Small doubling in groups

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial.  This is a short survey of the known results on classifying finite subsets of an (abelian) additive group or a (not necessarily abelian) multiplicative group that have small doubling in the sense that the sum set or product set is small.  Such sets behave approximately like finite subgroups of (and there is a closely related notion of an approximate group in which the analogy is even tighter) , and so this subject can be viewed as a sort of approximate version of finite group theory.  (Unfortunately, thus far the theory does not have much new to say about the classification of actual finite groups; progress has been largely made instead on classifying the (highly restricted) number of ways in which approximate groups can differ from a genuine group.)

In the classical case when is the integers , these sets were classified (in a qualitative sense, at least) by a celebrated theorem of Freiman, which roughly speaking says that such sets are necessarily “commensurate” in some sense with a (generalised) arithmetic progression of bounded rank.   There are a number of essentially equivalent ways to define what “commensurate” means here; for instance, in the original formulation of the theorem, one asks that be a dense subset of , but in modern formulations it is often more convenient to require instead that be of comparable size to and be covered by a bounded number of translates of , or that and have an intersection that is of comparable size to both and (cf. the notion of commensurability in group theory).

Freiman’s original theorem was extended to more general abelian groups in a sequence of papers culminating in the paper of Green and Ruzsa that handled arbitrary abelian groups.   As such groups now contain non-trivial finite subgroups, the conclusion of the theorem must be  modified by allowing for “coset progressions” , which can be viewed as “extensions”  of generalized arithmetic progressions by genuine finite groups .

The proof methods in these abelian results were Fourier-analytic in nature (except in the cases of sets of very small doubling, in which more combinatorial approaches can be applied, and there were also some geometric or combinatorial methods that gave some weaker structural results).  As such, it was a challenge to extend these results to nonabelian groups, although for various important special types of ambient group (such as an linear group over a finite or infinite field) it turns out that one can use tools exploiting the special structure of those groups (e.g. for linear groups one would use tools from Lie theory and algebraic geometry) to obtain quite satisfactory results; see e.g. this survey of  Pyber and Szabo for the linear case.   When the ambient group is completely arbitrary, it turns out the problem is closely related to the classical Hilbert’s fifth problem of determining the minimal requirements of a topological group in order for such groups to have Lie structure; this connection was first observed and exploited by Hrushovski, and then used by Breuillard, Green, and myself to obtain the analogue of Freiman’s theorem for an arbitrary nonabelian group.

This survey is too short to discuss in much detail the proof techniques used in these results (although the abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining open questions in the subject (in particular, there is substantial work left to be done in making the estimates more quantitative, particularly in the nonabelian setting).

254A, Notes 5: The structure of locally compact groups, and Hilbert’s fifth problem

In the previous notes, we established the Gleason-Yamabe theorem:

Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.

Roughly speaking, this theorem asserts the “mesoscopic” structure of a locally compact group (after restricting to an open subgroup to remove the macroscopic structure, and quotienting out by to remove the microscopic structure) is always of Lie type.

In this post, we combine the Gleason-Yamabe theorem with some additional tools from point-set topology to improve the description of locally compact groups in various situations.

We first record some easy special cases of this. If the locally compact group has the no small subgroups property, then one can take to be trivial; thus is Lie, which implies that is locally Lie and thus Lie as well. Thus the assertion that all locally compact NSS groups are Lie (Theorem 10 from Notes 4) is a special case of the Gleason-Yamabe theorem.

In a similar spirit, if the locally compact group is connected, then the only open subgroup of is the full group ; in particular, by arguing as in the treatment of the compact case (Exercise 19 of Notes 3), we conclude that any connected locally compact Hausdorff group is the inverse limit of Lie groups.

Now we return to the general case, in which need not be connected or NSS. One slight defect of Theorem 1 is that the group can depend on the open neighbourhood . However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make independent of :

Theorem 2 (Gleason-Yamabe theorem, stronger version) Let be a locally compact group. Then there exists an open subgoup of such that, for any open neighbourhood of the identity in , there exists a compact normal subgroup of in such that is isomorphic to a Lie group.

