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I recently finished the first draft of the the first of my books, entitled “Hilbert’s fifth problem and related topics“, based on the lecture notes for my graduate course of the same name.    The PDF of this draft is available here.  As always, comments and corrections are welcome.

This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other sets of lecture notes on this blog). When reviewing the zeroth set of notes in particular, I found that I had made a claim (Proposition 11 from those notes) which asserted, roughly speaking, that any sufficiently large nilprogression was an approximate group, and promised to prove it later in the course when we had developed the ability to calculate efficiently in nilpotent groups. As it turned out, I managed finish the course without the need to develop these calculations, and so the proposition remained unproven. In order to rectify this, I will use this post to lay out some of the basic algebra of nilpotent groups, and use it to prove the above proposition, which turns out to be a bit tricky. (In my paper with Breuillard and Green, we avoid the need for this proposition by restricting attention to a special type of nilprogression, which we call a nilprogression in ${C}$-normal form, for which the computations are simpler.)

There are several ways to think about nilpotent groups; for instance one can use the model example of the Heisenberg group

$\displaystyle H(R) :=\begin{pmatrix} 1 & R & R \\ 0 & 1 & R\\ 0 & 0 & 1 \end{pmatrix}$

over an arbitrary ring ${R}$ (which need not be commutative), or more generally any matrix group consisting of unipotent upper triangular matrices, and view a general nilpotent group as being an abstract generalisation of such concrete groups. (In the case of nilpotent Lie groups, at least, this is quite an accurate intuition, thanks to Engel’s theorem.) Or, one can adopt a Lie-theoretic viewpoint and try to think of nilpotent groups as somehow arising from nilpotent Lie algebras; this intuition is rigorous when working with nilpotent Lie groups (at least when the characteristic is large, in order to avoid issues coming from the denominators in the Baker-Campbell-Hausdorff formula), but also retains some conceptual value in the non-Lie setting. In particular, nilpotent groups (particularly finitely generated ones) can be viewed in some sense as “nilpotent Lie groups over ${{\bf Z}}$“, even though Lie theory does not quite work perfectly when the underlying scalars merely form an integral domain instead of a field.

Another point of view, which arises naturally both in analysis and in algebraic geometry, is to view nilpotent groups as modeling “infinitesimal” perturbations of the identity, where the infinitesimals have a certain finite order. For instance, given a (not necessarily commutative) ring ${R}$ without identity (representing all the “small” elements of some larger ring or algebra), we can form the powers ${R^j}$ for ${j=1,2,\ldots}$, defined as the ring generated by ${j}$-fold products ${r_1 \ldots r_j}$ of elements ${r_1,\ldots,r_j}$ in ${R}$; this is an ideal of ${R}$ which represents the elements which are “${j^{th}}$ order” in some sense. If one then formally adjoins an identity ${1}$ onto the ring ${R}$, then for any ${s \geq 1}$, the multiplicative group ${G := 1+R \hbox{ mod } R^{s+1}}$ is a nilpotent group of step at most ${s}$. For instance, if ${R}$ is the ring of strictly upper ${s \times s}$ matrices (over some base ring), then ${R^{s+1}}$ vanishes and ${G}$ becomes the group of unipotent upper triangular matrices over the same ring, thus recovering the previous matrix-based example. In analysis applications, ${R}$ might be a ring of operators which are somehow of “order” ${O(\epsilon)}$ or ${O(\hbar)}$ for some small parameter ${\epsilon}$ or ${\hbar}$, and one wishes to perform Taylor expansions up to order ${O(\epsilon^s)}$ or ${O(\hbar^s)}$, thus discarding (i.e. quotienting out) all errors in ${R^{s+1}}$.

From a dynamical or group-theoretic perspective, one can also view nilpotent groups as towers of central extensions of a trivial group. Finitely generated nilpotent groups can also be profitably viewed as a special type of polycylic group; this is the perspective taken in this previous blog post. Last, but not least, one can view nilpotent groups from a combinatorial group theory perspective, as being words from some set of generators of various “degrees” subject to some commutation relations, with commutators of two low-degree generators being expressed in terms of higher degree objects, and all commutators of a sufficiently high degree vanishing. In particular, generators of a given degree can be moved freely around a word, as long as one is willing to generate commutator errors of higher degree.

With this last perspective, in particular, one can start computing in nilpotent groups by adopting the philosophy that the lowest order terms should be attended to first, without much initial concern for the higher order errors generated in the process of organising the lower order terms. Only after the lower order terms are in place should attention then turn to higher order terms, working successively up the hierarchy of degrees until all terms are dealt with. This turns out to be a relatively straightforward philosophy to implement in many cases (particularly if one is not interested in explicit expressions and constants, being content instead with qualitative expansions of controlled complexity), but the arguments are necessarily recursive in nature and as such can become a bit messy, and require a fair amount of notation to express precisely. So, unfortunately, the arguments here will be somewhat cumbersome and notation-heavy, even if the underlying methods of proof are relatively simple.

In the last set of notes, we obtained the following structural theorem concerning approximate groups:

Theorem 1 Let ${A}$ be a finite ${K}$-approximate group. Then there exists a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ contained in ${A^4}$, such that ${A}$ is covered by ${O_K(1)}$ left-translates of ${P}$ (and hence also by ${O_K(1)}$ right-translates of ${P}$).

Remark 1 Under some mild additional hypotheses (e.g. if the dimensions of ${P}$ are sufficiently large, or if ${P}$ is placed in a certain “normal form”, details of which may be found in this paper), a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ will be an ${O_K(1)}$-approximate group, thus giving a partial converse to Theorem 1. (It is not quite a full converse though, even if one works qualitatively and forgets how the constants depend on ${K}$: if ${A}$ is covered by a bounded number of left- and right-translates ${gP, Pg}$ of ${P}$, one needs the group elements ${g}$ to “approximately normalise” ${P}$ in some sense if one wants to then conclude that ${A}$ is an approximate group.) The mild hypotheses alluded to above can be enforced in the statement of the theorem, but we will not discuss this technicality here, and refer the reader to the above-mentioned paper for details.

By placing the coset nilprogression in a virtually nilpotent group, we have the following corollary in the global case:

Corollary 2 Let ${A}$ be a finite ${K}$-approximate group in an ambient group ${G}$. Then ${A}$ is covered by ${O_K(1)}$ left cosets of a virtually nilpotent subgroup ${G'}$ of ${G}$.

