In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:
Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.
In the next set of notes, we will combine the Gleason-Yamabe theorem with some topological analysis (and in particular, using the invariance of domain theorem) to establish some further control on locally compact groups, and in particular obtaining a solution to Hilbert’s fifth problem.
To prove the Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a Gleason metric:
Definition 2 Let be a topological group. A Gleason metric on is a left-invariant metric which generates the topology on and obeys the following properties for some constant , writing for :
- (Escape property) If and is such that , then .
- (Commutator estimate) If are such that , then
where is the commutator of and .
The second tool is the existence of a left-invariant Haar measure on any locally compact group; see Theorem 3 from Notes 3. Finally, we will also need the compact case of the Gleason-Yamabe theorem (Theorem 8 from Notes 3), which was proven via the Peter-Weyl theorem:
Theorem 4 (Gleason-Yamabe theorem for compact groups) Let be a compact Hausdorff group, and let be a neighbourhood of the identity. Then there exists a compact normal subgroup of contained in such that is isomorphic to a linear group (i.e. a closed subgroup of a general linear group ).
To finish the proof of the Gleason-Yamabe theorem, we have to somehow use the available structures on locally compact groups (such as Haar measure) to build good metrics on those groups (or on suitable subgroups or quotient groups). The basic construction is as follows:
Definition 5 (Building metrics out of test functions) Let be a topological group, and let be a bounded non-negative function. Then we define the pseudometric by the formula
and the semi-norm by the formula
Note that one can also write
where is the “derivative” of in the direction .
Exercise 1 Let the notation and assumptions be as in the above definition. For any , establish the metric-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
- (Left-invariance) . In particular, .
In particular, we have the norm-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
We remark that the first three properties of in the above exercise ensure that is indeed a pseudometric.
To get good metrics (such as Gleason metrics) on groups , it thus suffices to obtain test functions that obey suitably good “regularity” properties. We will achieve this primarily by means of two tricks. The first trick is to obtain high-regularity test functions by convolving together two low-regularity test functions, taking advantage of the existence of a left-invariant Haar measure on . The second trick is to obtain low-regularity test functions by means of a metric-like object on . This latter trick may seem circular, as our whole objective is to get a metric on in the first place, but the key point is that the metric one starts with does not need to have as many “good properties” as the metric one ends up with, thanks to the regularity-improving properties of convolution. As such, one can use a “bootstrap argument” (or induction argument) to create a good metric out of almost nothing. It is this bootstrap miracle which is at the heart of the proof of the Gleason-Yamabe theorem (and hence to the solution of Hilbert’s fifth problem).
The arguments here are based on the nonstandard analysis arguments used to establish Hilbert’s fifth problem by Hirschfeld and by Goldbring (and also some unpublished lecture notes of Goldbring and van den Dries). However, we will not explicitly use any nonstandard analysis in this post.
— 1. Warmup: the Birkhoff-Kakutani theorem —
To illustrate the basic idea of using test functions to build metrics, let us first establish a classical theorem on topological groups, which gives a necessary and sufficient condition for metrisability. Recall that a topological space is metrisable if there is a metric on that space that generates the topology.
Theorem 6 (Birkhoff-Kakutani theorem) A topology group is metrisable if and only if it is Hausdorff and first countable.
Remark 1 The group structure is crucial; for instance, the long line is Hausdorff and first countable, but not metrisable.
We now prove this theorem (following the arguments in this book of Montgomery and Zippin). The “only if” direction is easy, so it suffices to establish the “if” direction. The key lemma is
- (Unique maximum) , and for all .
- (Neighbourhood base) The sets for form a neighbourhood base at the identity.
- (Uniform continuity) For every , there exists an open neighbourhood of the identity such that for all and .
Note that if had a left-invariant metric, then the function would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.
Exercise 2 Let be a Hausdorff first countable group, and let be as in Lemma 7. Show that is a metric on (so in particular, only vanishes when ) and that generates the topology of (thus every set which is open with respect to is open in , and vice versa).
In view of the above exercise, we see that to prove the Birkhoff-Kakutani theorem, it suffices to prove Lemma 7, which we now do. By first countability, we can find a countable neighbourhood base
of the identity. As is Hausdorff, we must have
For every dyadic rational in , we can now define the open sets by setting
for all and .
