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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:

  • ({S} small) One has

    \displaystyle  (S^{A^4})^{1000K^3} \subset A. \ \ \ \ \ (4)

  • (First trapping condition) If {g, g^2, \ldots, g^{1000} \in A^{100}}, then {g \in A}.
  • (Second trapping condition) If {g, g^2, \ldots, g^{10^6 K^3} \in A}, then {g \in S}.

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.

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