We prove this theorem below the fold. As in previous notes, if is Hausdorff, the group is thus an inverse limit of Lie groups (and if (and hence ) is first countable, it is the inverse limit of a sequence of Lie groups).

It remains to analyse inverse limits of Lie groups. To do this, it helps to have some control on the dimensions of the Lie groups involved. A basic tool for this purpose is the invariance of domain theorem:

Theorem 3 (Brouwer invariance of domain theorem) Let be an open subset of , and let be a continuous injective map. Then is also open.

We prove this theorem below the fold. It has an important corollary:

Corollary 4 (Topological invariance of dimension) If , and is a non-empty open subset of , then there is no continuous injective mapping from to . In particular, and are not homeomorphic.

Exercise 1 (Uniqueness of dimension) Let be a non-empty topological space. If is a manifold of dimension , and also a manifold of dimension , show that . Thus, we may define the dimension of a non-empty manifold in a well-defined manner.

If are non-empty manifolds, and there is a continuous injection from to , show that .

Remark 1 Note that the analogue of the above exercise for surjections is false: the existence of a continuous surjection from one non-empty manifold to another does not imply that , thanks to the existence of space-filling curves. Thus we see that invariance of domain, while intuitively plausible, is not an entirely trivial observation.

As we shall see, we can use Corollary 4 to bound the dimension of the Lie groups in an inverse limit by the “dimension” of the inverse limit . Among other things, this can be used to obtain a positive resolution to Hilbert’s fifth problem:

Theorem 5 (Hilbert’s fifth problem) Every locally Euclidean group is isomorphic to a Lie group.

Again, this will be shown below the fold.

Another application of this machinery is the following variant of Hilbert’s fifth problem, which was used in Gromov’s original proof of Gromov’s theorem on groups of polynomial growth, although we will not actually need it this course:

Proposition 6 Let be a locally compact -compact group that acts transitively, faithfully, and continuously on a connected manifold . Then is isomorphic to a Lie group.

Recall that a continuous action of a topological group on a topological space is a continuous map which obeys the associativity law for and , and the identity law for all . The action is transitive if, for every , there is a with , and faithful if, whenever are distinct, one has for at least one .

The -compact hypothesis is a technical one, and can likely be dropped, but we retain it for this discussion (as in most applications we can reduce to this case).

Exercise 2 Show that Proposition 6 implies Theorem 5.

Remark 2 It is conjectured that the transitivity hypothesis in Proposition 6 can be dropped; this is known as the Hilbert-Smith conjecture. It remains open; the key difficulty is to figure out a way to eliminate the possibility that is a -adic group . See this previous blog post for further discussion.

254A, Notes 4: Building metrics on groups, and the Gleason-Yamabe theorem

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

1. (Identity) , with equality when .
2. (Symmetry) .
3. (Triangle inequality) .
4. (Continuity) If , then the map is continuous.
5. (Boundedness) One has . If is supported in a set , then equality occurs unless .
6. (Left-invariance) . In particular, .

In particular, we have the norm-like properties

1. (Identity) , with equality when .
2. (Symmetry) .
3. (Triangle inequality) .
4. (Continuity) If , then the map is continuous.
5. (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.

254A, Notes 2: Building Lie structure from representations and metrics

Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group to ensure that it is actually a Lie group. In the previous set of notes, we saw that one could reduce the regularity hypothesis imposed on to a “” condition, namely that there was an open neighbourhood of that was isomorphic (as a local group) to an open subset of a Euclidean space with identity element , and with group operation obeying the asymptotic

for sufficiently small . We will call such local groups local groups.

We now reduce the regularity hypothesis further, to one in which there is no explicit Euclidean space that is initially attached to . Of course, Lie groups are still locally Euclidean, so if the hypotheses on do not involve any explicit Euclidean spaces, then one must somehow build such spaces from other structures. One way to do so is to exploit an ambient space with Euclidean or Lie structure that is embedded or immersed in. A trivial example of this is provided by the following basic fact from linear algebra:

Lemma 1 If is a finite-dimensional vector space (i.e. it is isomorphic to for some ), and is a linear subspace of , then is also a finite-dimensional vector space.