We now set
with the understanding that if the supremum is over the empty set. One easily verifies using (4) that is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 7, and the Birkhoff-Kakutani theorem follows.
Exercise 3 Let be a topological group. Show that is completely regular, that is to say for every closed subset in and every , there exists a continuous function that equals on and vanishes on .
- (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
In this section we will show
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
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
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 .
Then by the triangle inequality
This implies that and have overlapping support, and hence lies in . By the escape property (5), this implies (if is small enough) that , and the claim follows.
Combining Claim 2 with (8) we see that
whenever are small enough. Now we use the identity
and the triangle inequality to conclude that
whenever are small enough. Theorem 8 then follows from Claim 1 and Claim 2.
— 3. Building metrics on NSS groups —
We will now be able to build metrics on groups using a set of hypotheses that do not explicitly involve any metric at all. The key hypothesis will be the no small subgroups (NSS) property:
Definition 9 (No small subgroups) A topological group has the no small subgroups (or NSS) property if there exists an open neighbourhood of the identity which does not contain any subgroup of other than the trivial group.
Exercise 7 Show that any group with a weak Gleason metric is NSS.
For an example of a group which is not NSS, consider the infinite-dimensional torus . From the definition of the product topology, we see that any neighbourhood of the identity in this torus contains an infinite-dimensional subtorus, and so this group is not NSS.
Exercise 8 Show that for any prime , the -adic groups and are not NSS. What about the solenoid group ?
Exercise 9 Show that an NSS group is automatically Hausdorff. (Hint: use Exercise 3 from Notes 3.)
Exercise 10 Show that an NSS locally compact group is automatically metrisable. (Hint: use Exercise 4.)
Exercise 11 (NSS implies escape property) Let be a locally compact NSS group. Show that if is a sufficiently small neighbourhood of the identity, then for every , there exists a positive integer such that . Furthermore, for any other neighbourhood of the identity, there exists a positive integer such that if , then .
We can now prove the following theorem (first proven in full generality by Yamabe), which is a key component in the proof of the Gleason-Yamabe theorem and in the wider theory of Hilbert’s fifth problem.
In view of this theorem and Exercise 6, we see that for locally compact groups, the property of being a Lie group is equivalent to the property of being an NSS group. This is a major advance towards both the Gleason-Yamabe theorem and Hilbert’s fifth problem, as it has reduced the property of being a Lie group into a condition that is almost purely algebraic in nature.
We now prove Theorem 10. An important concept will be that of an escape norm associated to an open neighbourhood of a group , defined by the formula
for any , where ranges over the natural numbers (thus, for instance , with equality iff ). Thus, the longer it takes for the orbit to escape , the smaller the escape norm.
Strictly speaking, the escape norm is not necessarily a norm, as it need not obey the symmetry, non-degeneracy, or triangle inequalities; however, we shall see that in many situations, the escape norm behaves similarly to a norm, even if it does not exactly obey the norm axioms. Also, as the name suggests, the escape norm will be well suited for establishing the escape property (5).
It is possible for the escape norm of a non-identity element to be zero, if contains the group generated by . But if the group has the NSS property, then we see that this cannot occur for all sufficiently small (where “sufficiently small” means “contained in a suitably chosen open neighbourhood of the identity”). In fact, more is true: if are two sufficiently small open neighbourhoods of the identity in a locally compact NSS group , then the two escape norms are comparable, thus we have
for all (where the implied constants can depend on ).
By symmetry, it suffices to prove the second inequality in (12). By (11), it suffices to find an integer such that whenever is such that , then . But this follows from Exercise 11. This concludes the proof of (12).
Exercise 12 Let be a locally compact group. Show that if is a left-invariant metric on obeying the escape property (5) that generates the topology, then is NSS, and is comparable to for all sufficiently small and for all sufficiently small . (In particular, any two left-invariant metrics obeying the escape property and generating the topology are locally comparable to each other.)
Henceforth is a locally compact NSS group. We now establish a metric-like property on the escape norm .
(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 ; 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
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 observe the telescoping identity
We 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 11.
Now we can finish the proof of Theorem 10. Let be a locally compact NSS group, and let be a sufficiently small neighbourhood of the identity. From Proposition 11, we see that the escape norm and the modified escape norm are comparable. We have seen is a left-invariant pseudometric. As is NSS and is small, there are no non-identity elements with zero escape norm, and hence no non-identity elements with zero modified escape norm either; thus is a genuine metric.
