In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:
Theorem 1 (Gleason-Yamabe theorem) Let
be a locally compact group. Then, for any open neighbourhood
of the identity, there exists an open subgroup
of
and a compact normal subgroup
of
in
such that
is isomorphic to a Lie group.
In the next set of notes, we will combine the Gleason-Yamabe theorem with some topological analysis (and in particular, using the invariance of domain theorem) to establish some further control on locally compact groups, and in particular obtaining a solution to Hilbert’s fifth problem.
To prove the Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a Gleason metric:
Definition 2 Let
be a topological group. A Gleason metric on
is a left-invariant metric
which generates the topology on
and obeys the following properties for some constant
, writing
for
:
- (Escape property) If
and
is such that
, then
.
- (Commutator estimate) If
are such that
, then
where
is the commutator of
and
.
Theorem 3 (Building Lie structure from Gleason metrics) Let
be a locally compact group that has a Gleason metric. Then
is isomorphic to a Lie group.
The second tool is the existence of a left-invariant Haar measure on any locally compact group; see Theorem 3 from Notes 3. Finally, we will also need the compact case of the Gleason-Yamabe theorem (Theorem 8 from Notes 3), which was proven via the Peter-Weyl theorem:
Theorem 4 (Gleason-Yamabe theorem for compact groups) Let
be a compact Hausdorff group, and let
be a neighbourhood of the identity. Then there exists a compact normal subgroup
of
contained in
such that
is isomorphic to a linear group (i.e. a closed subgroup of a general linear group
).
To finish the proof of the Gleason-Yamabe theorem, we have to somehow use the available structures on locally compact groups (such as Haar measure) to build good metrics on those groups (or on suitable subgroups or quotient groups). The basic construction is as follows:
Definition 5 (Building metrics out of test functions) Let
be a topological group, and let
be a bounded non-negative function. Then we define the pseudometric
by the formula
and the semi-norm
by the formula
Note that one can also write
where is the “derivative” of
in the direction
.
Exercise 6 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 7 (Birkhoff-Kakutani theorem) A topology group is metrisable if and only if it is Hausdorff and first countable.
Remark 8 The group structure is crucial; for instance, the long line is Hausdorff and first countable, but not metrisable.
We now prove this theorem (following the arguments in this book of Montgomery and Zippin). The “only if” direction is easy, so it suffices to establish the “if” direction. The key lemma is
Lemma 9 (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 10 Let
be a Hausdorff first countable group, and let
be as in Lemma 9. 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 9, 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
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 9, and the Birkhoff-Kakutani theorem follows.
Exercise 11 Let
be a topological group. Show that
is completely regular, that is to say for every closed subset
in
and every
, there exists a continuous function
that equals
on
and vanishes on
.
Exercise 12 (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 13 (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
Theorem 14 Every weak Gleason metric is a Gleason metric (possibly after adjusting the constant
).
We now prove this theorem. The key idea here is to involve a bump function formed by convolving together two Lipschitz functions. The escape property (5) will be crucial in obtaining quantitative control of the metric geometry at very small scales, as one can study the size of a group element
very close to the origin through its powers
, which are further away from the origin.
Specifically, let be a small quantity to be chosen later, and let
be a non-negative Lipschitz function supported on the ball
which is not identically zero. For instance, one could use the explicit function
where , although the exact form of
will not be important for our argument. Being Lipschitz, we see that
for all (where we allow implied constants to depend on
,
, and
), where
denotes the sup norm.
Let be a left-invariant Haar measure on
, the existence of which was established in Theorem 3 from Notes 3. We then form the convolution
, with convolution defined using the formula
This is a continuous function supported in , and gives a metric
and a norm
as usual.
We now prove a variant of the commutator estimate (1), namely that
whenever . To see this, we first use the left-invariance of Haar measure to write
thus
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 for all
we have
and so
. Conjugating by
, this implies that
for all
, 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 14 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 15 (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 17 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 18 Show that for any prime
, the
-adic groups
and
are not NSS. What about the solenoid group
?