We will establish a non-linear version of this statement, known as Cartan’s theorem. Recall that a subset of a -dimensional smooth manifold is a -dimensional smooth (embedded) submanifold of for some if for every point there is a smooth coordinate chart of a neighbourhood of in that maps to , such that , where we identify with a subspace of . Informally, locally sits inside the same way that sits inside .

Theorem 2 (Cartan’s theorem) If is a (topologically) closed subgroup of a Lie group , then is a smooth submanifold of , and is thus also a Lie group.

Note that the hypothesis that is closed is essential; for instance, the rationals are a subgroup of the (additive) group of reals , but the former is not a Lie group even though the latter is.

Exercise 1 Let be a subgroup of a locally compact group . Show that is closed in if and only if it is locally compact.

A variant of the above results is provided by using (faithful) representations instead of embeddings. Again, the linear version is trivial:

Lemma 3 If is a finite-dimensional vector space, and is another vector space with an injective linear transformation from to , then is also a finite-dimensional vector space.

Here is the non-linear version:

Theorem 4 (von Neumann’s theorem) If is a Lie group, and is a locally compact group with an injective continuous homomorphism , then also has the structure of a Lie group.

Actually, it will suffice for the homomorphism to be locally injective rather than injective; related to this, von Neumann’s theorem localises to the case when is a local group rather a group. The requirement that be locally compact is necessary, for much the same reason that the requirement that be closed was necessary in Cartan’s theorem.

Example 1 Let be the two-dimensional torus, let , and let be the map , where is a fixed real number. Then is a continuous homomorphism which is locally injective, and is even globally injective if is irrational, and so Theorem 4 is consistent with the fact that is a Lie group. On the other hand, note that when is irrational, then is not closed; and so Theorem 4 does not follow immediately from Theorem 2 in this case. (We will see, though, that Theorem 4 follows from a local version of Theorem 2.)

As a corollary of Theorem 4, we observe that any locally compact Hausdorff group with a faithful linear representation, i.e. a continuous injective homomorphism from into a linear group such as or , is necessarily a Lie group. This suggests a representation-theoretic approach to Hilbert’s fifth problem. While this approach does not seem to readily solve the entire problem, it can be used to establish a number of important special cases with a well-understood representation theory, such as the compact case or the abelian case (for which the requisite representation theory is given by the Peter-Weyl theorem and Pontryagin duality respectively). We will discuss these cases further in later notes.

In all of these cases, one is not really building up Euclidean or Lie structure completely from scratch, because there is already a Euclidean or Lie structure present in another object in the hypotheses. Now we turn to results that can create such structure assuming only what is ostensibly a weaker amount of structure. In the linear case, one example of this is is the following classical result in the theory of topological vector spaces.

Theorem 5 Let be a locally compact Hausdorff topological vector space. Then is isomorphic (as a topological vector space) to for some finite .

Remark 1 The Banach-Alaoglu theorem asserts that in a normed vector space , the closed unit ball in the dual space is always compact in the weak-* topology. Of course, this dual space may be infinite-dimensional. This however does not contradict the above theorem, because the closed unit ball is not a neighbourhood of the origin in the weak-* topology (it is only a neighbourhood with respect to the strong topology).

The full non-linear analogue of this theorem would be the Gleason-Yamabe theorem, which we are not yet ready to prove in this set of notes. However, by using methods similar to that used to prove Cartan’s theorem and von Neumann’s theorem, one can obtain a partial non-linear analogue which requires an additional hypothesis of a special type of metric, which we will call a Gleason metric:

Definition 6 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 .

Exercise 2 Let be a topological group that contains a neighbourhood of the identity isomorphic to a local group. Show that admits at least one Gleason metric.

