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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 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.
One of the fundamental structures in modern mathematics is that of a group. Formally, a group is a set equipped with an identity element , a multiplication operation , and an inversion operation obeying the following axioms:
- (Closure) If , then and are well-defined and lie in . (This axiom is redundant from the above description, but we include it for emphasis.)
- (Associativity) If , then .
- (Identity) If , then .
- (Inverse) If , then .
One can also consider additive groups instead of multiplicative groups, with the obvious changes of notation. By convention, additive groups are always understood to be abelian, so it is convenient to use additive notation when one wishes to emphasise the abelian nature of the group structure. As usual, we often abbreviate by (and by ) when there is no chance of confusion.
If furthermore is equipped with a topology, and the group operations are continuous in this topology, then is a topological group. Any group can be made into a topological group by imposing the discrete topology, but there are many more interesting examples of topological groups, such as Lie groups, in which is not just a topological space, but is in fact a smooth manifold (and the group operations are not merely continuous, but also smooth).
There are many naturally occuring group-like objects that obey some, but not all, of the axioms. For instance, monoids are required to obey the closure, associativity, and identity axioms, but not the inverse axiom. If we also drop the identity axiom, we end up with a semigroup. Groupoids do not necessarily obey the closure axiom, but obey (versions of) the associativity, identity, and inverse axioms. And so forth.
Another group-like concept is that of a local topological group (or local group, for short), which is essentially a topological group with the closure axiom omitted (but do not obey the same axioms set as groupoids); they arise primarily in the study of local properties of (global) topological groups, and also in the study of approximate groups in additive combinatorics. Formally, a local group is a topological space equipped with an identity element , a partially defined but continuous multiplication operation for some domain , and a partially defined but continuous inversion operation , where , obeying the following axioms:
- (Local closure) is an open neighbourhood of , and is an open neighbourhood of .
- (Local associativity) If are such that and are both well-defined, then they are equal. (Note however that it may be possible for one of these products to be defined but not the other, in contrast for instance with groupoids.)
- (Identity) For all , .
- (Local inverse) If and is well-defined, then . (In particular this, together with the other axioms, forces .)
We will often refer to ordinary groups as global groups (and topological groups as global topological groups) to distinguish them from local groups. Every global topological group is a local group, but not conversely.
One can consider discrete local groups, in which the topology is the discrete topology; in this case, the openness and continuity axioms in the definition are automatic and can be omitted. At the other extreme, one can consider local Lie groups, in which the local group has the structure of a smooth manifold, and the group operations are smooth. We can also consider symmetric local groups, in which (i.e. inverses are always defined). Symmetric local groups have the advantage of local homogeneity: given any , the operation of left-multiplication is locally inverted by near the identity, thus giving a homeomorphism between a neighbourhood of and a neighbourhood of the identity; in particular, we see that given any two group elements in a symmetric local group , there is a homeomorphism between a neighbourhood of and a neighbourhood of . (If the symmetric local group is also Lie, then these homeomorphisms are in fact diffeomorphisms.) This local homogeneity already simplifies a lot of the possible topology of symmetric local groups, as it basically means that the local topological structure of such groups is determined by the local structure at the origin. (For instance, all connected components of a local Lie group necessarily have the same dimension.) It is easy to see that any local group has at least one symmetric open neighbourhood of the identity, so in many situations we can restrict to the symmetric case without much loss of generality.
A prime example of a local group can be formed by restricting any global topological group to an open neighbourhood of the identity, with the domains
one easily verifies that this gives the structure of a local group (which we will sometimes call to emphasise the original group ). If is symmetric (i.e. ), then we in fact have a symmetric local group. One can also restrict local groups to open neighbourhoods to obtain a smaller local group by the same procedure (adopting the convention that statements such as or are considered false if the left-hand side is undefined). (Note though that if one restricts to non-open neighbourhoods of the identity, then one usually does not get a local group; for instance is not a local group (why?).)
Finite subsets of (Hausdorff) groups containing the identity can be viewed as local groups. This point of view turns out to be particularly useful for studying approximate groups in additive combinatorics, a point which I hope to expound more on later. Thus, for instance, the discrete interval is an additive symmetric local group, which informally might model an adding machine that can only handle (signed) one-digit numbers. More generally, one can view a local group as an object that behaves like a group near the identity, but for which the group laws (and in particular, the closure axiom) can start breaking down once one moves far enough away from the identity.