Exercise 19 Show that an NSS group is automatically Hausdorff. (Hint: use Exercise 3 from Notes 3.)
Exercise 20 Show that an NSS locally compact group is automatically metrisable. (Hint: use Exercise 12.)
Exercise 21 (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 22 Every NSS locally compact group admits a weak Gleason metric. In particular, by Theorem 14 and Theorem 3, every NSS locally compact group is isomorphic to a Lie group.
In view of this theorem and Exercise 16, 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 22. 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 21. This concludes the proof of (12).
Exercise 23 Let
be a locally compact group. Show that if
is a left-invariant metric on
obeying the escape property (5) that generates the topology, then
is NSS, and
is comparable to
for all sufficiently small
and for all sufficiently small
. (In particular, any two left-invariant metrics obeying the escape property and generating the topology are locally comparable to each other.)
Henceforth is a locally compact NSS group. We now establish a metric-like property on the escape norm
.
Proposition 24 (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 and 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
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
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 24.
Now we can finish the proof of Theorem 22. Let be a locally compact NSS group, and let
be a sufficiently small neighbourhood of the identity. From Proposition 24, 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 22, 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 22, 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 25 (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 26 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 27 (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 22,
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 27.
We now prove Proposition 27, 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 27.
— 5. The subgroup trapping property —
In view of Theorem 22, Proposition 27, and Exercise 12, we see that the Gleason-Yamabe theorem (Theorem 1) now reduces to the following claim.
Proposition 28 Every locally compact metrisable group has the subgroup trapping property.
We now prove this proposition, which is the hardest step of the entire proof and uses almost all the tools already developed. In particular, it requires both Theorem 4 and Gleason’s convolution trick, as well as some of the basic theory of Hausdorff distance; as such, this is perhaps the most “infinitary” of all the steps in the argument.
The Gleason-type arguments can be encapsulated in the following proposition, which is a weak version of the subgroup trapping property:
Proposition 29 (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 29 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 28. 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
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
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 30 In
with the usual metric, the finite sets
converge in Hausdorff distance to the closed interval
.
Exercise 31 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 28. 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 29, we can also find
such that
.
The sets are closed subsets of
; by Exercise 31, 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
are 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 29 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 32 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.
Exercise 33 Show that the metrisability condition in Proposition 28 can be dropped; in other words, show that every locally compact group has the subgroup trapping property.
— 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 34 (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 35 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 36 (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 37 In the global group
, the open interval
is a normal sub-local subgroup if one takes (say)
as the normalising neighbourhood.
Example 38 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 39 (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 40 (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 14, 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 14 (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 22 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 27. 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 41. However, this difficulty can be easily avoided by rewording the definition:
Definition 41 (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 27 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 40, one has to establish the analogue of Proposition 28, 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 12 to the local group setting to reduce to the metrisable case.) The first step is to prove the local group analogue of Proposition 29 (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 28 to hold in the local setting as well (note, as in the proof of Proposition 27, 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.
18 comments
Comments feed for this article
4 October, 2011 at 2:13 pm
Mateusz Kwaśnicki
It seems that Theorem 1 is missing “contained in U” phrase.
[Corrected, thanks – T.]
5 October, 2011 at 12:48 am
Escape property of the Gleason metric and sub-riemannian distances again | chorasimilarity
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5 October, 2011 at 12:52 am
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8 October, 2011 at 12:57 pm
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24 October, 2011 at 6:46 pm
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27 October, 2011 at 8:37 pm
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6 November, 2011 at 9:22 am
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6 November, 2011 at 10:49 am
Ben Hayes
In Exercise 12, for comparability I think we need to assume that $g$ is sufficiently small as well. For example, one could consider $GL(n,\RR)$ and then any escape norm is bounded by 1, whereas there is no bound for the operator norm of an invertible operator.