We now claim that generates the topology of . Given the left-invariance of , it suffices to establish two things: firstly, that any open neighbourhood of the identity contains a ball around the identity in the metric; and conversely, any such ball contains an open neighbourhood around the identity.
To prove the first claim, let be an open neighbourhood around the identity, and let be a smaller neighbourhood of the identity. From (12) we see (if is small enough) that is comparable to , and contains a small ball around the origin in the metric, giving the claim. To prove the second claim, consider a ball in the metric. For any positive integer , we can find an open neighbourhood of the identity such that , and hence for all . For large enough, this implies that , and the claim follows.
To finish the proof of Theorem 10, we need to verify the escape property (5). Thus, we need to show that if , are such that is sufficiently small, then we have . We may of course assume that is not the identity, as the claim is trivial otherwise. As is comparable to , we know that there exists a natural number such that . Let be a neighbourhood of the identity small enough that . We have for all , so and hence . Let be the first multiple of larger than , then and so . Since , this implies . Since is divisible by , we conclude that , and the claim follows from (12).
— 4. NSS from subgroup trapping —
In view of Theorem 10, the only remaining task in the proof of the Gleason-Yamabe theorem is to locate “big” subquotients of a locally compact group with the NSS property. We will need some further notation. Given a neighbourhood of the identity in a topological group , let denote the union of all the subgroups of that are contained in . Thus, a group is NSS if is trivial for all sufficiently small .
We will need a property that is weaker than NSS:
Definition 12 (Subgroup trapping) A topological group has the subgroup trapping property if, for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that generates a subgroup contained in .
Clearly, every NSS group has the subgroup trapping property. Informally, groups with the latter property do have small subgroups, but one cannot get very far away from the origin just by combining together such subgroups.
Example 1 The infinite-dimensional torus does not have the NSS property, but it does have the subgroup trapping property.
It is difficult to produce an example of a group that does not have the subgroup trapping property; the reason for this will be made clear in the next section. For now, we establish the following key result.
Proposition 13 (From subgroup trapping to NSS) Let be a locally compact group with the subgroup trapping property, and let be an open neighbourhood of the identity in . Then there exists an open subgroup of , and a compact subgroup of contained in , such that is locally compact and NSS. In particular, by Theorem 10, is isomorphic to a Lie group.
Intuitively, the idea is to use the subgroup trapping property to find a small compact normal subgroup that contains for some small , and then quotient this group out to get an NSS group. Unfortunately, because is not necessarily contained in , this quotienting operation may create some additional small subgroups. To fix this, we need to pass from the compact subgroup to a smaller one. In order to understand the subgroups of compact groups, the main tool will be Gleason-Yamabe theorem for compact groups (Theorem 4).
For us, the main reason why we need the compact case of the Gleason-Yamabe theorem is that Lie groups automatically have the NSS property, even though need not. Thus, one can view Theorem 4 as giving the compact case of Proposition 13.
We now prove Proposition 13, using an argument of Yamabe. Let be a locally compact group with the subgroup trapping property, and let be an open neighbourhood of the identity. We may find a smaller neighbourhood of the identity with , which in particular implies that ; by shrinking if necessary, we may assume that is compact. By the subgroup trapping property, one can find an open neighbourhood of the identity such that is contained in , and thus is a compact subgroup of contained in . By shrinking if necessary we may assume .
Ideally, if were normal and contained in , then the quotient group would have the NSS property. Unfortunately need not be normal, and need not be contained in , but we can fix this as follows. Applying Theorem 4, we can find a compact normal subgroup of contained in such that is isomorphic to a Lie group, and in particular is NSS. In particular, we can find an open symmetric neighbourhood of the identity in such that and that the quotient space has no non-trivial subgroups in , where is the quotient map.
We now claim that is normalised by . Indeed, if , then the conjugate of is contained in and hence in . As is a group, it must thus be contained in and hence in . But then is a subgroup of that is contained in , and is hence trivial by construction. Thus , and so is normalised by . If we then let be the subgroup of generated by and , we see that is an open subgroup of , with a compact normal subgroup of .
To finish the job, we need to show that has the NSS property. It suffices to show that has no nontrivial subgroups. But any subgroup in pulls back to a subgroup in , hence in , hence in , hence in ; since has no nontrivial subgroups, the claim follows. This concludes the proof of Proposition 13.
