty closed subsets of a compact metric space is itself a compact metric space (with the Hausdorff distance as the metric).

*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.

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Theorem 1 (Hilbert’s fifth problem)Let be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem)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 isomorphic to a Lie group.

The deduction of Theorem 1 from Theorem 2 proceeds using the Brouwer invariance of domain theorem and is discussed in this previous post. In this post, I would like to discuss the proof of Theorem 2. We can split this proof into three parts, by introducing two additional concepts. The first is the property of having no small subgroups:

Definition 3 (NSS)A topological group is said to haveno small subgroups, or isNSSfor short, if there is an open neighbourhood of the identity in that contains no subgroups of other than the trivial subgroup .

An equivalent definition of an NSS group is one which has an open neighbourhood of the identity that every non-identity element *escapes* in finite time, in the sense that for some positive integer . It is easy to see that all Lie groups are NSS; we shall shortly see that the converse statement (in the locally compact case) is also true, though significantly harder to prove.

Another useful property is that of having what I will call a *Gleason metric*:

Definition 4Let be a topological group. AGleason metricon 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 .

For instance, the unitary group with the operator norm metric can easily verified to be a Gleason metric, with the commutator estimate (1) coming from the inequality

Similarly, any left-invariant Riemannian metric on a (connected) Lie group can be verified to be a Gleason metric. From the escape property one easily sees that all groups with Gleason metrics are NSS; again, we shall see that there is a partial converse.

Remark 1The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

The proof of Theorem 2 can then be split into three subtheorems:

Theorem 5 (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, locally compact, and metrisable.

Theorem 6 (Gleason’s lemma)Let be a locally compact metrisable NSS group. Then has a Gleason metric.

Theorem 7 (Building a Lie structure)Let be a locally compact group with a Gleason metric. Then is isomorphic to a Lie group.

Clearly, by combining Theorem 5, Theorem 6, and Theorem 7 one obtains Theorem 2 (and hence Theorem 1).

Theorem 5 and Theorem 6 proceed by some elementary combinatorial analysis, together with the use of Haar measure (to build convolutions, and thence to build “smooth” bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en route to the reduction to the NSS case), which was discussed previously on this blog.

In this post I would like to detail the final component to the proof of Theorem 2, namely Theorem 7. (I plan to discuss the other two steps, Theorem 5 and Theorem 6, in a separate post.) The strategy is similar to that used to prove von Neumann’s theorem, as discussed in this previous post (and von Neumann’s theorem is also used in the proof), but with the Gleason metric serving as a substitute for the faithful linear representation. Namely, one first gives the space of one-parameter subgroups of enough of a structure that it can serve as a proxy for the “Lie algebra” of ; specifically, it needs to be a vector space, and the “exponential map” needs to cover an open neighbourhood of the identity. This is enough to set up an “adjoint” representation of , whose image is a Lie group by von Neumann’s theorem; the kernel is essentially the centre of , which is abelian and can also be shown to be a Lie group by a similar analysis. To finish the job one needs to use arguments of Kuranishi and of Gleason, as discussed in this previous post.

The arguments here can be phrased either in the standard analysis setting (using sequences, and passing to subsequences often) or in the nonstandard analysis setting (selecting an ultrafilter, and then working with infinitesimals). In my view, the two approaches have roughly the same level of complexity in this case, and I have elected for the standard analysis approach.

Remark 2From Theorem 7 we see that a Gleason metric structure is a good enough substitute for smooth structure that it can actually be used to reconstruct the entire smooth structure; roughly speaking, the commutator estimate (1) allows for enough “Taylor expansion” of expressions such as that one can simulate the fundamentals of Lie theory (in particular, construction of the Lie algebra and the exponential map, and its basic properties. The advantage of working with a Gleason metric rather than a smoother structure, though, is that it is relatively undemanding with regards to regularity; in particular, the commutator estimate (1) is roughly comparable to the imposition structure on the group , as this is the minimal regularity to get the type of Taylor approximation (with quadratic errors) that would be needed to obtain a bound of the form (1). We will return to this point in a later post.

** — 1. Proof of theorem — **

We now prove Theorem 7. Henceforth, is a locally compact group with a Gleason metric (and an associated “norm” ). In particular, by the Heine-Borel theorem, is complete with this metric.

We use the asymptotic notation in place of for some constant that can vary from line to line (in particular, need not be the constant appearing in the definition of a Gleason metric), and write for . We also let be a sufficiently small constant (depending only on the constant in the definition of a Gleason metric) to be chosen later.