In this final set of notes, we give some applications of the above results. The first application is to replace “${K}$-approximate group” by “sets of bounded doubling”:

Proposition 3 Let ${A}$ be a finite non-empty subset of a (global) group ${G}$ such that ${|A^2| \leq K |A|}$. Then there exists a coset nilprogression ${P}$ of rank and step ${O_K(1)}$ and cardinality ${|P| \gg_K |A|}$ such that ${A}$ can be covered by ${O_K(1)}$ left-translates of ${P}$, and also by ${O_K(1)}$ right-translates of ${P}$.

We will also establish (a strengthening of) a well-known theorem of Gromov on groups of polynomial growth, as promised back in Notes 0, as well as a variant result (of a type known as a “generalised Margulis lemma”) controlling the almost stabilisers of discrete actions of isometries.

The material here is largely drawn from my recent paper with Emmanuel Breuillard and Ben Green.

A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat vague and imprecise, and their interpretation depends on the context and also on one’s choice of normalisations, but if one uses a “macroscopic” normalisation, “macroscopic” scales correspond to scales that are comparable to unit size (i.e. bounded above and below by absolute constants), while “microscopic” scales are much smaller, being the minimal scale at which nontrivial behaviour occurs. (Other normalisations are possible, e.g. making the microscopic scale a unit scale, and letting the macroscopic scale go off to infinity; for instance, such a normalisation is often used, at least initially, in the study of groups of polynomial growth. However, for the theory of approximate groups, a macroscopic scale normalisation is more convenient.)

One can also consider “mesoscopic” scales which are intermediate between microscopic and macroscopic scales, or large-scale behaviour at scales that go off to infinity (and in particular are larger than the macroscopic range of scales), although the behaviour of these scales will not be the main focus of this post. Finally, one can divide the macroscopic scales into “local” macroscopic scales (less than ${\epsilon}$ for some small but fixed ${\epsilon>0}$) and “global” macroscopic scales (scales that are allowed to be larger than a given large absolute constant ${C}$). For instance, given a finite approximate group ${A}$:

• Sets such as ${A^m}$ for some fixed ${m}$ (e.g. ${A^{10}}$) can be considered to be sets at a global macroscopic scale. Sending ${m}$ to infinity, one enters the large-scale regime.
• Sets such as the sets ${S}$ that appear in the Sanders lemma from the previous set of notes (thus ${S^m \subset A^4}$ for some fixed ${m}$, e.g. ${m=100}$) can be considered to be sets at a local macroscopic scale. Sending ${m}$ to infinity, one enters the mesoscopic regime.
• The non-identity element ${u}$ of ${A}$ that is “closest” to the identity in some suitable metric (cf. the proof of Jordan’s theorem from Notes 0) would be an element associated to the microscopic scale. The orbit ${u, u^2, u^3, \ldots}$ starts out at microscopic scales, and (assuming some suitable “escape” axioms) will pass through mesoscopic scales and finally entering the macroscopic regime. (Beyond this point, the orbit may exhibit a variety of behaviours, such as periodically returning back to the smaller scales, diverging off to ever larger scales, or filling out a dense subset of some macroscopic set; the escape axioms we will use do not exclude any of these possibilities.)

For comparison, in the theory of locally compact groups, properties about small neighbourhoods of the identity (e.g. local compactness, or the NSS property) would be properties at the local macroscopic scale, whereas the space ${L(G)}$ of one-parameter subgroups can be interpreted as an object at the microscopic scale. The exponential map then provides a bridge connecting the microscopic and macroscopic scales.

We return now to approximate groups. The macroscopic structure of these objects is well described by the Hrushovski Lie model theorem from the previous set of notes, which informally asserts that the macroscopic structure of an (ultra) approximate group can be modeled by a Lie group. This is already an important piece of information about general approximate groups, but it does not directly reveal the full structure of such approximate groups, because these Lie models are unable to see the microscopic behaviour of these approximate groups.

To illustrate this, let us review one of the examples of a Lie model of an ultra approximate group, namely Exercise 28 from Notes 7. In this example one studied a “nilbox” from a Heisenberg group, which we rewrite here in slightly different notation. Specifically, let ${G}$ be the Heisenberg group

$\displaystyle G := \{ (a,b,c): a,b,c \in {\bf Z} \}$

with group law

$\displaystyle (a,b,c) \ast (a',b',c') := (a+a', b+b', c+c'+ab') \ \ \ \ \ (1)$

and let ${A = \prod_{n \rightarrow \alpha} A_n}$, where ${A_n \subset G}$ is the box

$\displaystyle A_n := \{ (a,b,c) \in G: |a|, |b| \leq n; |c| \leq n^{10} \};$

thus ${A}$ is the nonstandard box

$\displaystyle A := \{ (a,b,c) \in {}^* G: |a|, |b| \leq N; |c| \leq N^{10} \}$

where ${N := \lim_{n \rightarrow \alpha} n}$. As the above exercise establishes, ${A \cup A^{-1}}$ is an ultra approximate group with a Lie model ${\pi: \langle A \rangle \rightarrow {\bf R}^3}$ given by the formula

$\displaystyle \pi( a, b, c ) := ( \hbox{st} \frac{a}{N}, \hbox{st} \frac{b}{N}, \hbox{st} \frac{c}{N^{10}} )$

for ${a,b = O(N)}$ and ${c = O(N^{10})}$. Note how the nonabelian nature of ${G}$ (arising from the ${ab'}$ term in the group law (1)) has been lost in the model ${{\bf R}^3}$, because the effect of that nonabelian term on ${\frac{c}{N^{10}}}$ is only ${O(\frac{N^2}{N^8})}$ which is infinitesimal and thus does not contribute to the standard part. In particular, if we replace ${G}$ with the abelian group ${G' := \{(a,b,c): a,b,c \in {\bf Z} \}}$ with the additive group law

$\displaystyle (a,b,c) \ast' (a',b',c') := (a+a',b+b',c+c')$

and let ${A'}$ and ${\pi'}$ be defined exactly as with ${A}$ and ${\pi}$, but placed inside the group structure of ${G'}$ rather than ${G}$, then ${A \cup A^{-1}}$ and ${A' \cup (A')^{-1}}$ are essentially “indistinguishable” as far as their models by ${{\bf R}^3}$ are concerned, even though the latter approximate group is abelian and the former is not. The problem is that the nonabelian-ness in the former example is so microscopic that it falls entirely inside the kernel of ${\pi}$ and is thus not detected at all by the model.

