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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 A of an (abelian) additive group G = (G,+) or a (not necessarily abelian) multiplicative group G = (G,\cdot) that have small doubling in the sense that the sum set A+A or product set A \cdot A is small.  Such sets behave approximately like finite subgroups of G (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 G is the integers {\mathbb Z}, these sets were classified (in a qualitative sense, at least) by a celebrated theorem of Freiman, which roughly speaking says that such sets A are necessarily “commensurate” in some sense with a (generalised) arithmetic progression P 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 A be a dense subset of P, but in modern formulations it is often more convenient to require instead that A be of comparable size to P and be covered by a bounded number of translates of P, or that A and P have an intersection that is of comparable size to both A and P (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” P+H, which can be viewed as “extensions”  of generalized arithmetic progressions P by genuine finite groups H.

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 G (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 G 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).

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

Theorem 1 (Gleason-Yamabe theorem) Let {G} be a locally compact group. Then, for any open neighbourhood {U} of the identity, there exists an open subgroup {G'} of {G} and a compact normal subgroup {K} of {G'} in {U} such that {G'/K} 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 {G'} to remove the macroscopic structure, and quotienting out by {K} 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 {G} has the no small subgroups property, then one can take {K} to be trivial; thus {G'} is Lie, which implies that {G} 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 {G} is connected, then the only open subgroup {G'} of {G} is the full group {G}; 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 {G} need not be connected or NSS. One slight defect of Theorem 1 is that the group {G'} can depend on the open neighbourhood {U}. However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make {G'} independent of {U}:

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

We prove this theorem below the fold. As in previous notes, if {G} is Hausdorff, the group {G'} is thus an inverse limit of Lie groups (and if {G} (and hence {G'}) 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 {U} be an open subset of {{\bf R}^n}, and let {f: U \rightarrow {\bf R}^n} be a continuous injective map. Then {f(U)} is also open.

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

Corollary 4 (Topological invariance of dimension) If {n > m}, and {U} is a non-empty open subset of {{\bf R}^n}, then there is no continuous injective mapping from {U} to {{\bf R}^m}. In particular, {{\bf R}^n} and {{\bf R}^m} are not homeomorphic.

Exercise 1 (Uniqueness of dimension) Let {X} be a non-empty topological space. If {X} is a manifold of dimension {d_1}, and also a manifold of dimension {d_2}, show that {d_1=d_2}. Thus, we may define the dimension {\hbox{dim}(X)} of a non-empty manifold in a well-defined manner.

If {X, Y} are non-empty manifolds, and there is a continuous injection from {X} to {Y}, show that {\hbox{dim}(X) \leq \hbox{dim}(Y)}.

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 {X} to another {Y} does not imply that {\hbox{dim}(X) \geq \hbox{dim}(Y)}, 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 {L_n} in an inverse limit {G = \lim_{n \rightarrow \infty} L_n} by the “dimension” of the inverse limit {G}. 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 {G} be a locally compact {\sigma}-compact group that acts transitively, faithfully, and continuously on a connected manifold {X}. Then {G} is isomorphic to a Lie group.

Recall that a continuous action of a topological group {G} on a topological space {X} is a continuous map {\cdot: G \times X \rightarrow X} which obeys the associativity law {(gh)x = g(hx)} for {g,h \in G} and {x \in X}, and the identity law {1x = x} for all {x \in X}. The action is transitive if, for every {x,y \in X}, there is a {g \in G} with {gx=y}, and faithful if, whenever {g, h \in G} are distinct, one has {gx \neq hx} for at least one {x}.

The {\sigma}-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 {G} is a {p}-adic group {{\bf Z}_p}. See this previous blog post for further discussion.

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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 {G} be a locally compact group. Then, for any open neighbourhood {U} of the identity, there exists an open subgroup {G'} of {G} and a compact normal subgroup {K} of {G'} in {U} such that {G'/K} 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 {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

Theorem 3 (Building Lie structure from Gleason metrics) Let {G} be a locally compact group that has a Gleason metric. Then {G} 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 {G} be a compact Hausdorff group, and let {U} be a neighbourhood of the identity. Then there exists a compact normal subgroup {H} of {G} contained in {U} such that {G/H} is isomorphic to a linear group (i.e. a closed subgroup of a general linear group {GL_n({\bf C})}).

