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 for some small but fixed ) and “global” macroscopic scales (scales that are allowed to be larger than a given large absolute constant ). For instance, given a finite approximate group :
- Sets such as for some fixed (e.g. ) can be considered to be sets at a global macroscopic scale. Sending to infinity, one enters the large-scale regime.
- Sets such as the sets that appear in the Sanders lemma from the previous set of notes (thus for some fixed , e.g. ) can be considered to be sets at a local macroscopic scale. Sending to infinity, one enters the mesoscopic regime.
- The non-identity element of 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 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 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 be the Heisenberg group
and let , where is the box
thus is the nonstandard box
where . As the above exercise establishes, is an ultra approximate group with a Lie model given by the formula
for and . Note how the nonabelian nature of (arising from the term in the group law (1)) has been lost in the model , because the effect of that nonabelian term on is only which is infinitesimal and thus does not contribute to the standard part. In particular, if we replace with the abelian group with the additive group law
and let and be defined exactly as with and , but placed inside the group structure of rather than , then and are essentially “indistinguishable” as far as their models by 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 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 (or equivalently, norms ) on one’s group, which obeyed useful “Gleason axioms” such as the commutator axiom
when was sufficiently small. Such axioms have important and non-trivial content even in the microscopic regime where or 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 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 , or more precisely a non-identity element of of minimal norm. The key point was that this microscopic element was virtually central in , and as such it restricted much of 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 and discussed earlier, the element 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 and 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 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 that allow one to understand the microscopic geometry of points close to the identity in terms of the (local) macroscopic geometry of points 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 that obeys the escape and commutator axioms (while being non-degenerate enough to adequately capture the geometry of 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 that only obeys the escape and commutator estimates macroscopically; roughly speaking, this means that one has a macroscopic commutator inequality
and a macroscopic escape property
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 in a group , and an element of , we can define the escape norm of by the formula
Thus, equals if lies outside of , equals if lies in but lies outside of , 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 which we will present shortly. However, it need not actually be a norm; in particular, the triangle inequality
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
where is a constant independent of . 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 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 . A strong -approximate group is a finite -approximate group in a group with a symmetric subset obeying the following axioms:
An ultra strong -approximate group is an ultraproduct of strong -approximate groups.
The first trapping condition can be rewritten as
and the second trapping condition can similarly be rewritten as
This makes the escape norms of , and 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 be a large natural number. Then the interval in the integers is a -approximate group, which is also a strong -approximate group (setting , for instance). On the other hand, if one places in rather than in the integers, then the first trapping condition is lost and one is no longer a strong -approximate group. Also, if one remains in the integers, but deletes a few elements from , e.g. deleting from ), then one is still a -approximate group, but is no longer a strong -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:
- (i) Let be an ultra approximate group with a good Lie model , and let be a symmetric convex body (i.e. a convex open bounded subset) in the Lie algebra . Show that if is a sufficiently small standard number, then there exists a strong ultra approximate group with
and with can be covered by finitely many left translates of . Furthermore, is also a good model for .
- (ii) If is a finite -approximate group, show that there is a strong -approximate group inside with the property that can be covered by left translates of . (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 will be convenient later, as it allows one to easily use the geometry of 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:
- (Symmetry) For any , one has .
- (Conjugacy bound) For any , one has .
- (Triangle inequality) For any , one has .
- (Escape property) One has whenever .
- (Commutator inequality) For any , one has .
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.
- Start with an (ultra) strong approximate group .
- From the Gleason lemma, the elements with zero escape norm form a normal subgroup of . Quotient these elements out. Show that all non-identity elements will have positive escape norm.
- Find the non-identity element in (the quotient of) of minimal escape norm. Use the commutator estimate (assuming it is inherited by the quotient) to show that will centralise (most of) this quotient. In particular, the orbit is (essentially) a central subgroup of .
- Quotient this orbit out; then find the next non-identity element in this new quotient of . Again, show that is essentially a central subgroup of this quotient.