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

We will rely on Theorem 7 to solve Hilbert’s fifth problem; this theorem reduces the task of establishing Lie structure on a locally compact group to that of building a metric with suitable properties. Thus, much of the remainder of the solution of Hilbert’s fifth problem will now be focused on the problem of how to construct good metrics on a locally compact group.

In all of the above results, a key idea is to use one-parameter subgroups to convert from the nonlinear setting to the linear setting. Recall from the previous notes that in a Lie group , the one-parameter subgroups are in one-to-one correspondence with the elements of the Lie algebra , which is a vector space. In a general topological group , the concept of a one-parameter subgroup (i.e. a continuous homomorphism from to ) still makes sense; the main difficulties are then to show that the space of such subgroups continues to form a vector space, and that the associated exponential map is still a local homeomorphism near the origin.

Exercise 3 The purpose of this exercise is to illustrate the perspective that a topological group can be viewed as a non-linear analogue of a vector space. Let be locally compact groups. For technical reasons we assume that are both -compact and metrisable.

• (i) (Open mapping theorem) Show that if is a continuous homomorphism which is surjective, then it is open (i.e. the image of open sets is open). (Hint: mimic the proof of the open mapping theorem for Banach spaces, as discussed for instance in these notes. In particular, take advantage of the Baire category theorem.)
• (ii) (Closed graph theorem) Show that if a homomorphism is closed (i.e. its graph is a closed subset of ), then it is continuous. (Hint: mimic the derivation of the closed graph theorem from the open mapping theorem in the Banach space case, as again discussed in these notes.)
• (iii) Let be a homomorphism, and let be a continuous injective homomorphism into another Hausdorff topological group . Show that is continuous if and only if is continuous.
• (iv) Relax the condition of metrisability to that of being Hausdorff. (Hint: Now one cannot use the Baire category theorem for metric spaces; but there is an analogue of this theorem for locally compact Hausdorff spaces.)

254A, Notes 0: Hilbert’s fifth problem and related topics

This fall (starting Monday, September 26), I will be teaching a graduate topics course which I have entitled “Hilbert’s fifth problem and related topics.” The course is going to focus on three related topics:

• Hilbert’s fifth problem on the topological description of Lie groups, as well as the closely related (local) classification of locally compact groups (the Gleason-Yamabe theorem).
• Approximate groups in nonabelian groups, and their classification via the Gleason-Yamabe theorem (this is very recent work of Emmanuel Breuillard, Ben Green, Tom Sanders, and myself, building upon earlier work of Hrushovski);
• Gromov’s theorem on groups of polynomial growth, as proven via the classification of approximate groups (as well as some consequences to fundamental groups of Riemannian manifolds).

I have already blogged about these topics repeatedly in the past (particularly with regard to Hilbert’s fifth problem), and I intend to recycle some of that material in the lecture notes for this course.

The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):

• (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
• (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)

Let me illustrate what I mean by these two principles with two simple examples, one in the continuous setting and one in the discrete setting. We begin with a continuous example. Given an complex matrix , define the matrix exponential of by the formula

which can easily be verified to be an absolutely convergent series.

Exercise 1 Show that the map is a real analytic (and even complex analytic) map from to , and obeys the restricted homomorphism property

for all and .

Proposition 1 (Rigidity and structure of matrix homomorphisms) Let be a natural number. Let be the group of invertible complex matrices. Let be a map obeying two properties:

• (Group-like object) is a homomorphism, thus for all .
• (Weak regularity) The map is continuous.

Then:

• (Strong regularity) The map is smooth (i.e. infinitely differentiable). In fact it is even real analytic.
• (Lie-type structure) There exists a (unique) complex matrix such that for all .

Proof: Let be as above. Let be a small number (depending only on ). By the homomorphism property, (where we use here to denote the identity element of ), and so by continuity we may find a small such that for all (we use some arbitrary norm here on the space of matrices, and allow implied constants in the notation to depend on ).