One can formalise this intuition as follows. Let us say that a word in a local group is well-defined in (or well-defined, for short) if every possible way of associating this word using parentheses is well-defined from applying the product operation. For instance, in order for to be well-defined, , , , , and must all be well-defined. In the preceding example , is not well-defined because one of the ways of associating this sum, namely , is not well-defined (even though is well-defined).
Exercise 1 (Iterating the associative law)
- Show that if a word in a local group is well-defined, then all ways of associating this word give the same answer, and so we can uniquely evaluate as an element in .
- Give an example of a word in a local group which has two ways of being associated that are both well-defined, but give different answers. (Hint: the local associativity axiom prevents this from happening for , so try . A small discrete local group will already suffice to give a counterexample; verifying the local group axioms are easier if one makes the domain of definition of the group operations as small as one can get away with while still having the counterexample.)
Exercise 2 Show that the number of ways to associate a word is given by the Catalan number .
Exercise 3 Let be a local group, and let be an integer. Show that there exists a symmetric open neighbourhood of the identity such that every word of length in is well-defined in (or more succinctly, is well-defined). (Note though that these words will usually only take values in , rather than in , and also the sets tend to become smaller as increases.)
In many situations (such as when one is investigating the local structure of a global group) one is only interested in the local properties of a (local or global) group. We can formalise this by the following definition. Let us call two local groups and locally identical if they have a common restriction, thus there exists a set such that (thus, , and the topology and group operations of and agree on ). This is easily seen to be an equivalence relation. We call an equivalence class of local groups a group germ.
Let be a property of a local group (e.g. abelianness, connectedness, compactness, etc.). We call a group germ locally if every local group in that germ has a restriction that obeys ; we call a local or global group locally if its germ is locally (or equivalently, every open neighbourhood of the identity in contains a further neighbourhood that obeys ). Thus, the study of local properties of (local or global) groups is subsumed by the study of group germs.
- Show that the above general definition is consistent with the usual definitions of the properties “connected” and “locally connected” from point-set topology.
- Strictly speaking, the above definition is not consistent with the usual definitions of the properties “compact” and “local compact” from point-set topology because in the definition of local compactness, the compact neighbourhoods are certainly not required to be open. Show however that the point-set topology notion of “locally compact” is equivalent, using the above conventions, to the notion of “locally precompact inside of an ambient local group”. Of course, this is a much more clumsy terminology, and so we shall abuse notation slightly and continue to use the standard terminology “locally compact” even though it is, strictly speaking, not compatible with the above general convention.
- Show that a local group is discrete if and only if it is locally trivial.
- Show that a connected global group is abelian if and only if it is locally abelian. (Hint: in a connected global group, the only open subgroup is the whole group.)
- Show that a global topological group is first-countable if and only if it is locally first countable. (By the Birkhoff-Kakutani theorem, this implies that such groups are metrisable if and only if they are locally metrisable.)
- Let be a prime. Show that the solenoid group , where is the -adic integers and is the diagonal embedding of inside , is connected but not locally connected.
Remark 1 One can also study the local properties of groups using nonstandard analysis. Instead of group germs, one works (at least in the case when is first countable) with the monad of the identity element of , defined as the nonstandard group elements in that are infinitesimally close to the origin in the sense that they lie in every standard neighbourhood of the identity. The monad is closely related to the group germ , but has the advantage of being a genuine (global) group, as opposed to an equivalence class of local groups. It is possible to recast most of the results here in this nonstandard formulation; see e.g. the classic text of Robinson. However, we will not adopt this perspective here.
A useful fact to know is that Lie structure is local. Call a (global or local) topological group Lie if it can be given the structure of a (global or local) Lie group.
We sketch a proof of this lemma below the fold. One direction is obvious, as the restriction a global Lie group to an open neighbourhood of the origin is clearly a local Lie group; for instance, the continuous interval is a symmetric local Lie group. The converse direction is almost as easy, but (because we are not assuming to be connected) requires one non-trivial fact, namely that local homomorphisms between local Lie groups are automatically smooth; details are provided below the fold.
As with so many other basic classes of objects in mathematics, it is of fundamental importance to specify and study the morphisms between local groups (and group germs). Given two local groups , we can define the notion of a (continuous) homomorphism between them, defined as a continuous map with
such that whenever are such that is well-defined, then is well-defined and equal to ; similarly, whenever is such that is well-defined, then is well-defined and equal to . (In abstract algebra, the continuity requirement is omitted from the definition of a homomorphism; we will call such maps discrete homomorphisms to distinguish them from the continuous ones which will be the ones studied here.)