The problem of not being able to “see” the microscopic structure of a group (or approximate group) also was a key difficulty in the theory surrounding Hilbert’s fifth problem that was discussed in previous notes. A key tool in being able to resolve such structure was to build left-invariant metrics ${d}$ (or equivalently, norms ${\| \|}$) on one’s group, which obeyed useful “Gleason axioms” such as the commutator axiom

$\displaystyle \| [g,h] \| \ll \|g\| \|h\| \ \ \ \ \ (2)$

for sufficiently small ${g,h}$, or the escape axiom

$\displaystyle \| g^n \| \gg |n| \|g\| \ \ \ \ \ (3)$

when ${|n| \|g\|}$ was sufficiently small. Such axioms have important and non-trivial content even in the microscopic regime where ${g}$ or ${h}$ are extremely close to the identity. For instance, in the proof of Jordan’s theorem from Notes 0, which showed that any finite unitary group ${G}$ was boundedly virtually abelian, a key step was to apply the commutator axiom (2) (for the distance to the identity in operator norm) to the most “microscopic” element of ${G}$, or more precisely a non-identity element of ${G}$ of minimal norm. The key point was that this microscopic element was virtually central in ${G}$, and as such it restricted much of ${G}$ to a lower-dimensional subgroup of the unitary group, at which point one could argue using an induction-on-dimension argument. As we shall see, a similar argument can be used to place “virtually nilpotent” structure on finite approximate groups. For instance, in the Heisenberg-type approximate groups ${A \cup A^{-1}}$ and ${A' \cup (A')^{-1}}$ discussed earlier, the element ${(0,0,1)}$ will be “closest to the origin” in a suitable sense to be defined later, and is centralised by both approximate groups; quotienting out (the orbit of) that central element and iterating the process two more times, we shall see that one can express both ${A \cup A^{-1}}$ and ${A'\cup (A')^{-1}}$ as a tower of central cyclic extensions, which in particular establishes the nilpotency of both groups.

The escape axiom (3) is a particularly important axiom in connecting the microscopic structure of a group ${G}$ to its macroscopic structure; for instance, as shown in Notes 2, this axiom (in conjunction with the closely related commutator axiom) tends to imply dilation estimates such as ${d( g^n, h^n ) \sim n d(g,h)}$ that allow one to understand the microscopic geometry of points ${g,h}$ close to the identity in terms of the (local) macroscopic geometry of points ${g^n, h^n}$ that are significantly further away from the identity.

It is thus of interest to build some notion of a norm (or left-invariant metrics) on an approximate group ${A}$ that obeys the escape and commutator axioms (while being non-degenerate enough to adequately capture the geometry of ${A}$ in some sense), in a fashion analogous to the Gleason metrics that played such a key role in the theory of Hilbert’s fifth problem. It is tempting to use the Lie model theorem to do this, since Lie groups certainly come with Gleason metrics. However, if one does this, one ends up, roughly speaking, with a norm on ${A}$ that only obeys the escape and commutator estimates macroscopically; roughly speaking, this means that one has a macroscopic commutator inequality

$\displaystyle \| [g,h] \| \ll \|g\| \|h\| + o(1)$

and a macroscopic escape property

$\displaystyle \| g^n \| \gg |n| \|g\| - o(|n|)$

but such axioms are too weak for analysis at the microscopic scale, and in particular in establishing centrality of the element closest to the identity.

Another way to proceed is to build a norm that is specifically designed to obey the crucial escape property. Given an approximate group ${A}$ in a group ${G}$, and an element ${g}$ of ${G}$, we can define the escape norm ${\|g\|_{e,A}}$ of ${g}$ by the formula

$\displaystyle \| g \|_{e,A} := \inf \{ \frac{1}{n+1}: n \in {\bf N}: g, g^2, \ldots, g^n \in A \}.$

Thus, ${\|g\|_{e,A}}$ equals ${1}$ if ${g}$ lies outside of ${A}$, equals ${1/2}$ if ${g}$ lies in ${A}$ but ${g^2}$ lies outside of ${A}$, and so forth. Such norms had already appeared in Notes 4, in the context of analysing NSS groups.

As it turns out, this expression will obey an escape axiom, as long as we place some additional hypotheses on ${A}$ which we will present shortly. However, it need not actually be a norm; in particular, the triangle inequality

$\displaystyle \|gh\|_{e,A} \leq \|g\|_{e,A} + \|h\|_{e,A}$

is not necessarily true. Fortunately, it turns out that by a (slightly more complicated) version of the Gleason machinery from Notes 4 we can establish a usable substitute for this inequality, namely the quasi-triangle inequality

$\displaystyle \|g_1 \ldots g_k \|_{e,A} \leq C (\|g_1\|_{e,A} + \ldots + \|g_k\|_{e,A}),$

where ${C}$ is a constant independent of ${k}$. As we shall see, these estimates can then be used to obtain a commutator estimate (2).

However, to do all this, it is not enough for ${A}$ to be an approximate group; it must obey two additional “trapping” axioms that improve the properties of the escape norm. We formalise these axioms (somewhat arbitrarily) as follows:

Definition 1 (Strong approximate group) Let ${K \geq 1}$. A strong ${K}$-approximate group is a finite ${K}$-approximate group ${A}$ in a group ${G}$ with a symmetric subset ${S}$ obeying the following axioms:

An ultra strong ${K}$-approximate group is an ultraproduct ${A = \prod_{n \rightarrow \alpha} A_n}$ of strong ${K}$-approximate groups.

The first trapping condition can be rewritten as

$\displaystyle \|g\|_{e,A} \leq 1000 \|g\|_{e,A^{100}}$

and the second trapping condition can similarly be rewritten as

$\displaystyle \|g\|_{e,S} \leq 10^6 K^3 \|g\|_{e,A}.$

This makes the escape norms of ${A, A^{100}}$, and ${S}$ comparable to each other, which will be needed for a number of reasons (and in particular to close a certain bootstrap argument properly). Compare this with equation (12) from Notes 4, which used the NSS hypothesis to obtain similar conclusions. Thus, one can view the strong approximate group axioms as being a sort of proxy for the NSS property.