The map is real analytic and (by the inverse function theorem) is a diffeomorphism near . Thus, by the inverse function theorem, we can (if is small enough) find a matrix of size such that . By the homomorphism property and (1), we thus have

On the other hand, by another application of the inverse function theorem we see that the squaring map is a diffeomorphism near in , and thus (if is small enough)

We may iterate this argument (for a fixed, but small, value of ) and conclude that

for all . By the homomorphism property and (1) we thus have

whenever is a dyadic rational, i.e. a rational of the form for some integer and natural number . By continuity we thus have

for all real . Setting we conclude that

for all real , which gives existence of the representation and also real analyticity and smoothness. Finally, uniqueness of the representation follows from the identity

Exercise 2 Generalise Proposition 1 by replacing the hypothesis that is continuous with the hypothesis that is Lebesgue measurable (Hint: use the Steinhaus theorem.). Show that the proposition fails (assuming the axiom of choice) if this hypothesis is omitted entirely.

Note how one needs both the group-like structure and the weak regularity in combination in order to ensure the strong regularity; neither is sufficient on its own. We will see variants of the above basic argument throughout the course. Here, the task of obtaining smooth (or real analytic structure) was relatively easy, because we could borrow the smooth (or real analytic) structure of the domain and range ; but, somewhat remarkably, we shall see that one can still build such smooth or analytic structures even when none of the original objects have any such structure to begin with.

Now we turn to a second illustration of the above principles, namely Jordan’s theorem, which uses a discreteness hypothesis to upgrade Lie type structure to nilpotent (and in this case, abelian) structure. We shall formulate Jordan’s theorem in a slightly stilted fashion in order to emphasise the adherence to the above-mentioned principles.

Theorem 2 (Jordan’s theorem) Let be an object with the following properties:

• (Group-like object) is a group.
• (Discreteness) is finite.
• (Lie-type structure) is contained in (the group of unitary matrices) for some .

Then there is a subgroup of such that

• ( is close to ) The index of in is (i.e. bounded by for some quantity depending only on ).
• (Nilpotent-type structure) is abelian.

A key observation in the proof of Jordan’s theorem is that if two unitary elements are close to the identity, then their commutator is even closer to the identity (in, say, the operator norm ). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have

and so by the triangle inequality

Now we can prove Jordan’s theorem.

Proof: We induct on , the case being trivial. Suppose first that contains a central element which is not a multiple of the identity. Then, by definition, is contained in the centraliser of , which by the spectral theorem is isomorphic to a product of smaller unitary groups. Projecting to each of these factor groups and applying the induction hypothesis, we obtain the claim.

Thus we may assume that contains no central elements other than multiples of the identity. Now pick a small (one could take in fact) and consider the subgroup of generated by those elements of that are within of the identity (in the operator norm). By considering a maximal -net of we see that has index at most in . By arguing as before, we may assume that has no central elements other than multiples of the identity.

If consists only of multiples of the identity, then we are done. If not, take an element of that is not a multiple of the identity, and which is as close as possible to the identity (here is where we crucially use that is finite). By (2), we see that if is sufficiently small depending on , and if is one of the generators of , then lies in and is closer to the identity than , and is thus a multiple of the identity. On the other hand, has determinant . Given that it is so close to the identity, it must therefore be the identity (if is small enough). In other words, is central in , and is thus a multiple of the identity. But this contradicts the hypothesis that there are no central elements other than multiples of the identity, and we are done.

Commutator estimates such as (2) will play a fundamental role in many of the arguments we will see in this course; as we saw above, such estimates combine very well with a discreteness hypothesis, but will also be very useful in the continuous setting.

Exercise 3 Generalise Jordan’s theorem to the case when is a finite subgroup of rather than of . (Hint: The elements of are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of . However, if one averages that inner product by the finite group , one obtains a new inner product on that is preserved by , which allows one to conjugate to a subgroup of . This averaging trick is (a small) part of Weyl’s unitary trick in representation theory.)