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 {G} be a topological group, and let {\psi: G \rightarrow {\bf R}^+} be a bounded non-negative function. Then we define the pseudometric {d_\psi: G \times G \rightarrow {\bf R}^+} by the formula

\displaystyle  d_\psi(g,h) := \sup_{x \in G} |\tau(g) \psi(x) - \tau(h) \psi(x)|

\displaystyle  = \sup_{x \in G} |\psi(g^{-1} x ) - \psi(h^{-1} x)|

and the semi-norm {\| \|_\psi: G \rightarrow {\bf R}^+} by the formula

\displaystyle  \|g\|_\psi := d_\psi(g, \hbox{id}).

Note that one can also write

\displaystyle  \|g\|_\psi = \sup_{x \in G} |\partial_g \psi(x)|

where {\partial_g \psi(x) := \psi(x) - \psi(g^{-1} x)} is the “derivative” of {\psi} in the direction {g}.

Exercise 1 Let the notation and assumptions be as in the above definition. For any {g,h,k \in G}, establish the metric-like properties

  1. (Identity) {d_\psi(g,h) \geq 0}, with equality when {g=h}.
  2. (Symmetry) {d_\psi(g,h) = d_\psi(h,g)}.
  3. (Triangle inequality) {d_\psi(g,k) \leq d_\psi(g,h) + d_\psi(h,k)}.
  4. (Continuity) If {\psi \in C_c(G)}, then the map {d_\psi: G \times G \rightarrow {\bf R}^+} is continuous.
  5. (Boundedness) One has {d_\psi(g,h) \leq \sup_{x \in G} |\psi(x)|}. If {\psi \in C_c(G)} is supported in a set {K}, then equality occurs unless {g^{-1} h \in K K^{-1}}.
  6. (Left-invariance) {d_\psi(g,h) = d_\psi(kg,kh)}. In particular, {d_\psi(g,h) = \| h^{-1} g \|_\psi = \| g^{-1} h \|_\psi}.

In particular, we have the norm-like properties

  1. (Identity) {\|g\|_\psi \geq 0}, with equality when {g=\hbox{id}}.
  2. (Symmetry) {\|g\|_\psi = \|g^{-1}\|_\psi}.
  3. (Triangle inequality) {\|gh\|_\psi \leq \|g\|_\psi + \|h\|_\psi}.
  4. (Continuity) If {\psi \in C_c(G)}, then the map {\|\|_\psi: G \rightarrow {\bf R}^+} is continuous.
  5. (Boundedness) One has {\|g\|_\psi \leq \sup_{x \in G} |\psi(x)|}. If {\psi \in C_c(G)} is supported in a set {K}, then equality occurs unless {g \in K K^{-1}}.

We remark that the first three properties of {d_\psi} in the above exercise ensure that {d_\psi} is indeed a pseudometric.

To get good metrics (such as Gleason metrics) on groups {G}, it thus suffices to obtain test functions {\psi} 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 {\mu} on {G}. The second trick is to obtain low-regularity test functions by means of a metric-like object on {G}. This latter trick may seem circular, as our whole objective is to get a metric on {G} 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.

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Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group {G} 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 {G} to a “{C^{1,1}}” condition, namely that there was an open neighbourhood of {G} that was isomorphic (as a local group) to an open subset {V} of a Euclidean space {{\bf R}^d} with identity element {0}, and with group operation {\ast} obeying the asymptotic

\displaystyle  x \ast y = x + y + O(|x| |y|)

for sufficiently small {x,y}. We will call such local groups {(V,\ast)} {C^{1,1}} local groups.