Example 1 Let ${N}$ be a large natural number. Then the interval ${A = [-N,N]}$ in the integers is a ${2}$-approximate group, which is also a strong ${2}$-approximate group (setting ${S = [10^{-6} N, 10^{-6} N]}$, for instance). On the other hand, if one places ${A}$ in ${{\bf Z}/5N{\bf Z}}$ rather than in the integers, then the first trapping condition is lost and one is no longer a strong ${2}$-approximate group. Also, if one remains in the integers, but deletes a few elements from ${A}$, e.g. deleting ${\pm \lfloor 10^{-10} N\rfloor}$ from ${A}$), then one is still a ${O(1)}$-approximate group, but is no longer a strong ${O(1)}$-approximate group, again because the first trapping condition is lost.

A key consequence of the Hrushovski Lie model theorem is that it allows one to replace approximate groups by strong approximate groups:

Exercise 1 (Finding strong approximate groups)

• (i) Let ${A}$ be an ultra approximate group with a good Lie model ${\pi: \langle A \rangle \rightarrow L}$, and let ${B}$ be a symmetric convex body (i.e. a convex open bounded subset) in the Lie algebra ${{\mathfrak l}}$. Show that if ${r>0}$ is a sufficiently small standard number, then there exists a strong ultra approximate group ${A'}$ with

$\displaystyle \pi^{-1}(\exp(rB)) \subset A' \subset \pi^{-1}(\exp(1.1 rB)) \subset A,$

and with ${A}$ can be covered by finitely many left translates of ${A'}$. Furthermore, ${\pi}$ is also a good model for ${A'}$.

• (ii) If ${A}$ is a finite ${K}$-approximate group, show that there is a strong ${O_K(1)}$-approximate group ${A'}$ inside ${A^4}$ with the property that ${A}$ can be covered by ${O_K(1)}$ left translates of ${A'}$. (Hint: use (i), Hrushovski’s Lie model theorem, and a compactness and contradiction argument.)

The need to compare the strong approximate group to an exponentiated small ball ${\exp(rB)}$ will be convenient later, as it allows one to easily use the geometry of ${L}$ to track various aspects of the strong approximate group.

As mentioned previously, strong approximate groups exhibit some of the features of NSS locally compact groups. In Notes 4, we saw that the escape norm for NSS locally compact groups was comparable to a Gleason metric. The following theorem is an analogue of that result:

Theorem 2 (Gleason lemma) Let ${A}$ be a strong ${K}$-approximate group in a group ${G}$.

• (Symmetry) For any ${g \in G}$, one has ${\|g^{-1}\|_{e,A} = \|g\|_{e,A}}$.
• (Conjugacy bound) For any ${g, h \in A^{10}}$, one has ${\|g^h\|_{e,A} \ll \|g\|_{e,A}}$.
• (Triangle inequality) For any ${g_1,\ldots,g_k \in G}$, one has ${\|g_1 \ldots g_k \|_{e,A} \ll_K (\|g_1\|_{e,A} + \ldots + \|g_k\|_{e,A})}$.
• (Escape property) One has ${\|g^n\|_{e,A} \gg |n| \|g\|_{e,A}}$ whenever ${|n| \|g\|_{e,A} < 1}$.
• (Commutator inequality) For any ${g,h \in A^{10}}$, one has ${\| [g,h] \|_{e,A} \ll_K \|g\|_{e,A} \|h\|_{e,A}}$.

The proof of this theorem will occupy a large part of the current set of notes. We then aim to use this theorem to classify strong approximate groups. The basic strategy (temporarily ignoring a key technical issue) follows the Bieberbach-Frobenius proof of Jordan’s theorem, as given in Notes 0, is as follows.

1. Start with an (ultra) strong approximate group ${A}$.
2. From the Gleason lemma, the elements with zero escape norm form a normal subgroup of ${A}$. Quotient these elements out. Show that all non-identity elements will have positive escape norm.
3. Find the non-identity element ${g_1}$ in (the quotient of) ${A}$ of minimal escape norm. Use the commutator estimate (assuming it is inherited by the quotient) to show that ${g_1}$ will centralise (most of) this quotient. In particular, the orbit ${\langle g_1 \rangle}$ is (essentially) a central subgroup of ${\langle A \rangle}$.
4. Quotient this orbit out; then find the next non-identity element ${g_2}$ in this new quotient of ${A}$. Again, show that ${\langle g_2 \rangle}$ is essentially a central subgroup of this quotient.
5. Repeat this process until ${A}$ becomes entirely trivial. Undoing all the quotients, this should demonstrate that ${\langle A \rangle}$ is virtually nilpotent, and that ${A}$ is essentially a coset nilprogression.

There are two main technical issues to resolve to make this strategy work. The first is to show that the iterative step in the argument terminates in finite time. This we do by returning to the Lie model theorem. It turns out that each time one quotients out by an orbit of an element that escapes, the dimension of the Lie model drops by at least one. This will ensure termination of the argument in finite time.

The other technical issue is that while the quotienting out all the elements of zero escape norm eliminates all “torsion” from ${A}$ (in the sense that the quotient of ${A}$ has no non-trivial elements of zero escape norm), further quotienting operations can inadvertently re-introduce such torsion. This torsion can be re-eradicated by further quotienting, but the price one pays for this is that the final structural description of ${\langle A \rangle}$ is no longer as strong as “virtually nilpotent”, but is instead a more complicated tower alternating between (ultra) finite extensions and central extensions.

Example 2 Consider the strong ${O(1)}$-approximate group

$\displaystyle A := \{ a N^{10} + 5 b: |a| \leq N; |b| \leq N^2 \}$

in the integers, where ${N}$ is a large natural number not divisible by ${5}$. As ${{\bf Z}}$ is torsion-free, all non-zero elements of ${A}$ have positive escape norm, and the nonzero element of minimal escape norm here is ${g=5}$ (or ${g=-5}$). But if one quotients by ${\langle g \rangle}$, ${A}$ projects down to ${{\bf Z}/5{\bf Z}}$, which now has torsion (and all elements in this quotient have zero escape norm). Thus torsion has been re-introduced by the quotienting operation. (A related observation is that the intersection of ${A}$ with ${\langle g \rangle = 5{\bf Z}}$ is not a simple progression, but is a more complicated object, namely a generalised arithmetic progression of rank two.)