— 5. The subgroup trapping property —
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
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 (19) one has
(note that and commute). For the first term, we can compute
Since , , so by (21) 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 2 In with the usual metric, the finite sets converge in Hausdorff distance to the closed interval .
Exercise 13 Show that the space of non-empty closed subsets of a compact metric space is itself a compact metric space (with the Hausdorff distance as the metric). (Hint: use the Heine-Borel theorem.)
Now we can prove Proposition 14. Let be a locally compact group endowed with some metric , and let be an open neighbourhood of the identity; by shrinking we may assume that is precompact. Let be a sequence of balls around the identity with radius going to zero, then is a symmetric set in that contains the identity. If, for some , for every , then and we are done. Thus, we may assume for sake of contradiction that there exists such that and ; since the go to zero, we have . By Proposition 15, we can also find such that .
The sets are closed subsets of ; by Exercise 13, we may pass to a subsequence and assume that they converge to some closed subset of . Since the are symmetric and contain the identity, is also symmetric and contains the identity. For any fixed , we have for all sufficiently large , which on taking Hausdorff limits implies that . In particular, the group is a compact subgroup of contained in .
Let be a small neighbourhood of the identity in to be chosen later. By Theorem 4, we can find a normal subgroup of contained in such that is NSS. Let be a neigbourhood of the identity in so small that has no small subgroups. A compactness argument then shows that there exists a natural number such that for any that is not in , at least one of must lie outside of .
Now let be a small parameter. Since , we see that does not lie in the -neighbourhood of if is small enough, where is the projection map. Let be the first integer for which does not lie in , then and as (for fixed ). On the other hand, as , we see from another application of Proposition 15 that if is sufficiently large depending on .
On the other hand, since converges to a subset of in the Hausdorff distance, we know that for large enough, and hence is contained in the -neighbourhood of . Thus we can find an element of that lies within of a group element of , but does not lie in ; thus lies inside . By construction of , we can find such that lies in . But also lies within of , which lies in and hence in , where denotes a quantity depending on that goes to zero as . We conclude that and are separated by , which leads to a contradiction if is sufficiently small (note that and are compact and disjoint, and hence separated by a positive distance), and the claim follows.
Exercise 14 Let be a compact metric space, denote the space of non-empty closed and connected subsets of . Show that with the Hausdorff metric is also a compact metric space.
— 6. The local group case —
In the thesis of Goldbring (and also the later paper of Goldbring and van den Dries), the above theory was extended to the setting of local groups. In fact, there is relatively little difficulty (other than some notational difficulties) in doing so, because the analysis in the previous sections can be made to take place on a small neighbourhood of the origin. This extension to local groups is not simply a generalisation for its own sake; it will turn out that it will be natural to work with local groups when we classify approximate groups in later notes.
One technical issue that comes up in the theory of local groups is that basic cancellation laws such as , which are easily verified for groups, are not always true for local groups. However, this is a minor issue as one can always recover the cancellation laws by passing to a slightly smaller local group, as follows.
Definition 16 (Cancellative local group) A local group is said to be symmetric if the inverse operation is always well-defined. It is said to be cancellative if it is symmetric, and the following axioms hold:
- (i) Whenever are such that and are well-defined and equal to each other, then . (Note that this implies in particular that .)
- (ii) Whenever are such that and are well-defined and equal to each other, then .
- (iii) Whenever are such that and are well-defined, then . (In particular, if is symmetric and is well-defined in for some , then is also symmetric.)
Clearly, all global groups are cancellative, and more generally the restriction of a global group to a symmetric neighbourhood of the identity s cancellative. While not all local groups are cancellative, we have the following substitute:
Exercise 15 Let be a local group. Show that there is a neighbourhood of the identity which is cancellative (thus, the restriction of to is cancellative).
Note that any symmetric neighbourhood of the identity in a cancellative local group is again a cancellative local group. Because of this, it turns out in practice that we may restrict to the cancellative setting without much loss of generality.
Next, we need to localise the notion of a quotient of a global group by a normal subgroup . Recall that in order for a subset og a global group to be a normal subgroup, it has to be symmetric, contain the identity, be closed under multiplication (thus whenever , and closed under conjugation (thus whenever and ). We now localise this concept as follows:
Definition 17 (Normal sublocal group) Let be a cancellative local group. A subset of is said to be a normal sublocal group if there is an open neighbourhood of (called a normalising neighbourhood of ) obeying the following axioms:
- (Identity and inverse) is symmetric and contains the identity.