We now reduce the regularity hypothesis further, to one in which there is no explicit Euclidean space that is initially attached to {G}. Of course, Lie groups are still locally Euclidean, so if the hypotheses on {G} 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 {G} is embedded or immersed in. A trivial example of this is provided by the following basic fact from linear algebra:

Lemma 1 If {V} is a finite-dimensional vector space (i.e. it is isomorphic to {{\bf R}^d} for some {d}), and {W} is a linear subspace of {V}, then {W} 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 {S} of a {d}-dimensional smooth manifold {M} is a {d'}-dimensional smooth (embedded) submanifold of {M} for some {0 \leq d' \leq d} if for every point {x \in S} there is a smooth coordinate chart {\phi: U \rightarrow V} of a neighbourhood {U} of {x} in {M} that maps {x} to {0}, such that {\phi(U \cap S) = V \cap {\bf R}^{d'}}, where we identify {{\bf R}^{d'} \equiv {\bf R}^{d'} \times \{0\}^{d-d'}} with a subspace of {{\bf R}^d}. Informally, {S} locally sits inside {M} the same way that {{\bf R}^{d'}} sits inside {{\bf R}^d}.

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

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

Exercise 1 Let {H} be a subgroup of a locally compact group {G}. Show that {H} is closed in {G} 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 {V} is a finite-dimensional vector space, and {W} is another vector space with an injective linear transformation {\rho: W \rightarrow V} from {W} to {V}, then {W} is also a finite-dimensional vector space.

Here is the non-linear version:

Theorem 4 (von Neumann’s theorem) If {G} is a Lie group, and {H} is a locally compact group with an injective continuous homomorphism {\rho: H \rightarrow G}, then {H} also has the structure of a Lie group.

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

Example 1 Let {G = ({\bf R}/{\bf Z})^2} be the two-dimensional torus, let {H = {\bf R}}, and let {\rho: H \rightarrow G} be the map {\rho(x) := (x,\alpha x)}, where {\alpha \in {\bf R}} is a fixed real number. Then {\rho} is a continuous homomorphism which is locally injective, and is even globally injective if {\alpha} is irrational, and so Theorem 4 is consistent with the fact that {H} is a Lie group. On the other hand, note that when {\alpha} is irrational, then {\rho(H)} 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 {H} with a faithful linear representation, i.e. a continuous injective homomorphism from {H} into a linear group such as {GL_n({\bf R})} or {GL_n({\bf C})}, 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 {V} be a locally compact Hausdorff topological vector space. Then {V} is isomorphic (as a topological vector space) to {{\bf R}^d} for some finite {d}.

Remark 1 The Banach-Alaoglu theorem asserts that in a normed vector space {V}, the closed unit ball in the dual space {V^*} is always compact in the weak-* topology. Of course, this dual space {V^*} 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 {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

Exercise 2 Let {G} be a topological group that contains a neighbourhood of the identity isomorphic to a {C^{1,1}} local group. Show that {G} admits at least one Gleason metric.

Theorem 7 (Building Lie structure from Gleason metrics) Let {G} be a locally compact group that has a Gleason metric. Then {G} 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 {G}, the one-parameter subgroups are in one-to-one correspondence with the elements of the Lie algebra {{\mathfrak g}}, which is a vector space. In a general topological group {G}, the concept of a one-parameter subgroup (i.e. a continuous homomorphism from {{\bf R}} to {G}) 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 {\exp: \phi \mapsto \phi(1)} 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 {G, H} be locally compact groups. For technical reasons we assume that {G, H} are both {\sigma}-compact and metrisable.

  • (i) (Open mapping theorem) Show that if {\phi: G \rightarrow H} 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 {\phi: G \rightarrow H} is closed (i.e. its graph {\{ (g, \phi(g)): g \in G \}} is a closed subset of {G \times H}), 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 {\phi: G \rightarrow H} be a homomorphism, and let {\rho: H \rightarrow K} be a continuous injective homomorphism into another Hausdorff topological group {K}. Show that {\phi} is continuous if and only if {\rho \circ \phi} 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.)