Note that the left-invariant metric properties of give the symmetry property

and the triangle inequality

From the commutator estimate (1) and the triangle inequality we also obtain a conjugation estimate

whenever . Since left-invariance gives

we then conclude an approximate right invariance

whenever . In a similar spirit, the commutator estimate (1) also gives

This has the following useful consequence, which asserts that the power maps behave like dilations:

and

*Proof:* We begin with the first inequality. By the triangle inequality, it suffices to show that

uniformly for all . By left-invariance and approximate right-invariance, the left-hand side is comparable to

which by (2) is bounded above by

as required.

Now we prove the second estimate. Write , then . We have

thanks to the escape property (shrinking if necessary). On the other hand, from the first inequality, we have

If is small enough, the claim now follows from the triangle inequality.

Remark 3Lemma 8 implies (by a standard covering argument) that the group is locally of bounded doubling, though we will not use this fact here.

Now we introduce the space of one-parameter subgroups, i.e. continuous homomorphisms . We give this space the compact-open topology, thus the topology is generated by balls of the form

for , , and compact . Actually, using the homomorphism property, one can use a single compact interval , such as , to generate the topology if desired, thus making a metric space.

Given that is eventually going to be shown to be a Lie group, must be isomorphic to a Euclidean space. We now move towards this goal by establishing various properties of that Euclidean spaces enjoy.

Lemma 9is locally compact.

*Proof:* It is easy to see that is complete. Let . As is continuous, we can find an interval small enough that for all . By the Heine-Borel theorem, it will suffice to show that the set

is totally bounded. By the Arzelá-Ascoli theorem, it suffices to show that the family of functions in is equicontinuous.

By construction, we have whenever . By the escape property, this implies (for small enough, of course) that for all and , thus whenever . From the homomorphism property, we conclude that whenever , which gives uniform Lipschitz control and hence equicontinuity as desired.

We observe for future reference that the proof of the above lemma also shows that all one-parameter subgroups are locally Lipschitz.

Now we put a vector space structure on , which we define by analogy with the Lie group case, in which each tangent vector generates a one-parameter subgroup . From this analogy, the scalar multiplication operation has an obvious definition: if and , we define to be the one-parameter subgroup

which is easily seen to actually be a one-parameter subgroup.

Now we turn to the addition operation. In the Lie group case, one can express the one-parameter subgroup in terms of the one-parameter subgroups , by the limiting formula

In view of this, we would like to define the sum of two one-parameter subgroups by the formula

*Proof:* To show well-definedness, it suffices to show that for each , the sequence is a Cauchy sequence. It suffices to show that

as . By the continuity of multiplication, it suffices to show that there is some such that

as .

Since are locally Lipschitz, we can find a quantity (depending on ) such that

for all . From Lemma 8, we conclude that

if and is sufficiently large. Another application of Lemma 8 then gives

if , is sufficiently large, , and is sufficiently small depending on . The claim follows.

The above argument in fact shows that is uniformly Cauchy for in a compact interval, and so the pointwise limit is in fact a uniform limit of continuous functions and is thus continuous. To prove that is a homomorphism, it suffices by density of the rationals to show that

and

for all and all positive integers . To prove the first claim, we observe that

and similarly for and , whence the claim. To prove the second claim, we see that

but is conjugated by , which goes to the identity; and the claim follows.

also has an obvious zero element, namely the trivial one-parameter subgroup .

Lemma 11is a topological vector space.

*Proof:* We first show that is a vector space. It is clear that the zero element of is an additive and scalar multiplication identity, and that scalar multiplication is associative. To show that addition is commutative, we again use the observation that is conjugated by an element that goes to the identity. A similar argument shows that , and a change of variables argument shows that for all positive integers , hence for all rational , and hence by continuity for all real . The only remaining thing to show is that addition is associative, thus if , that for all . By the homomorphism property, it suffices to show this for all sufficiently small .

An inspection of the argument used to establish (10) reveals that there is a constant such that

for all small and all large , and hence also that

(thanks to Lemma 8). Similarly we have (after adjusting if necessary)

From Lemma 8 we have

and thus

Similarly for . By the triangle inequality we conclude that

sending to zero, the claim follows.

Finally, we need to show that the vector space operations are continuous. It is easy to see that scalar multiplication is continuous, as are the translation operations; the only remaining thing to verify is that addition is continuous at the origin. Thus, for every we need to find a such that whenever and . But if are as above, then by the escape property (assuming small enough) we conclude that for , and then from the triangle inequality we conclude that for , giving the claim.