[Corrected, thanks – T.]
25 January, 2012 at 12:51 pm
Hans
If I could just ask a silly question about beginning of Prop. 15:
Why can we choose V small enough, such that
? The fact that we don’t know a lot about
worries me.(especially when I think about xy and
)? Thank you.
25 January, 2012 at 2:12 pm
Terence Tao
Oops, I forgot to add the hypothesis that U (and hence
) was precompact, in which case one can make
as small as desired by a compactness argument.
25 January, 2012 at 2:24 pm
Hans
Yes, that’s what I secretly suspected. Thank you for the prompt reply!
13 April, 2012 at 8:01 am
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9 April, 2019 at 3:22 pm
Arturo Rodríguez Fanlo
Dear Prof. Tao,
I have a question about the proof of theorem 8. There, to prove {\|g^y\|\ll\|g\|}, you say
“We can achieve this by the escape property (5). Let {n} be a natural number such that {n \|g\| \leq \epsilon}, then {\|g^n\| \leq \epsilon} and so {g^n \in B(0,\epsilon)}. Conjugating by {y}, this implies that {(g^y)^n \in B(0,5\epsilon)}, and so by (5), we have {\|g^y\| \ll \frac{1}{n}} (if {\epsilon} is small enough), and the claim follows.”
For me, it seems that you are using the escape property under the hypothesis that {\|g^n\|} is small. However, the escape property is given when {n\|g\|} is small.
Similarly, in Notes 2, proving that the one-parameter subgroups are Lipschitz, it also seems that you used the escape property when {\|\phi(\frac{t}{n})^n\|} is small, while it should be used when {n\|\phi(\frac{t}{n})\|} is small.
Of course, when {n\|g\|} is small, we have {\|g^n\|\leq n\|g\|}, concluding that {\|g^n\|} is small. However, I do not see a way to conclude that {n\|g\|} is small assuming that {\|g^n\|} is small. It is particularly annoying, since the escape property is actually giving that control.
In other words, my problem could be rewritten as follows:
Why is not possible to have a sequence {(g_k)_{k\in {\bf N}} } and a constant {C} such that {\|g^n_k\|\rightarrow 0} while {k\rightarrow \infty} but {n\|g_k\|>C} for each {k\in{\bf N}}?
Best wishes,
Arturo
9 April, 2019 at 5:46 pm
Terence Tao
I guess strictly speaking one should not just use the fact that
, but rather that
for all
. This is then incompatible with the escape property if
for any
, and since
, this only leaves the possibility that
.
17 April, 2019 at 3:19 pm
Arturo Rodríguez Fanlo
Dear Prof. Tao,
Thank you very much. That was really useful.
Please, would you mind to help me again with my next questions?
First of all, in proposition 11, with the dilatation given by
, to prove that
is well defined we need to show that
and
are
. Clearly,
is simple so it is in
. However, I do not see how to argue in the case of
. Furthermore, I expect that it is continuous, but I do not know how to show that.
Secondly, in the proof of proposition 14, I do not understand the following sentence:
“Since
, we see that
does not lie in the
-neighbourhood
of
if
is small enough, where
is the projection map.”
Why we get that? I do not see trivially a contradiction assuming that
is contained in the closure of
.
Also, in proposition 14, what about
? You express that it will be chosen later, but I do not see how and where we need to choose
.
Finally, if it is possible, I would sincerely appreciate any hint to show the lower bound of point (vii) of exercise 9 of notes 1.
Again, thank you very much.
Best wishes,
Arturo
18 April, 2019 at 7:33 am
Terence Tao
“
small enough” should really be “
small enough”, to ensure that
lies in
.
(Regarding the Notes 1 exercise, please post a detailed question in that post regarding the exercise, stating what you have tried and where precisely you are stuck.)
29 April, 2019 at 10:25 am
Anonymous
Dear Prof. Tao,
Thank you very much for your attention. That was really useful.
Best wishes,
Arturo