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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 {n \times n} complex matrix {A \in M_n({\bf C})}, define the matrix exponential {\exp(A)} of {A} by the formula

\displaystyle  \exp(A) := \sum_{k=0}^\infty \frac{A^k}{k!} = 1 + A + \frac{1}{2!} A^2 + \frac{1}{3!} A^3 + \ldots

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

Exercise 1 Show that the map {A \mapsto \exp(A)} is a real analytic (and even complex analytic) map from {M_n({\bf C})} to {M_n({\bf C})}, and obeys the restricted homomorphism property

\displaystyle  \exp(sA) \exp(tA) = \exp((s+t)A) \ \ \ \ \ (1)

for all {A \in M_n({\bf C})} and {s,t \in {\bf C}}.

Proposition 1 (Rigidity and structure of matrix homomorphisms) Let {n} be a natural number. Let {GL_n({\bf C})} be the group of invertible {n \times n} complex matrices. Let {\Phi: {\bf R} \rightarrow GL_n({\bf C})} be a map obeying two properties:

  • (Group-like object) {\Phi} is a homomorphism, thus {\Phi(s) \Phi(t) = \Phi(s+t)} for all {s,t \in {\bf R}}.
  • (Weak regularity) The map {t \mapsto \Phi(t)} is continuous.

Then:

  • (Strong regularity) The map {t \mapsto \Phi(t)} is smooth (i.e. infinitely differentiable). In fact it is even real analytic.
  • (Lie-type structure) There exists a (unique) complex {n \times n} matrix {A} such that {\Phi(t) = \exp(tA)} for all {t \in {\bf R}}.

Proof: Let {\Phi} be as above. Let {\epsilon > 0} be a small number (depending only on {n}). By the homomorphism property, {\Phi(0) = 1} (where we use {1} here to denote the identity element of {GL_n({\bf C})}), and so by continuity we may find a small {t_0>0} such that {\Phi(t) = 1 + O(\epsilon)} for all {t \in [-t_0,t_0]} (we use some arbitrary norm here on the space of {n \times n} matrices, and allow implied constants in the {O()} notation to depend on {n}).

The map {A \mapsto \exp(A)} is real analytic and (by the inverse function theorem) is a diffeomorphism near {0}. Thus, by the inverse function theorem, we can (if {\epsilon} is small enough) find a matrix {B} of size {B = O(\epsilon)} such that {\Phi(t_0) = \exp(B)}. By the homomorphism property and (1), we thus have

\displaystyle  \Phi(t_0/2)^2 = \Phi(t_0) = \exp(B) = \exp(B/2)^2.

On the other hand, by another application of the inverse function theorem we see that the squaring map {A \mapsto A^2} is a diffeomorphism near {1} in {GL_n({\bf C})}, and thus (if {\epsilon} is small enough)

\displaystyle  \Phi(t_0/2) = \exp(B/2).

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

\displaystyle  \Phi(t_0/2^k) = \exp(B/2^k)

for all {k = 0,1,2,\ldots}. By the homomorphism property and (1) we thus have

\displaystyle  \Phi(qt_0) = \exp(qB)

whenever {q} is a dyadic rational, i.e. a rational of the form {a/2^k} for some integer {a} and natural number {k}. By continuity we thus have

\displaystyle  \Phi(st_0) = \exp(sB)

for all real {s}. Setting {A := B/t_0} we conclude that

\displaystyle  \Phi(t) = \exp(tA)

for all real {t}, which gives existence of the representation and also real analyticity and smoothness. Finally, uniqueness of the representation {\Phi(t) = \exp(tA)} follows from the identity

\displaystyle  A = \frac{d}{dt} \exp(tA)|_{t=0}.

\Box

Exercise 2 Generalise Proposition 1 by replacing the hypothesis that {\Phi} is continuous with the hypothesis that {\Phi} 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 {{\bf R}} and range {M_n({\bf C})}; 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 {G} be an object with the following properties:

  • (Group-like object) {G} is a group.
  • (Discreteness) {G} is finite.
  • (Lie-type structure) {G} is contained in {U_n({\bf C})} (the group of unitary {n \times n} matrices) for some {n}.