- Repeat this process until becomes entirely trivial. Undoing all the quotients, this should demonstrate that is virtually nilpotent, and that 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 (in the sense that the quotient of 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 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 -approximate group
in the integers, where is a large natural number not divisible by . As is torsion-free, all non-zero elements of have positive escape norm, and the nonzero element of minimal escape norm here is (or ). But if one quotients by , projects down to , 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 with 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 generated by the element of minimal escape norm, but rather by an arithmetic progression , where is a natural number comparable to the reciprocal 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 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 is in the global case when 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.
— 1. Gleason’s lemma —
Throughout this section, is a strong -approximate group in a global group . We will prove the various estimates in Theorem 2. The arguments will be very close to those in Notes 4; indeed, it is possible to unify the results here with the results in those notes by a suitable modification of the notation, but we will not do so here.
We begin with the easy estimates. The symmetry property is immediate from the symmetry of . Now we turn to the escape property. By symmetry, we may take to be positive (the case is trivial). We may of course assume that is strictly positive, say equal to ; thus and , and . By the first trapping property, this implies that for some .
Let be the first multiple of larger than or equal to , then . Since is less than , we have ; since , we conclude that . In particular this shows that , and the claim follows.
The escape property implies the conjugacy bound:
Exercise 2 Establish the conjugacy bound. (Hint: one can mimic the arguments establishing a nearly identical bound in Section 2 of Notes 4.)
Now we turn to the triangle inequality, which (as in Notes 4) is the most difficult property to establish. Our arguments will closely resemble the proof of Proposition 11 from these notes, with and playing the roles of and from that argument. As in that theorem, we will initially assume that we have an a priori bound of the form
for all , and some (large) independent of , and remove this hypothesis later. We then introduce the norm
for all .
We introduce the function by
for all , thanks to the triangle inequality for and (6). We also introduce the function by
for all and .
Now we form the convolution by the formula
Similarly, from the identity
and (7) we have the Lipschitz bound
Finally, from the identity
for and .
We can use this to improve the bound (10). Indeed, using the telescoping identity
we see that
whenever . Using the second trapping property, this implies that
In the converse direction, if , then
and thus from the support of , for all . But then by the first trapping property, this implies that for all . We conclude that
The triangle inequality for then implies a triangle inequality for ,
which is (5) with replaced by . If we knew (5) for some large but finite , we could iterate this argument and conclude that (5) held with replaced by , which would give the triangle inequality. Now it is not immediate that (5) holds for any finite , but we can avoid this problem with the usual regularisation trick of replacing with throughout the argument for some small , which makes (5) automatically true with , run the above iteration argument, and then finally send to zero.
Exercise 3 Verify that the modifications to the above argument sketched above actually do establish the triangle inequality.
A final application of the Gleason convolution machinery then gives the final estimate in Gleason’s lemma:
Exercise 4 Use the properties of the escape norm already established (and in particular, the escape property and the triangle inequality) to establish the commutator inequality. (Hint: adapt the argument from Section 2 of Notes 4.)
The proof of Theorem 2 is now complete.
Exercise 5 Generalise Theorem 2 to the setting where is not necessarily finite, but is instead an open precompact subset of a locally compact group . (Hint: replace cardinality by left-invariant Haar measure and follow the arguments in Notes 4 closely.) Note that this already gives most of one of the key results from that set of notes, namely that any NSS group admits a Gleason metric, since it is not difficult to show that NSS groups contain open precompact strong approximate groups.
— 2. A cheap version of the structure theorem —
In this section we use Theorem 2 to establish a “cheap” version of the structure theorem for ultra approximate groups. We begin by eliminating the elements of zero escape norm. Let us say that an approximate group is NSS if it contains no non-trivial subgroups of the ambient group, or equivalently if every non-identity element of has a positive escape norm. We say that an ultra approximate group is NSS if it is the ultralimit of NSS approximate groups.
Using the Gleason lemma, we can easily reduce to the NSS case:
Exercise 6 (Reduction to the NSS case) Let be a connected Lie group with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let , and let be an ultra strong approximate group which has a good model with
Let be the set of all elements in of zero (nonstandard) escape norm. Show that is a normal nonstandard finite subgroup of that lies in . If is the quotient map, and is the map factored through , show that there exists an ultra strong NSS approximate group in which has as a good model with
and such that is covered by finitely many left-translates of .