It is often more convenient to work locally: define a local (continuous) homomorphism from to to be a homomorphism from an open neighbourhood of the identity to . Given two local homomorphisms , from one pair of locally identical groups to another pair , we say that are locally identical if they agree on some open neighbourhood of the identity in (note that it does not matter here whether we require openness in , in , or both). An equivalence class of local homomorphisms will be called a germ homomorphism (or morphism for short) from the group germ to the group germ .
Exercise 5 Show that the class of group germs, equipped with the germ homomorphisms, becomes a category. (Strictly speaking, because group germs are themselves classes rather than sets, the collection of all group germs is a second-order class rather than a class, but this set-theoretic technicality can be resolved in a number of ways (e.g. by restricting all global and local groups under consideration to some fixed “universe”) and should be ignored for this exercise.)
As is usual in category theory, once we have a notion of a morphism, we have a notion of an isomorphism: two group germs are isomorphic if there are germ homomorphisms , that invert each other. Lifting back to local groups, the associated notion is that of local isomorphism: two local groups are locally isomorphic if there exist local isomorphisms and from to and from to that locally invert each other, thus for sufficiently close to , and for sufficiently close to . Note that all local properties of (global or local) groups that can be defined purely in terms of the group and topological structures will be preserved under local isomorphism. Thus, for instance, if are locally isomorphic local groups, then is locally connected iff is, is locally compact iff is, and (by Lemma 1) is Lie iff is.
Show that the additive global groups and are locally isomorphic. Show that every locally path-connected group is locally isomorphic to a path-connected, simply connected group.
— 1. Lie’s third theorem —
Lie’s fundamental theorems of Lie theory link the Lie group germs to Lie algebras. Observe that if is a locally Lie group germ, then the tangent space at the identity of this germ is well-defined, and is a finite-dimensional vector space. If we choose to be symmetric, then can also be identified with the left-invariant (say) vector fields on , which are first-order differential operators on . The Lie bracket for vector fields then endows with the structure of a Lie algebra. It is easy to check that every morphism of locally Lie germs gives rise (via the derivative map at the identity) to a morphism of the associated Lie algebras. From the Baker-Campbell-Hausdorff formula (which is valid for local Lie groups, as discussed in this previous post) we conversely see that uniquely determines the germ homomorphism . Thus the derivative map provides a covariant functor from the category of locally Lie group germs to the category of (finite-dimensional) Lie algebras. In fact, this functor is an isomorphism, which is part of a fact known as Lie’s third theorem:
Theorem 2 (Lie’s third theorem) For this theorem, all Lie algebras are understood to be finite dimensional (and over the reals).
- Every Lie algebra is the Lie algebra of a local Lie group germ , which is unique up to germ isomorphism (fixing ).
- Every Lie algebra is the Lie algebra of some global connected, simply connected Lie group , which is unique up to Lie group isomorphism (fixing ).
- Every homomorphism between Lie algebras is the derivative of a unique germ homomorphism between the associated local Lie group germs.
- Every homomorphism between Lie algebras is the derivative of a unique Lie group homomorphism between the associated global connected, simply connected, Lie groups.
- Every local Lie group germ is the germ of a global connected, simply connected Lie group , which is unique up to Lie group isomorphism. In particular, every local Lie group is locally isomorphic to a global Lie group.
We record the (standard) proof of this theorem below the fold, which is ultimately based on Ado’s theorem and the Baker-Campbell-Hausdorff formula. Lie’s third theorem (which, actually, was proven in full generality by Cartan) demonstrates the equivalence of three categories: the category of finite-dimensonal Lie algebras, the category of local Lie group germs, and the category of connected, simply connected Lie groups.
— 2. Globalising a local group —
Many properties of a local group improve after passing to a smaller neighbourhood of the identity. Here are some simple examples:
Exercise 7 Let be a local group.
- Give an example to show that does not necessarily obey the cancellation laws
for (with the convention that statements such as are false if either side is undefined). However, show that there exists an open neighbourhood of within which the cancellation law holds.
- Repeat the previous part, but with the cancellation law (1) replaced by the inversion law
for any for which both sides are well-defined.
- Repeat the previous part, but with the inversion law replaced by the involution law
for any for which the left-hand side is well-defined.