Then there is a subgroup {G'} of {G} such that

  • ({G'} is close to {G}) The index {|G/G'|} of {G'} in {G} is {O_n(1)} (i.e. bounded by {C_n} for some quantity {C_n} depending only on {n}).
  • (Nilpotent-type structure) {G'} is abelian.

A key observation in the proof of Jordan’s theorem is that if two unitary elements {g, h \in U_n({\bf C})} are close to the identity, then their commutator {[g,h] = g^{-1}h^{-1}gh} is even closer to the identity (in, say, the operator norm {\| \|_{op}}). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have

\displaystyle  \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}

\displaystyle  = \| (g-1)(h-1) - (h-1)(g-1) \|_{op}

and so by the triangle inequality

\displaystyle  \| [g,h] - 1 \|_{op} \leq 2 \|g-1\|_{op} \|h-1\|_{op}. \ \ \ \ \ (2)

Now we can prove Jordan’s theorem.

Proof: We induct on {n}, the case {n=1} being trivial. Suppose first that {G} contains a central element {g} which is not a multiple of the identity. Then, by definition, {G} is contained in the centraliser {Z(g)} of {g}, which by the spectral theorem is isomorphic to a product {U_{n_1}({\bf C}) \times \ldots \times U_{n_k}({\bf C})} of smaller unitary groups. Projecting {G} to each of these factor groups and applying the induction hypothesis, we obtain the claim.

Thus we may assume that {G} contains no central elements other than multiples of the identity. Now pick a small {\epsilon > 0} (one could take {\epsilon=\frac{1}{10d}} in fact) and consider the subgroup {G'} of {G} generated by those elements of {G} that are within {\epsilon} of the identity (in the operator norm). By considering a maximal {\epsilon}-net of {G} we see that {G'} has index at most {O_{n,\epsilon}(1)} in {G}. By arguing as before, we may assume that {G'} has no central elements other than multiples of the identity.

If {G'} consists only of multiples of the identity, then we are done. If not, take an element {g} of {G'} 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 {G} is finite). By (2), we see that if {\epsilon} is sufficiently small depending on {n}, and if {h} is one of the generators of {G'}, then {[g,h]} lies in {G'} and is closer to the identity than {g}, and is thus a multiple of the identity. On the other hand, {[g,h]} has determinant {1}. Given that it is so close to the identity, it must therefore be the identity (if {\epsilon} is small enough). In other words, {g} is central in {G'}, 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. \Box

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 {G} is a finite subgroup of {GL_n({\bf C})} rather than of {U_n({\bf C})}. (Hint: The elements of {G} are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of {{\bf C}^n}. However, if one averages that inner product by the finite group {G}, one obtains a new inner product on {{\bf C}^n} that is preserved by {G}, which allows one to conjugate {G} to a subgroup of {U_n({\bf C})}. 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 {n \geq 3}, then the orthogonal group {O_n({\bf R})} cannot contain arbitrarily dense finite subgroups, in the sense that there exists an {\epsilon = \epsilon_n > 0} depending only on {n} such that for every finite subgroup {G} of {O_n({\bf R})}, there exists a ball of radius {\epsilon} in {O_n({\bf R})} (with, say, the operator norm metric) that is disjoint from {G}. What happens in the {n=2} case?

Remark 1 More precise classifications of the finite subgroups of {U_n({\bf C})} are known, particularly in low dimensions. For instance, one can show that the only finite subgroups of {SO_3({\bf R})} (which {SU_2({\bf C})} 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.

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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 {G} be a topological group which is locally Euclidean. Then {G} 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 {G} be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold {X}. Then {G} is isomorphic to a Lie group.

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

The hypothesis that the action is faithful (i.e. each non-identity group element {g \in G \backslash \{\hbox{id}\}} acts non-trivially on {X}) cannot be completely eliminated, as any group {G} will have a trivial action on any space {X}. The requirement that {G} be locally compact is similarly necessary: consider for instance the diffeomorphism group {\hbox{Diff}(S^1)} of, say, the unit circle {S^1}, which acts on {S^1} but is infinite dimensional and is not locally compact (with, say, the uniform topology). Finally, the connectedness of {X} is also important: the infinite torus {G = ({\bf R}/{\bf Z})^{\bf N}} (with the product topology) acts faithfully on the disconnected manifold {X := {\bf R}/{\bf Z} \times {\bf N}} by the action

\displaystyle  (g_n)_{n \in {\bf N}} (\theta, m) := (\theta + g_m, m).