Exercise 4 (Inability to discretise nonabelian Lie groups) Show that if , then the orthogonal group cannot contain arbitrarily dense finite subgroups, in the sense that there exists an depending only on such that for every finite subgroup of , there exists a ball of radius in (with, say, the operator norm metric) that is disjoint from . What happens in the case?

Remark 1 More precise classifications of the finite subgroups of are known, particularly in low dimensions. For instance, one can show that the only finite subgroups of (which is a double cover of) are isomorphic to either a cyclic group, a dihedral group, or the symmetry group of one of the Platonic solids.

The Hilbert-Smith conjecture

The classical formulation of Hilbert’s fifth problem asks whether topological groups that have the topological structure of a manifold, are necessarily Lie groups. This is indeed, the case, thanks to following theorem of Gleason and Montgomery-Zippin:

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

We have discussed the proof of this result, and of related results, in previous posts. There is however a generalisation of Hilbert’s fifth problem which remains open, namely the Hilbert-Smith conjecture, in which it is a space acted on by the group which has the manifold structure, rather than the group itself:

Conjecture 2 (Hilbert-Smith conjecture) Let be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold . Then is isomorphic to a Lie group.

Note that Conjecture 2 easily implies Theorem 1 as one can pass to the connected component of a locally Euclidean group (which is clearly locally compact), and then look at the action of on itself by left-multiplication.

The hypothesis that the action is faithful (i.e. each non-identity group element acts non-trivially on ) cannot be completely eliminated, as any group will have a trivial action on any space . The requirement that be locally compact is similarly necessary: consider for instance the diffeomorphism group of, say, the unit circle , which acts on but is infinite dimensional and is not locally compact (with, say, the uniform topology). Finally, the connectedness of is also important: the infinite torus (with the product topology) acts faithfully on the disconnected manifold by the action

The conjecture in full generality remains open. However, there are a number of partial results. For instance, it was observed by Montgomery and Zippin that the conjecture is true for transitive actions, by a modification of the argument used to establish Theorem 1. This special case of the Hilbert-Smith conjecture (or more precisely, a generalisation thereof in which “finite-dimensional manifold” was replaced by “locally connected locally compact finite-dimensional”) was used in Gromov’s proof of his famous theorem on groups of polynomial growth. I record the argument of Montgomery and Zippin below the fold.

Another partial result is the reduction of the Hilbert-Smith conjecture to the -adic case. Indeed, it is known that Conjecture 2 is equivalent to

Conjecture 3 (Hilbert-Smith conjecture for -adic actions) It is not possible for a -adic group to act continuously and effectively on a connected finite-dimensional manifold .

The reduction to the -adic case follows from the structural theory of locally compact groups (specifically, the Gleason-Yamabe theorem discussed in previous posts) and some results of Newman that sharply restrict the ability of periodic actions on a manifold to be close to the identity. I record this argument (which appears for instance in this paper of Lee) below the fold also.

Gleason’s lemma

This is another installment of my my series of posts on Hilbert’s fifth problem. One formulation of this problem is answered by the following theorem of Gleason and Montgomery-Zippin:

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 have no small subgroups, or is NSS for short, if there is an open neighbourhood of the identity in that contains no subgroups of other than the trivial subgroup .

Definition 3 (Gleason metric) Let be a topological group. A Gleason metric on is a left-invariant metric which generates the topology on and obeys the following properties for some constant , writing for :

• (Escape property) If and is such that , then

  

• (Commutator estimate) If are such that , then

  

  where is the commutator of and .

The remaining steps in the resolution of Hilbert’s fifth problem are then as follows:

Theorem 4 (Reduction to the NSS case) Let be a locally compact group, and let be an open neighbourhood of the identity in . Then there exists an open subgroup of , and a compact subgroup of contained in , such that is NSS and locally compact.

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 the subgroup trapping property: for every open neighbourhood of the identity, there exists another open neighbourhood of the identity such that generates a subgroup contained in .