Let us now analyse the NSS case. Let be a connected Lie group, with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let be an ultra strong NSS approximate group which has a good model with
If is zero-dimensional, then by connectedness it is trivial, and then (by the properties of a good model) is a nonstandard finite group; since it is NSS, it is also trivial. Now suppose that is not zero-dimensional. Then contains non-identity elements whose image under is arbitrarily close to the identity of ; in particular, does not consist solely of the identity element, and thus contains elements of positive escape norm by the NSS assumption. Let be a non-identity element of with minimal escape norm , then must be the identity (so in particular is infinitesimal). (Note that any non-trivial NSS finite approximate group will contain non-identity elements of minimal escape norm, and the extension of this claim to the ultra approximate group case follows from Los’s theorem.) From Theorem 2 one has
for all . (Here we are now using the nonstandard asymptotic notation, thus means that for some standard .) In particular, from the minimality of , we see that there is a standard such that commutes with all elements with . In particular, if is a sufficiently small standard real number, we can find an ultra approximate subgroup of with
which is centralised by .
Now we show that generates a one-parameter subgroup of the model Lie group .
- Show that whenever .
- Show that the map is a one-parameter subgroup in (i.e. a continuous homomorphism from to ).
- Show that there exists an element of such that for all .
Similar statements hold with , replaced by .
We can now quotient out by the centraliser of and reduce the dimension of the Lie model:
Exercise 8 Let be the centraliser of in , and let be the nonstandard cyclic group generated by . (Thus, by the preceding discussion, contains , and is a central subgroup of containing . It will be important for us that and are both nonstandard sets, i.e. ultraproducts of standard sets.)
- (i) Show that is a compact subset of for each standard .
- (ii) Show that is a central subgroup of that contains .
- (iii) Show that is a central subgroup of that is a Lie group of dimension at least one, and so the quotient group is a Lie group of dimension strictly less than the dimension of .
- (iv) Let be the quotient map, and let be the obvious quotient of . Let be a convex symmetric body in the Lie algebra of . Show that for sufficiently small standard , there exists an ultra strong approximate group
with as a good model, with , and with covered by finitely many left-translates of .
Note that the quotient approximate group obtained by the above procedure is not necessarily NSS. However, it can be made NSS by Exercise 6. As such, one can iterate the above exercise until the dimension of the Lie model shrinks all the way to zero, at which point the NSS approximate group one is working with becomes trivial. This leads to a “cheap” structure theorem for approximate groups:
- (i) Show that if has a good model by a connected Lie group , then is nilpotent. (Hint: first use Exercise 1, and then induct on the dimension of .)
- (ii) Show that is covered by finitely many left translates of a nonstandard subgroup of which admits a normal series
for some standard , where for every , is a normal nonstandard subgroup of , and is either a nonstandard finite group or a nonstandard central subgroup of . Furthermore, if is not central, then it is contained in the image of in . (Hint: first use the Lie model theorem and Exercise 1, and then induct on the dimension of .)
Exercise 10 (Cheap structure theorem, finite version) Let be a finite -approximate group in a group . Show that is covered by left-translates of a subgroup of which admits a normal series
for some , where for every , is a normal subgroup of , and is either finite or central in . Furthermore, if is not central, then it is contained in the image of in .
One can push the cheap structure theorem a bit further by controlling the dimension of the nilpotent Lie group in terms of the covering number of the ultra approximate group, as laid out in the following exercise.
Exercise 11 (Nilpotent groups) A Lie algebra is said to be nilpotent if the derived series , , becomes trivial after a finite number of steps.
- (i) Show that a connected Lie group is nilpotent if and only if its Lie algebra is nilpotent.
- (ii) If is a finite-dimensional nilpotent Lie algebra, show that there is a simply connected Lie group with Lie algebra , for which the exponential map is a (global) homeomorphism. Furthermore, any other connected Lie group with Lie algebra is a quotient of by a discrete central subgroup of .
- (iii) If and are as in (ii), show that the pushforward of a Haar measure (or Lebesgue measure) on is a bi-invariant Haar measure on . (Recall from Exercise 6 of Notes 3 that connected nilpotent Lie groups are unimodular.)