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 {p}-adic case. Indeed, it is known that Conjecture 2 is equivalent to

Conjecture 3 (Hilbert-Smith conjecture for {p}-adic actions) It is not possible for a {p}-adic group {{\bf Z}_p} to act continuously and effectively on a connected finite-dimensional manifold {X}.

The reduction to the {p}-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 {X} to be close to the identity. I record this argument (which appears for instance in this paper of Lee) below the fold also.

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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 {G} be a topological group which is locally Euclidean. Then {G} 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 {G} is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood {U} of the identity in {G} that contains no subgroups of {G} other than the trivial subgroup {\{ \hbox{id}\}}.

Definition 3 (Gleason metric) Let {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then

    \displaystyle  \|g^n\| \geq \frac{1}{C} n \|g\|. \ \ \ \ \ (1)

  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (2)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

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

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

Theorem 5 (Gleason’s lemma) Let {G} be a locally compact NSS group. Then {G} 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 {G} by some amount:

Proposition 6 (From locally compact to metrisable) Let {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is locally compact and metrisable.

For any open neighbourhood {U} of the identity in {G}, let {Q(U)} be the union of all the subgroups of {G} that are contained in {U}. (Thus, for instance, {G} is NSS if and only if {Q(U)} is trivial for all sufficiently small {U}.)

Proposition 7 (From metrisable to subgroup trapping) Let {G} be a locally compact metrisable group. Then {G} has the subgroup trapping property: for every open neighbourhood {U} of the identity, there exists another open neighbourhood {V} of the identity such that {Q(V)} generates a subgroup {\langle Q(V) \rangle} contained in {U}.

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

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

Proposition 10 (From escape to the commutator estimate) Let {G} be a locally compact group with a left-invariant metric {d} that obeys the escape property (1). Then {d} 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 {G} by starting with a suitable “bump function” {\phi \in C_c(G)} (i.e. a continuous, compactly supported function from {G} to {{\bf R}}) and pulling back the metric structure on {C_c(G)} by using the translation action {\tau_g \phi(x) := \phi(g^{-1} x)}, thus creating a (semi-)metric

\displaystyle  d_\phi( g, h ) := \| \tau_g \phi - \tau_h \phi \|_{C_c(G)} := \sup_{x \in G} |\phi(g^{-1} x) - \phi(h^{-1} x)|. \ \ \ \ \ (3)

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 {d_\phi(g,h) = \|g^{-1} h \|_\phi = \| h^{-1} g \|_\phi}, where

\displaystyle  \| g \|_\phi = d_\phi(g,\hbox{id}) = \| \partial_g \phi \|_{C_c(G)}

where {\partial_g} is the difference operator {\partial_g = 1 - \tau_g},

\displaystyle  \partial_g \phi(x) = \phi(x) - \phi(g^{-1} x).

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 {\phi} 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 {\phi} to have “{C^{1,1}} regularity” with respect to the “right” smooth structure on {G} By {C^{1,1}} regularity, we mean here something like a bound of the form

\displaystyle  \| \partial_g \partial_h \phi \|_{C_c(G)} \ll \|g\|_\phi \|h\|_\phi \ \ \ \ \ (4)

for all {g,h \in G}. Here we use the usual asymptotic notation, writing {X \ll Y} or {X=O(Y)} if {X \leq CY} for some constant {C} (which can vary from line to line).

The following lemma illustrates how {C^{1,1}} regularity can be used to build Gleason metrics.

Lemma 11 Suppose that {\phi \in C_c(G)} obeys (4). Then the (semi-)metric {d_\phi} (and associated (semi-)norm {\|\|_\phi}) obey the escape property (1) and the commutator property (2).