Proposition 8 (From subgroup trapping to NSS) Let be a locally compact group with the subgroup trapping property, and let be an open neighbourhood of the identity in . Then there exists an open subgroup of , and a compact subgroup of contained in , such that is locally compact and NSS.

Proposition 9 (From NSS to the escape property) Let be a locally compact NSS group. Then there exists a left-invariant metric on generating the topology on which obeys the escape property (1) for some constant .

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 11 Suppose 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.

The C^{1,1} Baker-Campbell-Hausdorff formula

Let be a Lie group with Lie algebra . As is well known, the exponential map is a local homeomorphism near the identity. As such, the group law on can be locally pulled back to an operation defined on a neighbourhood of the identity in , defined as

where is the local inverse of the exponential map. One can view as the group law expressed in local exponential coordinates around the origin.

An asymptotic expansion for is provided by the Baker-Campbell-Hausdorff (BCH) formula

for all sufficiently small , where is the Lie bracket. More explicitly, one has the Baker-Campbell-Hausdorff-Dynkin formula

for all sufficiently small , where , is the adjoint representation , and is the function

which is real analytic near and can thus be applied to linear operators sufficiently close to the identity. One corollary of this is that the multiplication operation is real analytic in local coordinates, and so every smooth Lie group is in fact a real analytic Lie group.

It turns out that one does not need the full force of the smoothness hypothesis to obtain these conclusions. It is, for instance, a classical result that regularity of the group operations is already enough to obtain the Baker-Campbell-Hausdorff formula. Actually, it turns out that we can weaken this a bit, and show that even regularity (i.e. that the group operations are continuously differentiable, and the derivatives are locally Lipschitz) is enough to make the classical derivation of the Baker-Campbell-Hausdorff formula work. More precisely, we have

Theorem 1 ( Baker-Campbell-Hausdorff formula) Let be a finite-dimensional vector space, and suppose one has a continuous operation defined on a neighbourhood around the origin, which obeys the following three axioms:

• (Approximate additivity) For sufficiently close to the origin, one has

  

  (In particular, for sufficiently close to the origin.)

• (Associativity) For sufficiently close to the origin, .
• (Radial homogeneity) For sufficiently close to the origin, one has

  

  for all . (In particular, for all sufficiently close to the origin.)

Then is real analytic (and in particular, smooth) near the origin. (In particular, gives a neighbourhood of the origin the structure of a local Lie group.)

Indeed, we will recover the Baker-Campbell-Hausdorff-Dynkin formula (after defining appropriately) in this setting; see below the fold.

The reason that we call this a Baker-Campbell-Hausdorff formula is that if the group operation has regularity, and has as an identity element, then Taylor expansion already gives (2), and in exponential coordinates (which, as it turns out, can be defined without much difficulty in the category) one automatically has (3).

We will record the proof of Theorem 1 below the fold; it largely follows the classical derivation of the BCH formula, but due to the low regularity one will rely on tools such as telescoping series and Riemann sums rather than on the fundamental theorem of calculus. As an application of this theorem, we can give an alternate derivation of one of the components of the solution to Hilbert’s fifth problem, namely the construction of a Lie group structure from a Gleason metric, which was covered in the previous post; we discuss this at the end of this article. With this approach, one can avoid any appeal to von Neumann’s theorem and Cartan’s theorem (discussed in this post), or the Kuranishi-Gleason extension theorem (discussed in this post).

Hilbert’s fifth problem and Gleason metrics

Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

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 have no small subgroups, or is NSS for short, if there is an open neighbourhood of the identity in that contains no subgroups of other than the trivial subgroup .

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 4 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 .

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 1 The 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 2 From 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.

Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem

We recall Brouwer’s famous fixed point theorem:

Theorem 1 (Brouwer fixed point theorem) Let be a continuous function on the unit ball in a Euclidean space . Then has at least one fixed point, thus there exists with .

This theorem has many proofs, most of which revolve (either explicitly or implicitly) around the notion of the degree of a continuous map of the unit sphere to itself, and more precisely around the stability of degree with respect to homotopy. (Indeed, one can view the Brouwer fixed point theorem as an assertion that some non-trivial degree-like invariant must exist, or more abstractly that the homotopy group is non-trivial.)