- (iv) If and are as in (ii), and is a bi-invariant Haar measure on , show that for all open precompact , where is the dimension of .
- (v) If is a connected (but not necessarily simply connected) nilpotent Lie group, and is the maximal compact normal subgroup of (which exists by Exercise 32 of Notes 7), show that is central, and is simply connected. As a consequence, conclude that if is a left-Haar measure of , then for all open precompact , where is the dimension of .
- (vi) Show that if is an ultra -approximate group which has a Lie model , and is the maximal compact normal subgroup of , then has dimension at most .
- (vii) Show that if is an ultra -approximate group, then there is an ultra -approximate group in that is modeled by a Lie group , such that is covered by finitely many left-translates of . (Hint: has a good model by a locally compact group ; by the Gleason-Yamabe theorem, has an open subgroup and a normal subgroup of inside with a Lie group. Set .)
- (viii) Show that if is an ultra -approximate group, then there is an ultra -approximate group in that is modeled by a nilpotent group of dimension , such that can be covered by finitely many left-translates of .
— 3. Local groups —
The main weakness of the cheap structure theorem in the preceding section is the continual reintroduction of torsion whenever one quotients out by the centraliser , which can destroy the NSS property. We now address the issue of how to fix this, by moving to the context of local groups rather than global groups. We will omit some details, referring to this recent paper for details.
We need to extend many of the notions we have been considering to the local group setting. We begin by generalising the concept of an approximate group.
Definition 3 (Approximate groups) A (local) -approximate group is a subset of a local group which is symmetric and contains the identity, such that is well-defined in , and for which is covered by left translates of (by elements in ). An ultra approximate group is an ultraproduct of -approximate groups.
Note that we make no topological requirements on or in this definition; in particular, we may as well give the local group the discrete topology. There are some minor technical advantages in requiring the local group to be symmetric (so that the inversion map is globally defined) and cancellative (so that or implies ), although these assumptions are essentially automatic in practice.
The exponent here is not terribly important in practice, thanks to the following variant of the Sanders lemma:
Exercise 12 Let be a finite -approximate group in a local group , except with only known to be well-defined rather than . Let . Show that there exists a finite -approximate subgroup in with well-defined and contained in , and with covered by left-translates of (by elements in ). (Hint: adapt the proof of Lemma 1 from Notes 7.)
Just as global approximate groups can be modeled by global locally compact groups (and in particular, global Lie groups), local approximate groups can be modeled by local locally compact groups:
Definition 4 (Good models) Let be a (local) ultra approximate group. A (local) good model for is a homomorphism from to a locally compact Hausdorff local group that obeys the following axioms:
- (Thick image) There exists a neighbourhood of the identity in such that and .
- (Compact image) is precompact.
- (Approximation by nonstandard sets) Suppose that , where is compact and is open. Then there exists a nonstandard finite set such that .
We make the pedantic remark that with our conventions, a global good model of a global approximate group only becomes a local good model of by after restricting the domain of to . It is also convenient for minor technical reasons to assume that the local group is symmetric (i.e. the inversion map is globally defined) but this hypothesis is not of major importance.
The Hrushovski Lie Model theorem can be localised:
Theorem 5 (Local Hrushovski Lie model theorem) Let be a (local) ultra approximate group. Then there is an ultra approximate subgroup of (thus ) with covered by finitely many left-translates of (by elements in ), which has a good model by a connected local Lie group .
The proof of this theorem is basically a localisation of the proof of the global Lie model theorem from Notes 7, and is omitted (see for details). One key replacement is that if is a local approximate group rather than a global one, then the global Gleason-Yamabe theorem (Theorem 1 from Notes 4) must be replaced by the local Gleason-Yamabe theorem of Goldbring, discussed in Section 6 of Notes 4.
One can define the notion of a strong -approximate group and ultra strong approximate group in the local setting without much difficulty, since strong approximate groups only need to work inside , which is well-defined. Using the local Lie model theorem, one can obtain a local version of Exercise 1. The Gleason lemma (Theorem 2) also localises without much difficulty to local strong approximate groups, as does the reduction to the NSS case in Exercise 6.