To deal with this issue, we will not quotient out by the entire cyclic group ${\langle g \rangle = \{g^n: n \in {\bf Z} \}}$ generated by the element ${g}$ of minimal escape norm, but rather by an arithmetic progression ${P = \{g^n: |n| \leq N\}}$, where ${N}$ is a natural number comparable to the reciprocal ${1/\|g\|_{e,A}}$ of the escape norm, as this will be enough to cut the dimension of the Lie model down by one without introducing any further torsion. Of course, this cannot be done in the category of global groups, since the arithmetic progression ${P}$ will not, in general, be a group. However, it is still a local group, and it turns out that there is an analogue of the quotient space construction in local groups. This fixes the problem, but at a cost: in order to make the inductive portion of the argument work smoothly, it is now more natural to place the entire argument inside the category of local groups rather than global groups, even though the primary interest in approximate groups ${A}$ is in the global case when ${A}$ lies inside a global group. This necessitates some technical modification to some of the preceding discussion (for instance, the Gleason-Yamabe theorem must be replaced by the local version of this theorem, due to Goldbring); details can be found in this recent paper of Emmanuel Breuillard, Ben Green, and myself, but will only be sketched here.

In the previous set of notes, we introduced the notion of an ultra approximate group – an ultraproduct ${A = \prod_{n \rightarrow\alpha} A_n}$ of finite ${K}$-approximate groups ${A_n}$ for some ${K}$ independent of ${n}$, where each ${K}$-approximate group ${A_n}$ may lie in a distinct ambient group ${G_n}$. Although these objects arise initially from the “finitary” objects ${A_n}$, it turns out that ultra approximate groups ${A}$ can be profitably analysed by means of infinitary groups ${L}$ (and in particular, locally compact groups or Lie groups ${L}$), by means of certain models ${\rho: \langle A \rangle \rightarrow L}$ of ${A}$ (or of the group ${\langle A \rangle}$ generated by ${A}$). We will define precisely what we mean by a model later, but as a first approximation one can view a model as a representation of the ultra approximate group ${A}$ (or of ${\langle A \rangle}$) that is “macroscopically faithful” in that it accurately describes the “large scale” behaviour of ${A}$ (or equivalently, that the kernel of the representation is “microscopic” in some sense). In the next section we will see how one can use “Gleason lemma” technology to convert this macroscopic control of an ultra approximate group into microscopic control, which will be the key to classifying approximate groups.

Models of ultra approximate groups can be viewed as the multiplicative combinatorics analogue of the more well known concept of an ultralimit of metric spaces, which we briefly review below the fold as motivation.

The crucial observation is that ultra approximate groups enjoy a local compactness property which allows them to be usefully modeled by locally compact groups (and hence, through the Gleason-Yamabe theorem from previous notes, by Lie groups also). As per the Heine-Borel theorem, the local compactness will come from a combination of a completeness property and a local total boundedness property. The completeness property turns out to be a direct consequence of the countable saturation property of ultraproducts, thus illustrating one of the key advantages of the ultraproduct setting. The local total boundedness property is more interesting. Roughly speaking, it asserts that “large bounded sets” (such as ${A}$ or ${A^{100}}$) can be covered by finitely many translates of “small bounded sets” ${S}$, where “small” is a topological group sense, implying in particular that large powers ${S^m}$ of ${S}$ lie inside a set such as ${A}$ or ${A^4}$. The easiest way to obtain such a property comes from the following lemma of Sanders:

Lemma 1 (Sanders lemma) Let ${A}$ be a finite ${K}$-approximate group in a (global) group ${G}$, and let ${m \geq 1}$. Then there exists a symmetric subset ${S}$ of ${A^4}$ with ${|S| \gg_{K,m} |A|}$ containing the identity such that ${S^m \subset A^4}$.

This lemma has an elementary combinatorial proof, and is the key to endowing an ultra approximate group with locally compact structure. There is also a closely related lemma of Croot and Sisask which can achieve similar results, and which will also be discussed below. (The locally compact structure can also be established more abstractly using the much more general methods of definability theory, as was first done by Hrushovski, but we will not discuss this approach here.)

By combining the locally compact structure of ultra approximate groups ${A}$ with the Gleason-Yamabe theorem, one ends up being able to model a large “ultra approximate subgroup” ${A'}$ of ${A}$ by a Lie group ${L}$. Such Lie models serve a number of important purposes in the structure theory of approximate groups. Firstly, as all Lie groups have a dimension which is a natural number, they allow one to assign a natural number “dimension” to ultra approximate groups, which opens up the ability to perform “induction on dimension” arguments. Secondly, Lie groups have an escape property (which is in fact equivalent to no small subgroups property): if a group element ${g}$ lies outside of a very small ball ${B_\epsilon}$, then some power ${g^n}$ of it will escape a somewhat larger ball ${B_1}$. Or equivalently: if a long orbit ${g, g^2, \ldots, g^n}$ lies inside the larger ball ${B_1}$, one can deduce that the original element ${g}$ lies inside the small ball ${B_\epsilon}$. Because all Lie groups have this property, we will be able to show that all ultra approximate groups ${A}$ “essentially” have a similar property, in that they are “controlled” by a nearby ultra approximate group which obeys a number of escape-type properties analogous to those enjoyed by small balls in a Lie group, and which we will call a strong ultra approximate group. This will be discussed in the next set of notes, where we will also see how these escape-type properties can be exploited to create a metric structure on strong approximate groups analogous to the Gleason metrics studied in previous notes, which can in turn be exploited (together with an induction on dimension argument) to fully classify such approximate groups (in the finite case, at least).

There are some cases where the analysis is particularly simple. For instance, in the bounded torsion case, one can show that the associated Lie model ${L}$ is necessarily zero-dimensional, which allows for a easy classification of approximate groups of bounded torsion.

Some of the material here is drawn from my recent paper with Ben Green and Emmanuel Breuillard, which is in turn inspired by a previous paper of Hrushovski.

Roughly speaking, mathematical analysis can be divided into two major styles, namely hard analysis and soft analysis. The precise distinction between the two types of analysis is imprecise (and in some cases one may use a blend the two styles), but some key differences can be listed as follows.