One of the many applications of this result is to prove Brouwer’s invariance of domain theorem:

Theorem 2 (Brouwer invariance of domain theorem) Let be an open subset of , and let be a continuous injective map. Then is also open.

This theorem in turn has an important corollary:

Corollary 3 (Topological invariance of dimension) If , and is a non-empty open subset of , then there is no continuous injective mapping from to . In particular, and are not homeomorphic.

This corollary is intuitively obvious, but note that topological intuition is not always rigorous. For instance, it is intuitively plausible that there should be no continuous surjection from to for , but such surjections always exist, thanks to variants of the Peano curve construction.

Theorem 2 or Corollary 3 can be proven by simple ad hoc means for small values of or (for instance, by noting that removing a point from will disconnect when , but not for ), but I do not know of any proof of these results in general dimension that does not require algebraic topology machinery that is at least as sophisticated as the Brouwer fixed point theorem. (Lebesgue, for instance, famously failed to establish the above corollary rigorously, although he did end up discovering the important concept of Lebesgue covering dimension as a result of his efforts.)

Nowadays, the invariance of domain theorem is usually proven using the machinery of singular homology. In this post I would like to record a short proof of Theorem 2 using Theorem 1 that I discovered in a paper of Kulpa, which avoids any use of algebraic topology tools beyond the fixed point theorem, though it is more ad hoc in its approach than the systematic singular homology approach.

Remark 1 A heuristic explanation as to why the Brouwer fixed point theorem is more or less a necessary ingredient in the proof of the invariance of domain theorem is that a counterexample to the former result could conceivably be used to create a counterexample to the latter one. Indeed, if the Brouwer fixed point theorem failed, then (as is well known) one would be able to find a continuous function that was the identity on (indeed, one could take to be the first point in which the ray from through hits ). If one then considered the function defined by , then this would be a continuous function which avoids the interior of , but which maps the origin to a point on the sphere (and maps to the dilate ). This could conceivably be a counterexample to Theorem 2, except that is not necessarily injective. I do not know if there is a more rigorous way to formulate this connection.

The reason I was looking for a proof of the invariance of domain theorem was that it comes up in the very last stage of the solution to Hilbert’s fifth problem, namely to establish the following fact:

Theorem 4 (Hilbert’s fifth problem) Every locally Euclidean group is isomorphic to a Lie group.

Recall that a locally Euclidean group is a topological group which is locally homeomorphic to an open subset of a Euclidean space , i.e. it is a continuous manifold. Note in contrast that a Lie group is a topological group which is locally diffeomorphic to an open subset of , it is a smooth manifold. Thus, Hilbert’s fifth problem is a manifestation of the “rigidity” of algebraic structure (in this case, group structure), which turns weak regularity (continuity) into strong regularity (smoothness).

It is plausible that something like Corollary 3 would need to be invoked in order to solve Hilbert’s fifth problem. After all, if Euclidean spaces , of different dimension were homeomorphic to each other, then the property of being locally Euclidean loses a lot of meaning, and would thus not be a particularly powerful hypothesis. Note also that it is clear that two Lie groups can only be isomorphic if they have the same dimension, so in view of Theorem 4, it becomes plausible that two Euclidean spaces can only be homeomorphic if they have the same dimension, although I do not know of a way to rigorously deduce this claim from Theorem 4.

Interestingly, Corollary 3 is the only place where algebraic topology enters into the solution of Hilbert’s fifth problem (although its cousin, point-set topology, is used all over the place). There are results closely related to Theorem 4, such as the Gleason-Yamabe theorem mentioned in a recent post, which do not use the notion of being locally Euclidean, and do not require algebraic topological methods in their proof. Indeed, one can deduce Theorem 4 from the Gleason-Yamabe theorem and invariance of domain; we sketch a proof of this (following Montgomery and Zippin) below the fold.