Now we once again analyse the NSS case. As before, let be a connected (local) Lie group, with Lie algebra , let be a bounded symmetric convex body in , let be a sufficiently small standard real. Let be a (local) ultra strong NSS approximate group which has a (local) good model with
Again, we assume has dimension at least , since is trivial otherwise. We let be a non-identity element of minimal escape norm. As before, will have an infinitesimal escape norm and lie in the kernel of . If we set , then is an unbounded natural number, and the map will be a local one-parameter subgroup, i.e. a continuous homomorphism from to . This one-parameter subgroup will be non-trivial and centralised by a neighbourhood of the identity in .
In the global setting, we quotiented (the group generated by a large portion of) by the centraliser of . In the local setting, we perform a more “gentle” quotienting, which roughly speaking arises by quotienting by the geometric progression , where is a sufficiently small standard quantity to be chosen later. However, is only a local group rather than a global one, and so we must now digress to introduce the notion of quotients of local groups. It is convenient to restrict attention to symmetric cancellative local groups:
- (i) Whenever are such that and are well-defined and equal to each other, then . (Note that this implies in particular that .)
- (ii) Whenever are such that and are well-defined and equal to each other, then .
- (iii) Whenever are such that and are well-defined, then . (In particular, if is symmetric and is well-defined in for some , then is also symmetric.)
Exercise 13 Show that every local group contains an open neighbourhood of the identity which is also a symmetric cancellative local group.
Definition 7 (Sub-local groups) Given two symmetric local groups and , we say that is a sub-local group of if is the restriction of to a symmetric neighbourhood of the identity, and there exists an open neighbourhood of with the property that whenever are such that is defined in , then ; we refer to as an associated neighbourhood for . If is also a global group, we say that is a subgroup of .
If is a sub-local group of , we say that is normal if there exists an associated neighbourhood for with the additional property that whenever are such that is well-defined and lies in , then . We call a normalising neighbourhood of .
Example 3 If are the (additive) local groups and , then is a sub-local group of (with associated neighbourhood ). Note that this is despite not being closed with respect to addition in ; thus we see why it is necessary to allow the associated neighbourhood to be strictly smaller than . In a similar vein, the open interval is a sub-local group of .
The interval is also a sub-local group of ; here, one can take for instance as the associated neighbourhood. As all these examples are abelian, they are clearly normal.
Example 4 Let be a linear transformation on a finite-dimensional vector space , and let be the associated semi-direct product. Let , where is a subspace of that is not preserved by . Then is not a normal subgroup of , but it is a normal sub-local group of , where one can take as a normalising neighbourhood of .
Observe that any sub-local group of a cancellative local group is again a cancellative local group.
One also easily verifies that if is a local homomorphism from to for some open neighbourhood of the identity in , then is a normal sub-local group of , and hence of . Note that the kernel of a local morphism is well-defined up to local identity. If is Hausdorff, then the kernel will also be closed.
Conversely, normal sub-local groups give rise to local homomorphisms into quotient spaces.
Exercise 14 (Quotient spaces) Let be a cancellative local group, and let be a normal sub-local group with normalising neighbourhood . Let be a symmetric open neighbourhood of the identity such that . Show that there exists a cancellative local group and a surjective continuous homomorphism such that, for any , one has if and only if , and for any , one has open if and only if is open.
Example 5 Let be the additive local group , and let be the sub-local group , with normalising neighbourhood . If we then set , then the hypotheses of Exercise 14 are obeyed, and can be identified with , with the projection map .
Example 6 Let be the torus , and let be the sub-local group , where is an irrational number, with normalising neighbourhood . Set . Then the hypotheses of Exercise 14 are again obeyed, and can be identified with the interval , with the projection map for . Note, in contrast, that if one quotiented by the global group generated by , the quotient would be a non-Hausdorff space (and would also contain a dense set of torsion points, in contrast to the interval which is “locally torsion free”). It is because of this pathological behaviour of quotienting by global groups that we need to work with local group quotients instead.
We now return to the analysis of the NSS ultra strong approximate group . We give the ambient local group the discrete topology.