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

\displaystyle  \tau_{[g,h]} = \tau_{hg}^{-1} \tau_{gh}

whence

\displaystyle  \partial_{[g,h]} = \tau_{hg}^{-1} ( \tau_{hg} - \tau_{gh} )

\displaystyle  = \tau_{hg}^{-1} ( \partial_h \partial_g - \partial_g \partial_h ).

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

\displaystyle  \partial_{g^n} = n \partial_g + \sum_{i=0}^{n-1} \partial_g \partial_{g^i}

for any {g \in G} and natural number {n}, and thus by the triangle inequality

\displaystyle  \| g^n \|_\phi = n \| g \|_\phi + O( \sum_{i=0}^{n-1} \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} ). \ \ \ \ \ (5)

But from (4) (and the triangle inequality) we have

\displaystyle  \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} \ll \|g\|_\phi \|g^i \|_\phi \ll i \|g\|_\phi^2

and thus we have the “Taylor expansion”

\displaystyle  \|g^n\|_\phi = n \|g\|_\phi + O( n^2 \|g\|_\phi^2 )

which gives (1). \Box

It remains to obtain {\phi} that have the desired {C^{1,1}} 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 “{C^{0,1}} 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 {\mu} on the locally compact group {G}. Here we exploit the basic fact that the convolution

\displaystyle  f_1 * f_2(x) := \int_G f_1(y) f_2(y^{-1} x)\ d\mu(y) \ \ \ \ \ (6)

of two functions {f_1,f_2 \in C_c(G)} tends to be smoother than either of the two factors {f_1,f_2}. This is easiest to see in the abelian case, since in this case we can distribute derivatives according to the law

\displaystyle  \partial_g (f_1 * f_2) = (\partial_g f_1) * f_2 = f_1 * (\partial_g f_2),

which suggests that the order of “differentiability” of {f_1*f_2} should be the sum of the orders of {f_1} and {f_2} 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.

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Let {G} be a Lie group with Lie algebra {{\mathfrak g}}. As is well known, the exponential map {\exp: {\mathfrak g} \rightarrow G} is a local homeomorphism near the identity. As such, the group law on {G} can be locally pulled back to an operation {*: U \times U \rightarrow {\mathfrak g}} defined on a neighbourhood {U} of the identity in {G}, defined as

\displaystyle  x * y := \log( \exp(x) \exp(y) )

where {\log} 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 {x*y} is provided by the Baker-Campbell-Hausdorff (BCH) formula

\displaystyle  x*y = x+y+ \frac{1}{2} [x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{12}[y,[x,y]] + \ldots

for all sufficiently small {x,y}, where {[,]: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}} is the Lie bracket. More explicitly, one has the Baker-Campbell-Hausdorff-Dynkin formula

\displaystyle  x * y = x + \int_0^1 F( \hbox{Ad}_x \hbox{Ad}_{ty} ) y\ dt \ \ \ \ \ (1)

for all sufficiently small {x,y}, where {\hbox{Ad}_x = \exp( \hbox{ad}_x )}, {\hbox{ad}_x: {\bf R}^d \rightarrow {\bf R}^d} is the adjoint representation {\hbox{ad}_x(y) := [x,y]}, and {F} is the function

\displaystyle  F( t ) := \frac{t \log t}{t-1}

which is real analytic near {t=1} 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 {C^2} 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 {C^{1,1}} 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 ({C^{1,1}} Baker-Campbell-Hausdorff formula) Let {{\bf R}^d} be a finite-dimensional vector space, and suppose one has a continuous operation {*: U \times U \rightarrow {\bf R}^d} defined on a neighbourhood {U} around the origin, which obeys the following three axioms:

  • (Approximate additivity) For {x,y} sufficiently close to the origin, one has

    \displaystyle  x*y = x+y+O(|x| |y|). \ \ \ \ \ (2)

    (In particular, {0*x=x*0=x} for {x} sufficiently close to the origin.)