• Hard analysis tends to be concerned with quantitative or effective properties such as estimates, upper and lower bounds, convergence rates, and growth rates or decay rates. In contrast, soft analysis tends to be concerned with qualitative or ineffective properties such as existence and uniqueness, finiteness, measurability, continuity, differentiability, connectedness, or compactness.
• Hard analysis tends to be focused on finitary, finite-dimensional or discrete objects, such as finite sets, finitely generated groups, finite Boolean combination of boxes or balls, or “finite-complexity” functions, such as polynomials or functions on a finite set. In contrast, soft analysis tends to be focused on infinitary, infinite-dimensional, or continuous objects, such as arbitrary measurable sets or measurable functions, or abstract locally compact groups.
• Hard analysis tends to involve explicit use of many parameters such as ${\epsilon}$, ${\delta}$, ${N}$, etc. In contrast, soft analysis tends to rely instead on properties such as continuity, differentiability, compactness, etc., which implicitly are defined using a similar set of parameters, but whose parameters often do not make an explicit appearance in arguments.
• In hard analysis, it is often the case that a key lemma in the literature is not quite optimised for the application at hand, and one has to reprove a slight variant of that lemma (using a variant of the proof of the original lemma) in order for it to be suitable for applications. In contrast, in soft analysis, key results can often be used as “black boxes”, without need of further modification or inspection of the proof.
• The properties in soft analysis tend to enjoy precise closure properties; for instance, the composition or linear combination of continuous functions is again continuous, and similarly for measurability, differentiability, etc. In contrast, the closure properties in hard analysis tend to be fuzzier, in that the parameters in the conclusion are often different from the parameters in the hypotheses. For instance, the composition of two Lipschitz functions with Lipschitz constant ${K}$ is still Lipschitz, but now with Lipschitz constant ${K^2}$ instead of ${K}$. These changes in parameters mean that hard analysis arguments often require more “bookkeeping” than their soft analysis counterparts, and are less able to utilise algebraic constructions (e.g. quotient space constructions) that rely heavily on precise closure properties.

In the lectures so far, focusing on the theory surrounding Hilbert’s fifth problem, the results and techniques have fallen well inside the category of soft analysis. However, we will now turn to the theory of approximate groups, which is a topic which is traditionally studied using the methods of hard analysis. (Later we will also study groups of polynomial growth, which lies on an intermediate position in the spectrum between hard and soft analysis, and which can be profitably analysed using both styles of analysis.)

Despite the superficial differences between hard and soft analysis, though, there are a number of important correspondences between results in hard analysis and results in soft analysis. For instance, if one has some sort of uniform quantitative bound on some expression relating to finitary objects, one can often use limiting arguments to then conclude a qualitative bound on analogous expressions on infinitary objects, by viewing the latter objects as some sort of “limit” of the former objects. Conversely, if one has a qualitative bound on infinitary objects, one can often use compactness and contradiction arguments to recover uniform quantitative bounds on finitary objects as a corollary.

Remark 1 Another type of correspondence between hard analysis and soft analysis, which is “syntactical” rather than “semantical” in nature, arises by taking the proofs of a soft analysis result, and translating such a qualitative proof somehow (e.g. by carefully manipulating quantifiers) into a quantitative proof of an analogous hard analysis result. This type of technique is sometimes referred to as proof mining in the proof theory literature, and is discussed in this previous blog post (and its comments). We will however not employ systematic proof mining techniques here, although in later posts we will informally borrow arguments from infinitary settings (such as the methods used to construct Gleason metrics) and adapt them to finitary ones.

Let us illustrate the correspondence between hard and soft analysis results with a simple example.

Proposition 1 Let ${X}$ be a sequentially compact topological space, let ${S}$ be a dense subset of ${X}$, and let ${f: X \rightarrow [0,+\infty]}$ be a continuous function (giving the extended half-line ${[0,+\infty]}$ the usual order topology). Then the following statements are equivalent:

• (i) (Qualitative bound on infinitary objects) For all ${x \in X}$, one has ${f(x) < +\infty}$.
• (ii) (Quantitative bound on finitary objects) There exists ${M < +\infty}$ such that ${f(x) \leq M}$ for all ${x \in S}$.

In applications, ${S}$ is typically a (non-compact) set of “finitary” (or “finite complexity”) objects of a certain class, and ${X}$ is some sort of “completion” or “compactification” of ${S}$ which admits additional “infinitary” objects that may be viewed as limits of finitary objects.

Proof: To see that (ii) implies (i), observe from density that every point ${x}$ in ${X}$ is adherent to ${S}$, and so given any neighbourhood ${U}$ of ${x}$, there exists ${y \in S \cap U}$. Since ${f(y) \leq M}$, we conclude from the continuity of ${f}$ that ${f(x) \leq M}$ also, and the claim follows.

Conversely, to show that (i) implies (ii), we use the “compactness and contradiction” argument. Suppose for sake of contradiction that (ii) failed. Then for any natural number ${n}$, there exists ${x_n \in S}$ such that ${f(x_n) \geq n}$. (Here we have used the axiom of choice, which we will assume throughout this course.) Using sequential compactness, and passing to a subsequence if necessary, we may assume that the ${x_n}$ converge to a limit ${x \in X}$. By continuity of ${f}$, this implies that ${f(x) = +\infty}$, contradicting (i). $\Box$

Remark 2 Note that the above deduction of (ii) from (i) is ineffective in that it gives no explicit bound on the uniform bound ${M}$ in (ii). Without any further information on how the qualitative bound (i) is proven, this is the best one can do in general (and this is one of the most significant weaknesses of infinitary methods when used to solve finitary problems); but if one has access to the proof of (i), one can often finitise or proof mine that argument to extract an effective bound for ${M}$, although often the bound one obtains in the process is quite poor (particularly if the proof of (i) relied extensively on infinitary tools, such as limits). See this blog post for some related discussion.