Exercise 15 If is a standard real that is sufficiently small depending on , show that there exists an ultra approximate group with
such that is a sub-local group of with normalising neighbourhood , that is also centralised by .
By Exercise 14, we may now form the quotient set . Show that this is an ultra approximate group that is modeled by , where is an open neighbourhood of the identity in and is the local one-parameter subgroup of introduced earlier. In particular, is modeled by a local Lie group of dimension one less than the dimension of .
Now we come to a key observation, which is the main reason why we work in the local groups category in the first place:
We will in fact prove a stronger claim:
Since is NSS, all non-identity elements of have non-zero escape norm, and so by the lifting lemma, all non-identity elements of also have non-zero escape norm, giving Lemma 8.
Proof: (Proof of Lemma 9) We choose to be a lift of (i.e. an element of in ) that minimises the escape norm . (Such a minimum exists since is nonstandard finite, thanks to Los’s theorem.) If is trivial, then so is and there is nothing to prove. Therefore we may assume that is not the identity and hence, since is NSS, that it has positive escape norm. Suppose, by way of contradiction, that . Our goal will be to reach a contradiction by finding another lift of with strictly smaller escape norm than . We will do this by setting for some suitably chosen .
We may assume that is infinitesimal, since otherwise there is nothing to prove; in particular lies in the kernel of the local model . We may thus find a lift of in the kernel of . In particular, we may assume that has infinitesimal escape norm.
Set , then is unbounded. By hypothesis, ; thus whenever . In particular, for every (standard) integer , . This implies that the group generated by lies in . In particular, lies in the kernel of , and hence lies in for all .
By (an appropriate local version of) Exercise 7, we can find such that
whenever (say) for some that is also parallel to . In particular, for some
Since is the minimal escape norm of non-identity elements of , we have , and thus for some ; in particular, . Comparing this with (11) we see that
By the Euclidean algorithm, we can thus find a nonstandard integer number such that the quantity
lies in the interval . In particular
If we set then (as commutes with ) we see for all that
for all . In particular, . Since is also a lift of , this contradicts the minimality of , and the claim follows.
Because the NSS property is preserved, it is possible to improve upon Exercise 9:
Exercise 16 Strengthen Exercise 9 by ensuring the final quotient is nonstandard finite, and all the other quotients are central in .
As a consequence, one obtains a stronger structure theorem than Exercise 9. Call a symmetric subset containing the identity in a local group nilpotent of step at most if every iterated commutator in of length is well-defined and trivial.
- (i) Let be a (local) NSS ultra strong approximate group. Show that there is a symmetric subset of containing the identity which is nilpotent of some finite step, such that is covered by a finite number of left translates of .
- (ii) Let be a global NSS ultra strong approximate group with ambient group . Show that there is a nonstandard nilpotent subgroup of such that is covered by a finite number of left translates of .
- (iii) Let be an NSS strong -approximate group in a global group . Show that there is a nilpotent subgroup of of step such that can be covered by a finite number of left translates of .
- (iv) Let be a -approximate group in a global group . Show that there exists a subgroup of and a normal subgroup of contained in , such that is covered by left-translates of , and is nilpotent of step .
In fact, a stronger statement is true, involving the nilprogressions defined in Notes 6:
- (i) If is an NSS ultra strong approximate group, then there is an ultra nilprogression in such that contains , and can be covered by finitely many left-translates of .
- (ii) If is an ultra approximate group, then there is an ultra coset nilprogression in such that contains , and can be covered by finitely many left-translates of .
- (iii) For all , there exists such that, given a finite -approximate group in a group , one can find a coset nilprogression in of rank at most and step at most such that contains , and can be covered by at most left-translates of .
This proposition is established in this paper. The key point is to use the lifting lemma to observe that if (with the notation of the preceding discussion) contains a large nilprogression, then also contains a large nilprogression. One consequence of this proposition is that there is essentially no difference between local and global approximate groups, at the qualitative level at least:
Corollary 11 Let be a local -approximate group. Then there exists a -approximate subgroup of , with covered by left-translates of , such that is isomorphic to a global -approximate subgroup.
This is because coset nilprogressions (or large fractions thereof) can be embedded into global groups; again, see this paper for details.