- (Local closure) If and is well-defined in , then .
- (Normality) If are such that is well-defined in , then .
(Strictly speaking, one should refer to the pair as the normal sublocal group, rather than just , but by abuse of notation we shall omit the normalising neighbourhood when referring to the normal sublocal group.)
It is easy to see that if is a normal sublocal group of , then is itself a cancellative local group, using the topology and group structure formed by restriction from . (Note how the open neighbourhood is needed to ensure that the domain of the multiplication map in remains open.)
Example 3 In the global group , the open interval is a normal sub-local subgroup if one takes (say) as the normalising neighbourhood.
Example 4 Let be the shift map , and let be the semidirect product of and . Then if is any (global) subgroup of , the set is a normal sub-local subgroup of (with normalising neighbourhood ). This is despite the fact that will, in general, not be normal in in the classical (global) sense.
As observed by Goldbring, one can define the operation of quotienting a local group by a normal sub-local group, provided that one restricts to a sufficiently small neighbourhood of the origin:
Exercise 16 (Quotient spaces) Let be a cancellative local group, and let be a normal sub-local group with normalising neighbourhood . Let be a symmetric open neighbourhood of the identity such that is well-defined and contained in . Show that there exists a cancellative local group and a surjective continuous homomorphism such that, for any , one has if and only if , and for any , one has open if and only if is open.
It is not difficult to show that the quotient defined by the above exercise is unique up to local isomorphism, so we will abuse notation and talk about “the” quotient space given by the above construction.
We can now state the local version of the Gleason-Yamabe theorem, first proven by Goldbring in his thesis, and later reproven by Goldbring and van den Dries by a slightly different method:
Theorem 18 (Local Gleason-Yamabe theorem) Let be a locally compact local group. Then there exists an open symmetric neighbourhood of the identity, and a compact global group in that is normalised by , such that is well-defined and isomorphic to a local Lie group.
The proofs of this theorem by Goldbring and Goldbring-van den Dries were phrased in the language of nonstandard analysis. However, it is possible to translate those arguments to standard analysis arguments, which closely follow the arguments given in previous sections and notes. (Actually, our arguments are not a verbatim translation of those in Goldbring and Goldbring-van den Dries, as we have made a few simplifications in which the role of Gleason metrics is much more strongly emphasised.) We briefly sketch the main points here.
As in the global case, the route to obtaining (local) Lie structure is via Gleason metrics. On a local group , we define a local Gleason metric to be a metric defined on some symmetric open neighbourhood of the identity with (say) well-defined (to avoid technical issues), which generates the topology of , and which obeys the following version of the left-invariance, escape and commutator properties:
- (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 the subgroup trapping property if, for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that is contained in a global subgroup that is in turn contained in . (Here, is, as before, the union of all the global subgroups contained in .)
Because is now contained in a global group , the group generated by is well-defined. As is in the open neighbourhood , one can then also form the closure ; if we choose small enough to be precompact, then this is a compact global group (and thus describable by the Gleason-Yamabe theorem for such groups, Theorem 4). Because of this, it is possible to adapt Proposition 13 without much difficulty to the local setting to conclude that given any locally compact local group with the subgroup trapping property, there exists an open symmetric neighbourhood of the identity, and a compact global group in that is normalised by , such that is well-defined and NSS (and thus locally isomorphic to a local Lie group).
Finally, to finish the proof of Theorem 18, one has to establish the analogue of Proposition 14, namely that one has to show that every locally compact metrisable local group has the subgroup trapping property. (It is not difficult to adapt Exercise 4 to the local group setting to reduce to the metrisable case.) The first step is to prove the local group analogue of Proposition 15 (again adopting the obvious convention that a statement such as is only considered true if is well-defined, and adding the additional hypothesis that is sufficiently small in order to ensure that all manipulations are justified). This can be done by a routine modification of the proof. But then one can modify the rest of the argument in Proposition 14 to hold in the local setting as well (note, as in the proof of Proposition 13, that the compact set generated in the course of this argument remains a global group rather than a local one, and so one can again use Theorem 4 without difficulty). Again, we omit the details.