Note that the counterexamples in the above exercise demonstrate that not every local group is the restriction of a global group, because global groups (and hence, their restrictions) always obey the cancellation law (1), the inversion law (2), and the involution law (3). Another way in which a local group can fail to come from a global group is if it contains relations which can interact in a “global’ way to cause trouble, in a fashion which is invisible at the local level. For instance, consider the open unit cube , and consider four points in this cube that are close to the upper four corners of this cube respectively. Define an equivalence relation on this cube by setting if and is equal to either or for some . Note that this indeed an equivalence relation if are close enough to the corners (as this forces all non-trivial combinations to lie outside the doubled cube ). The quotient space (which is a cube with bits around opposite corners identified together) can then be seen to be a symmetric additive local Lie group, but will usually not come from a global group. Indeed, it is not hard to see that if is the restriction of a global group , then must be a Lie group with Lie algebra (by Lemma 1), and so the connected component of containing the identity is isomorphic to for some sublattice of that contains ; but for generic , there is no such lattice, as the will generate a dense subset of . (The situation here is somewhat analogous to a number of famous Escher prints, such as Ascending and Descending, in which the geometry is locally consistent but globally inconsistent.) We will give this sort of argument in more detail below the fold (see the proof of Proposition 7).
Nevertheless, the space is still locally isomorphic to a global Lie group, namely ; for instance, the open neighbourhood is isomorphic to , which is an open neighbourhood of . More generally, Lie’s third theorem tells us that any local Lie group is locally isomorphic to a global Lie group.
Let us call a local group globalisable if it is locally isomorphic to a global group; thus Lie’s third theorem tells us that every local Lie group is globalisable. Thanks to Goldbring’s solution to the local version of Hilbert’s fifth problem, we also know that locally Euclidean local groups are globalisable. A modification of this argument by van den Dries and Goldbring shows in fact that every locally compact local group is globalisable.
In view of these results, it is tempting to conjecture that all local groups are globalisable;; among other things, this would simplify the proof of Lie’s third theorem (and of the local version of Hilbert’s fifth problem). Unfortunately, this claim as stated is false:
The counterexamples used to establish Theorem 3 are remarkably delicate; the first example I know of is due to van Est and Korthagen. One reason for this, of course, is that the previous results prevents one from using any local Lie group, or even a locally compact group as a counterexample. We will present a (somewhat complicated) example below, based on the unit ball in the infinite-dimensional Banach space .
However, there are certainly many situations in which we can globalise a local group. For instance, this is the case if one has a locally faithful representation of that local group inside a global group:
Lemma 4 (Faithful representation implies globalisability) Let be a local group, and suppose there exists an injective local homomorphism from into a global topological group with symmetric. Then is isomorphic to the restriction of a global topological group to an open neighbourhood of the identity; in particular, is globalisable.
A topological space is said to be metrisable if one can find a metric on it whose open balls generate the topology.
There are some obvious necessary conditions on the space in order for it to be metrisable. For instance, it must be Hausdorff, since all metric spaces are Hausdorff. It must also be first countable, because every point in a metric space has a countable neighbourhood base of balls , .
In the converse direction, being Hausdorff and first countable is not always enough to guarantee metrisability, for a variety of reasons. For instance the long line is not metrisable despite being both Hausdorff and first countable, due to a failure of paracompactness, which prevents one from gluing together the local metric structures on this line into a global one. Even after adding in paracompactness, this is still not enough; the real line with the lower limit topology (also known as the Sorgenfrey line) is Hausdorff, first countable, and paracompact, but still not metrisable (because of a failure of second countability despite being separable).
However, there is one important setting in which the Hausdorff and first countability axioms do suffice to give metrisability, and that is the setting of topological groups:
Theorem 1 (Birkhoff-Kakutani theorem) Let be a topological group (i.e. a topological space that is also a group, such that the group operations and are continuous). Then is metrisable if and only if it is both Hausdorff and first countable.
Remark 1 It is not hard to show that a topological group is Hausdorff if and only if the singleton set is closed. More generally, in an arbitrary topological group, it is a good exercise to show that the closure of is always a closed normal subgroup of , whose quotient is then a Hausdorff topological group. Because of this, the study of topological groups can usually be reduced immediately to the study of Hausdorff topological groups. (Indeed, in many texts, topological groups are automatically understood to be an abbreviation for “Hausdorff topological group”.)
The standard proof of the Birkhoff-Kakutani theorem (which we have taken from this book of Montgomery and Zippin) relies on the following Urysohn-type lemma:
- (Unique maximum) , and for all .