  • (Associativity) For {x,y,z} sufficiently close to the origin, {(x*y)*z = x*(y*z)}.
  • (Radial homogeneity) For {x} sufficiently close to the origin, one has

    \displaystyle  (sx) * (tx) = (s+t)x \ \ \ \ \ (3)

    for all {s,t \in [-1,1]}. (In particular, {x * (-x) = (-x) * x = 0} for all {x} 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 {\hbox{Ad}_x} appropriately) in this setting; see below the fold.

The reason that we call this a {C^{1,1}} Baker-Campbell-Hausdorff formula is that if the group operation {*} has {C^{1,1}} regularity, and has {0} 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 {C^{1,1}} 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).

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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 {G} be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then {G} 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 {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} 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 {G} is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood {U} of the identity in {G} that contains no subgroups of {G} other than the trivial subgroup {\{ \hbox{id}\}}.

An equivalent definition of an NSS group is one which has an open neighbourhood {U} of the identity that every non-identity element {g \in G \backslash \{\hbox{id}\}} escapes in finite time, in the sense that {g^n \not \in U} for some positive integer {n}. 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 {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

For instance, the unitary group {U(n)} with the operator norm metric {d(g,h) := \|g-h\|_{op}} can easily verified to be a Gleason metric, with the commutator estimate (1) coming from the inequality

\displaystyle  \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}

\displaystyle  = \| (g-1) (h-1) - (h-1) (g-1) \|_{op}

\displaystyle  \leq 2 \|g-1\|_{op} \|g-1\|_{op}.

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 {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is NSS, locally compact, and metrisable.

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

Theorem 7 (Building a Lie structure) Let {G} be a locally compact group with a Gleason metric. Then {G} 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 {L(G)} of one-parameter subgroups of {G} enough of a structure that it can serve as a proxy for the “Lie algebra” of {G}; 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 {G}, whose image is a Lie group by von Neumann’s theorem; the kernel is essentially the centre of {G}, 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 {g^n h^n} 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 {C^{1,1}} structure on the group {G}, 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.

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We recall Brouwer’s famous fixed point theorem:

Theorem 1 (Brouwer fixed point theorem) Let {f: B^n \rightarrow B^n} be a continuous function on the unit ball {B^n := \{ x \in {\bf R}^n: \|x\| \leq 1 \}} in a Euclidean space {{\bf R}^n}. Then {f} has at least one fixed point, thus there exists {x \in B^n} with {f(x)=x}.

This theorem has many proofs, most of which revolve (either explicitly or implicitly) around the notion of the degree of a continuous map {f: S^{n-1} \rightarrow S^{n-1}} of the unit sphere {S^{n-1} := \{ x \in {\bf R}^n: \|x\|=1\}} 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 {\pi_{n-1}(S^{n-1})} 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 {U} be an open subset of {{\bf R}^n}, and let {f: U \rightarrow {\bf R}^n} be a continuous injective map. Then {f(U)} is also open.

This theorem in turn has an important corollary:

Corollary 3 (Topological invariance of dimension) If {n > m}, and {U} is a non-empty open subset of {{\bf R}^n}, then there is no continuous injective mapping from {U} to {{\bf R}^m}. In particular, {{\bf R}^n} and {{\bf R}^m} 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 {{\bf R}^m} to {{\bf R}^n} for {n>m}, 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 {n} or {m} (for instance, by noting that removing a point from {{\bf R}^n} will disconnect {{\bf R}^n} when {n=1}, but not for {n>1}), 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 {F: B^n \rightarrow S^{n-1}} that was the identity on {S^{n-1}} (indeed, one could take {F(x)} to be the first point in which the ray from {f(x)} through {x} hits {S^{n-1}}). If one then considered the function {G: B^n \rightarrow {\bf R}^n} defined by {G(x) := (1+\|x\|) F(x)}, then this would be a continuous function which avoids the interior of {B^n}, but which maps the origin {0} to a point on the sphere {S^{n-1}} (and maps {S^{n-1}} to the dilate {2 \cdot S^{n-1}}). This could conceivably be a counterexample to Theorem 2, except that {G} 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 {{\bf R}^n}, 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 {{\bf R}^n}, 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 {{\bf R}^n}, {{\bf R}^m} 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.

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