The above simple example illustrates that in order to get from an “infinitary” statement such as (i) to a “finitary” statement such as (ii), a key step is to be able to take a sequence ${(x_n)_{n \in {\bf N}}}$ (or in some cases, a more general net ${(x_\alpha)_{\alpha \in A}}$) of finitary objects and extract a suitable infinitary limit object ${x}$. In the literature, there are three main ways in which one can extract such a limit:

• (Topological limit) If the ${x_n}$ are all elements of some topological space ${S}$ (e.g. an incomplete function space) which has a suitable “compactification” or “completion” ${X}$ (e.g. a Banach space), then (after passing to a subsequence if necessary) one can often ensure the ${x_n}$ converge in a topological sense (or in a metrical sense) to a limit ${x}$. The use of this type of limit to pass between quantitative/finitary and qualitative/infinitary results is particularly common in the more analytical areas of mathematics (such as ergodic theory, asymptotic combinatorics, or PDE), due to the abundance of useful compactness results in analysis such as the (sequential) Banach-Alaoglu theorem, Prokhorov’s theorem, the Helly selection theorem, the Arzelá-Ascoli theorem, or even the humble Bolzano-Weierstrass theorem. However, one often has to take care with the nature of convergence, as many compactness theorems only guarantee convergence in a weak sense rather than in a strong one.
• (Categorical limit) If the ${x_n}$ are all objects in some category (e.g. metric spaces, groups, fields, etc.) with a number of morphisms between the ${x_n}$ (e.g. morphisms from ${x_{n+1}}$ to ${x_n}$, or vice versa), then one can often form a direct limit ${\lim_{\rightarrow} x_n}$ or inverse limit ${\lim_{\leftarrow} x_n}$ of these objects to form a limiting object ${x}$. The use of these types of limits to connect quantitative and qualitative results is common in subjects such as algebraic geometry that are particularly amenable to categorical ways of thinking. (We have seen inverse limits appear in the discussion of Hilbert’s fifth problem, although in that context they were not really used to connect quantitative and qualitative results together.)
• (Logical limit) If the ${x_n}$ are all distinct spaces (or elements or subsets of distinct spaces), with few morphisms connecting them together, then topological and categorical limits are often unavailable or unhelpful. In such cases, however, one can still tie together such objects using an ultraproduct construction (or similar device) to create a limiting object ${\lim_{n \rightarrow \alpha} x_n}$ or limiting space ${\prod_{n \rightarrow \alpha} x_n}$ that is a logical limit of the ${x_n}$, in the sense that various properties of the ${x_n}$ (particularly those that can be phrased using the language of first-order logic) are preserved in the limit. As such, logical limits are often very well suited for the task of connecting finitary and infinitary mathematics together. Ultralimit type constructions are of course used extensively in logic (particularly in model theory), but are also popular in metric geometry. They can also be used in many of the previously mentioned areas of mathematics, such as algebraic geometry (as discussed in this previous post).

The three types of limits are analogous in many ways, with a number of connections between them. For instance, in the study of groups of polynomial growth, both topological limits (using the metric notion of Gromov-Hausdorff convergence) and logical limits (using the ultralimit construction) are commonly used, and to some extent the two constructions are at least partially interchangeable in this setting. (See also these previous posts for the use of ultralimits as a substitute for topological limits.) In the theory of approximate groups, though, it was observed by Hrushovski that logical limits (and in particular, ultraproducts) are the most useful type of limit to connect finitary approximate groups to their infinitary counterparts. One reason for this is that one is often interested in obtaining results on approximate groups ${A}$ that are uniform in the choice of ambient group ${G}$. As such, one often seeks to take a limit of approximate groups ${A_n}$ that lie in completely unrelated ambient groups ${G_n}$, with no obvious morphisms or metrics tying the ${G_n}$ to each other. As such, the topological and categorical limits are not easily usable, whereas the logical limits can still be employed without much difficulty.

Logical limits are closely tied with non-standard analysis. Indeed, by applying an ultraproduct construction to standard number systems such as the natural numbers ${{\bf N}}$ or the reals ${{\bf R}}$, one can obtain nonstandard number systems such as the nonstandard natural numbers ${{}^* {\bf N}}$ or the nonstandard real numbers (or hyperreals) ${{}^* {\bf R}}$. These nonstandard number systems behave very similarly to their standard counterparts, but also enjoy the advantage of containing the standard number systems as proper subsystems (e.g. ${{\bf R}}$ is a subring of ${{}^* {\bf R}}$), which allows for some convenient algebraic manipulations (such as the quotient space construction to create spaces such as ${{}^* {\bf R} / {\bf R}}$) which are not easily accessible in the purely standard universe. Nonstandard spaces also enjoy a useful completeness property, known as countable saturation, which is analogous to metric completeness (as discussed in this previous blog post) and which will be particularly useful for us in tying together the theory of approximate groups with the theory of Hilbert’s fifth problem. See this previous post for more discussion on ultrafilters and nonstandard analysis.

In these notes, we lay out the basic theory of ultraproducts and ultralimits (in particular, proving Los’s theorem, which roughly speaking asserts that ultralimits are limits in a logical sense, as well as the countable saturation property alluded to earlier). We also lay out some of the basic foundations of nonstandard analysis, although we will not rely too heavily on nonstandard tools in this course. Finally, we apply this general theory to approximate groups, to connect finite approximate groups to an infinitary type of approximate group which we will call an ultra approximate group. We will then study these ultra approximate groups (and models of such groups) in more detail in the next set of notes.

Remark 3 Throughout these notes (and in the rest of the course), we will assume the axiom of choice, in order to easily use ultrafilter-based tools. If one really wanted to expend the effort, though, one could eliminate the axiom of choice from the proofs of the final “finitary” results that one is ultimately interested in proving, at the cost of making the proofs significantly lengthier. Indeed, there is a general result of Gödel that any result which can be stated in the language of Peano arithmetic (which, roughly speaking, means that the result is “finitary” in nature), and can be proven in set theory using the axiom of choice (or more precisely, in the ZFC axiom system), can also be proven in set theory without the axiom of choice (i.e. in the ZF system). As this course is not focused on foundations, we shall simply assume the axiom of choice henceforth to avoid further distraction by such issues.

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.

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.

In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with the minimal assumption of local compactness on a topological group ${G}$, and seeing what kind of structures one can build using this assumption. (For simplicity we shall mostly confine our discussion to global groups rather than local groups for now.) In view of the preceding notes, we would like to see two types of structures emerge in particular:

• representations of ${G}$ into some more structured group, such as a matrix group ${GL_n({\bf C})}$; and
• metrics on ${G}$ that capture the escape and commutator structure of ${G}$ (i.e. Gleason metrics).