- (Neighbourhood base) The sets for form a neighbourhood base at the identity.
- (Uniform continuity) For every , there exists an open neighbourhood of the identity such that for all and .
Note that if had a left-invariant metric, then the function would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.
Let us assume Lemma 2 for now and finish the proof of the Birkhoff-Kakutani theorem. We only prove the difficult direction, namely that a Hausdorff first countable topological group is metrisable. We let be the function from Lemma 2, and define the function by the formula
where is the space of bounded continuous functions on (with the supremum norm) and is the left-translation operator .
Clearly obeys the the identity and symmetry axioms, and the triangle inequality is also immediate. This already makes a pseudometric. In order for to be a genuine metric, what is needed is that have no non-trivial translation invariances, i.e. one has for all . But this follows since attains its maximum at exactly one point, namely the group identity .
To put it another way: because has no non-trivial translation invariances, the left translation action gives an embedding , and then inherits a metric from the metric structure on .
Now we have to check whether the metric actually generates the topology. This amounts to verifying two things. Firstly, that every ball in this metric is open; and secondly, that every open neighbourhood of a point contains a ball .
To verify the former claim, it suffices to show that the map from to is continuous, follows from the uniform continuity hypothesis. The second claim follows easily from the neighbourhood base hypothesis, since if then .
Remark 2 The above argument in fact shows that if a group is metrisable, then it admits a left-invariant metric. The idea of using a suitable continuous function to generate a useful metric structure on a topological group is a powerful one, for instance underlying the Gleason lemmas which are fundamental to the solution of Hilbert’s fifth problem. I hope to return to this topic in a future post.
Now we prove Lemma 2. By first countability, we can find a countable neighbourhood base
of the identity. As is Hausdorff, we must have
For every dyadic rational in , we can now define the open sets by setting
for all and .
We now set
with the understanding that if the supremum is over the empty set. One easily verifies using (4) that is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 2.
Remark 3 A very similar argument to the one above also establishes that every topological group is completely regular.
Notice that the function constructed in the above argument was localised to the set . As such, it is not difficult to localise the Birkhoff-Kakutani theorem to local groups. A local group is a topological space equipped with an identity , a partially defined inversion operation , and a partially defined product operation , where , are open subsets of and , obeying the following restricted versions of the group axioms:
- (Continuity) and are continuous on their domains of definition.
- (Identity) For any , and are well-defined and equal to .
- (Inverse) For any , and are well-defined and equal to . is well-defined and equal to .
- (Local associativity) If are such that , , , and are all well-defined, then .
Informally, one can view a local group as a topological group in which the closure axiom has been almost completely dropped, but with all the other axioms retained. A basic way to generate a local group is to start with an ordinary topological group and restrict it to an open neighbourhood of the identity, with and . However, this is not quite the only way to generate local groups (ultimately because the local associativity axiom does not necessarily imply a (stronger) global associativity axiom in which one considers two different ways to multiply more than three group elements together).
Remark 4 Another important example of a local group is that of a group chunk, in which the sets and are somehow “generic”; for instance, could be an algebraic variety, Zariski-open, and the group operations birational on their domains of definition. This is somewhat analogous to the notion of a “ group” in additive combinatorics. There are a number of group chunk theorems, starting with a theorem of Weil in the algebraic setting, which roughly speaking assert that a generic portion of a group chunk can be identified with the generic portion of a genuine group.
We then have
Theorem 3 (Birkhoff-Kakutani theorem for local groups) Let be a local group which is Hausdorff and first countable. Then there exists an open neighbourhood of the identity which is metrisable.
Proof: (Sketch) It is not difficult to see that in a local group , one can find a symmetric neighbourhood of the identity such that the product of any (say) elements of (multiplied together in any order) are well-defined, which effectively allows us to treat elements of as if they belonged to a group for the purposes of simple algebraic manipulation, such as applying the cancellation laws for . Inside this , one can then repeat the previous arguments and eventually end up with a continuous function supported in obeying the conclusions of Lemma 2 (but in the uniform continuity conclusion, one has to restrict to, say, , to avoid issues of ill-definedness). The definition (1) then gives a metric on with the required properties, where we make the convention that vanishes for (say) and .
My motivation for studying local groups is that it turns out that there is a correspondence (first observed by Hrushovski) between the concept of an approximate group in additive combinatorics, and a locally compact local group in topological group theory; I hope to discuss this correspondence further in a subsequent post.