As is both locally compact, metrisable, and a topological vector space, it must be isomorphic to a finite-dimensional vector space with the usual topology (see this blog post for a proof).

In analogy with the Lie algebra setting, we define the *exponential map* by setting . Given the topology on , it is clear that this is a continuous map. Using Lemma 8 one can see that the exponential map is locally injective near the origin, although we will not actually need this fact.

We have proved a number of useful things about , but at present we have not established that is *large* in any substantial sense; indeed, at present, could be completely trivial even if was large. In particular, the image of the exponential map could conceivably be quite small. We now address this issue. As a warmup, we show that is at least non-trivial if is non-trivial:

Proposition 12Suppose that is not a discrete group. Then is non-trivial.

Of course, the converse is obvious; discrete groups do not admit any non-trivial one-parameter subgroups.

*Proof:* As is not discrete, there is a sequence of non-identity elements of such that as . Writing for the integer part of , then as , and we conclude from the escape property that for all .

We define the approximate one-parameter subgroups by setting

Then we have for , and we have the approximate homomorphism property

uniformly whenever . As a consequence, is asymptotically equicontinuous on , and so by (a slight generalisation of) the Arzéla-Ascoli theorem, we may pass to a subsequence in which converges uniformly to a limit , which is a genuine homomorphism that is genuinely continuous, and is thus can be extended to a one-parameter subgroup. Also, for all , and thus ; in particular, is non-trivial, and the claim follows.

We now generalise the above proposition to a more useful result.

Proposition 13For any neighbourhood of the origin in , is a neighbourhood of the identity in .

*Proof:* We use an argument of Hirschfeld (communicated to me by van den Dries and Goldbring). By shrinking if necessary, we may assume that is a compact star-shaped neighbourhood, with contained in the ball of radius around the origin. As is compact, is compact also.

Suppose for contradiction that is not a neighbourhood of the identity, then there is a sequence of elements of such that as . By the compactness of , we can find an element of that minimises the distance . If we then write , then

and hence as .

Let be the integer part of , then as , and for all .

Let be the approximate one-parameter subgroups defined as

As before, we may pass to a subsequence such that converges uniformly to a limit , which extends to a one-parameter subgroup .

In a similar vein, since , we can find such that , which by the escape property (and the smallness of implies that for . In particular, goes to zero in .

We now claim that is close to . Indeed, from Lemma 8 we see that

Since , we conclude from the triangle inequality and left-invariance that

But from Lemma 8 again, one has

and thus

But for large enough, lies in , and so the distance from to is . But this contradicts the minimality of for large enough, and the claim follows.

We have some easy corollaries of this result:

Corollary 14is locally connected. In particular, the connected component of the identity is an open subgroup of .

Corollary 15 (Abelian case)If is abelian, then is isomorphic to a Lie group. In particular, in the non-abelian setting, the centre of is a Lie group.

*Proof:* In the abelian case one easily sees that is a homomorphism. Thus we see from Proposition 13 that has locally the structure of a vector space, and the claim clearly follows in that case.

We are now finally ready to prove Theorem 7. By Corollary 14 we may assume without loss of generality that is connected. (Note that if a topological group is locally connected, and the connected component of the identity is a Lie group, then the entire group a Lie group, because all outer automorphisms of are necessarily smooth, as discussed here.)

Now we consider the adjoint action of on . If and , we can define another one-parameter subgroup by setting

As conjugation by is an automorphism, one easily verifies that is linear, thus is a map from to the finite-dimensional linear group . One easily verifies that this map is continuous, and so is a finite-dimensional linear representation of . If is in the kernel of this representation, then by construction, centralises , and thus by Proposition 13, centralises an open neighbourhood of the identity in . As we are assuming to be connected, we conclude that is central. Thus we see that the kernel of is the center , thus giving a short exact sequence

The adjoint representation is a faithful finite-dimensional linear representation of , and so is a Lie group by a theorem of von Neumann (discussed here). By Corollary 15, is a central Lie group. By a result of Kuranishi and Gleason (discussed here), this implies that is itself a Lie group, as required.

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Remark 4An alternate approach to Theorem 7 would be to construct a Lie bracket on , and then show that the multiplication law on is locally given by the Baker-Campbell-Hausdorff formula; we will discuss this approach in a sequel to this post.