To build either of these structures, a fundamentally useful tool is that of (left-) Haar measure – a left-invariant Radon measure ${\mu}$ on ${G}$. (One can of course also consider right-Haar measures; in many cases (such as for compact or abelian groups), the two concepts are the same, but this is not always the case.) This concept generalises the concept of Lebesgue measure on Euclidean spaces ${{\bf R}^d}$, which is of course fundamental in analysis on those spaces.

Haar measures will help us build useful representations and useful metrics on locally compact groups ${G}$. For instance, a Haar measure ${\mu}$ gives rise to the regular representation ${\tau: G \rightarrow U(L^2(G,d\mu))}$ that maps each element ${g \in G}$ of ${G}$ to the unitary translation operator ${\rho(g): L^2(G,d\mu) \rightarrow L^2(G,d\mu)}$ on the Hilbert space ${L^2(G,d\mu)}$ of square-integrable measurable functions on ${G}$ with respect to this Haar measure by the formula

$\displaystyle \tau(g) f(x) := f(g^{-1} x).$

(The presence of the inverse ${g^{-1}}$ is convenient in order to obtain the homomorphism property ${\tau(gh) = \tau(g)\tau(h)}$ without a reversal in the group multiplication.) In general, this is an infinite-dimensional representation; but in many cases (and in particular, in the case when ${G}$ is compact) we can decompose this representation into a useful collection of finite-dimensional representations, leading to the Peter-Weyl theorem, which is a fundamental tool for understanding the structure of compact groups. This theorem is particularly simple in the compact abelian case, where it turns out that the representations can be decomposed into one-dimensional representations ${\chi: G \rightarrow U({\bf C}) \equiv S^1}$, better known as characters, leading to the theory of Fourier analysis on general compact abelian groups. With this and some additional (largely combinatorial) arguments, we will also be able to obtain satisfactory structural control on locally compact abelian groups as well.

The link between Haar measure and useful metrics on ${G}$ is a little more complicated. Firstly, once one has the regular representation ${\tau: G\rightarrow U(L^2(G,d\mu))}$, and given a suitable “test” function ${\psi: G \rightarrow {\bf C}}$, one can then embed ${G}$ into ${L^2(G,d\mu)}$ (or into other function spaces on ${G}$, such as ${C_c(G)}$ or ${L^\infty(G)}$) by mapping a group element ${g \in G}$ to the translate ${\tau(g) \psi}$ of ${\psi}$ in that function space. (This map might not actually be an embedding if ${\psi}$ enjoys a non-trivial translation symmetry ${\tau(g)\psi=\psi}$, but let us ignore this possibility for now.) One can then pull the metric structure on the function space back to a metric on ${G}$, for instance defining an ${L^2(G,d\mu)}$-based metric

$\displaystyle d(g,h) := \| \tau(g) \psi - \tau(h) \psi \|_{L^2(G,d\mu)}$

if ${\psi}$ is square-integrable, or perhaps a ${C_c(G)}$-based metric

$\displaystyle d(g,h) := \| \tau(g) \psi - \tau(h) \psi \|_{C_c(G)} \ \ \ \ \ (1)$

if ${\psi}$ is continuous and compactly supported (with ${\|f \|_{C_c(G)} := \sup_{x \in G} |f(x)|}$ denoting the supremum norm). These metrics tend to have several nice properties (for instance, they are automatically left-invariant), particularly if the test function is chosen to be sufficiently “smooth”. For instance, if we introduce the differentiation (or more precisely, finite difference) operators

$\displaystyle \partial_g := 1-\tau(g)$

(so that ${\partial_g f(x) = f(x) - f(g^{-1} x)}$) and use the metric (1), then a short computation (relying on the translation-invariance of the ${C_c(G)}$ norm) shows that

$\displaystyle d([g,h], \hbox{id}) = \| \partial_g \partial_h \psi - \partial_h \partial_g \psi \|_{C_c(G)}$

for all ${g,h \in G}$. This suggests that commutator estimates, such as those appearing in the definition of a Gleason metric in Notes 2, might be available if one can control “second derivatives” of ${\psi}$; informally, we would like our test functions ${\psi}$ to have a “${C^{1,1}}$” type regularity.

If ${G}$ was already a Lie group (or something similar, such as a ${C^{1,1}}$ local group) then it would not be too difficult to concoct such a function ${\psi}$ by using local coordinates. But of course the whole point of Hilbert’s fifth problem is to do without such regularity hypotheses, and so we need to build ${C^{1,1}}$ test functions ${\psi}$ by other means. And here is where the Haar measure comes in: it provides the fundamental tool of convolution

$\displaystyle \phi * \psi(x) := \int_G \phi(x y^{-1}) \psi(y) d\mu(y)$

between two suitable functions ${\phi, \psi: G \rightarrow {\bf C}}$, which can be used to build smoother functions out of rougher ones. For instance:

Exercise 1 Let ${\phi, \psi: {\bf R}^d \rightarrow {\bf C}}$ be continuous, compactly supported functions which are Lipschitz continuous. Show that the convolution ${\phi * \psi}$ using Lebesgue measure on ${{\bf R}^d}$ obeys the ${C^{1,1}}$-type commutator estimate

$\displaystyle \| \partial_g \partial_h (\phi * \psi) \|_{C_c({\bf R}^d)} \leq C \|g\| \|h\|$

for all ${g,h \in {\bf R}^d}$ and some finite quantity ${C}$ depending only on ${\phi, \psi}$.

This exercise suggests a strategy to build Gleason metrics by convolving together some “Lipschitz” test functions and then using the resulting convolution as a test function to define a metric. This strategy may seem somewhat circular because one needs a notion of metric in order to define Lipschitz continuity in the first place, but it turns out that the properties required on that metric are weaker than those that the Gleason metric will satisfy, and so one will be able to break the circularity by using a “bootstrap” or “induction” argument.

We will discuss this strategy – which is due to Gleason, and is fundamental to all currently known solutions to Hilbert’s fifth problem – in later posts. In this post, we will construct Haar measure on general locally compact groups, and then establish the Peter-Weyl theorem, which in turn can be used to obtain a reasonably satisfactory structural classification of both compact groups and locally compact abelian groups.

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.)