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The wave equation is usually expressed in the form
where is a function of both time and space , with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on . We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy
which we can rewrite using integration by parts and the inner product on as
A key feature of the wave equation is finite speed of propagation: if, at time (say), the initial position and initial velocity are both supported in a ball , then at any later time , the position and velocity are supported in the larger ball . This can be seen for instance (formally, at least) by inspecting the exterior energy
and observing (after some integration by parts and differentiation under the integral sign) that it is non-increasing in time, non-negative, and vanishing at time .
The wave equation is second order in time, but one can turn it into a first order system by working with the pair rather than just the single field , where is the velocity field. The system is then
Finite speed of propagation then tells us that if are both supported on , then are supported on for all . One also has time reversal symmetry: if is a solution, then is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times using this symmetry.
If one has an eigenfunction
of the Laplacian, then we have the explicit solutions
of the wave equation, which formally can be used to construct all other solutions via the principle of superposition.
When one has vanishing initial velocity , the solution is given via functional calculus by
and the propagator can be expressed as the average of half-wave operators:
One can view as a minor of the full wave propagator
which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that
Viewing the contraction as a minor of a unitary operator is an instance of the “dilation trick“.
It turns out (as I learned from Yuval Peres) that there is a useful discrete analogue of the wave equation (and of all of the above facts), in which the time variable now lives on the integers rather than on , and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form
where is now an integer, take values in some Hilbert space (e.g. functions on a graph ), and is some operator on that Hilbert space (which in applications will usually be a self-adjoint contraction). To connect this with the classical wave equation, let us first consider a rescaling of this system
where is a small parameter (representing the discretised time step), now takes values in the integer multiples of , and is the wave propagator operator or the heat propagator (the two operators are different, but agree to fourth order in ). One can then formally verify that the wave equation emerges from this rescaled system in the limit . (Thus, is not exactly the direct analogue of the Laplacian , but can be viewed as something like in the case of small , or if we are not rescaling to the small case. The operator is sometimes known as the diffusion operator)
Assuming is self-adjoint, solutions to the system (3) formally conserve the energy
This energy is positive semi-definite if is a contraction. We have the same time reversal symmetry as before: if solves the system (3), then so does . If one has an eigenfunction
to the operator , then one has an explicit solution
to (3), and (in principle at least) this generates all other solutions via the principle of superposition.
Finite speed of propagation is a lot easier in the discrete setting, though one has to offset the support of the “velocity” field by one unit. Suppose we know that has unit speed in the sense that whenever is supported in a ball , then is supported in the ball . Then an easy induction shows that if are supported in respectively, then are supported in .
where and are the Chebyshev polynomials of the first and second kind, thus
As before, is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers and are discrete analogues of the heat propagators and wave propagators respectively.
One nice application of all this formalism, which I learned from Yuval Peres, is the Varopoulos-Carne inequality:
Theorem 1 (Varopoulos-Carne inequality) Let be a (possibly infinite) regular graph, let , and let be vertices in . Then the probability that the simple random walk at lands at at time is at most , where is the graph distance.
This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers . Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length concentrate on the ball of radius or so centred at the origin of the random walk.
Proof: Let be the graph Laplacian, thus
for any , where is the degree of the regular graph and sum is over the vertices that are adjacent to . This is a contraction of unit speed, and the probability that the random walk at lands at at time is
where are the Dirac deltas at . Using (5), we can rewrite this as
where we are now using the energy form (4). We can write
where is the simple random walk of length on the integers, that is to say where are independent uniform Bernoulli signs. Thus we wish to show that
By finite speed of propagation, the inner product here vanishes if . For we can use Cauchy-Schwarz and the unitary nature of to bound the inner product by . Thus the left-hand side may be upper bounded by
and the claim now follows from the Chernoff inequality.
This inequality has many applications, particularly with regards to relating the entropy, mixing time, and concentration of random walks with volume growth of balls; see this text of Lyons and Peres for some examples.
For sake of comparison, here is a continuous counterpart to the Varopoulos-Carne inequality:
Theorem 2 (Continuous Varopoulos-Carne inequality) Let , and let be supported on compact sets respectively. Then
where is the Euclidean distance between and .
Proof: By Fourier inversion one has
for any real , and thus
By finite speed of propagation, the inner product vanishes when ; otherwise, we can use Cauchy-Schwarz and the contractive nature of to bound this inner product by . Thus
Bounding by , we obtain the claim.
Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than .
A core foundation of the subject now known as arithmetic combinatorics (and particularly the subfield of additive combinatorics) are the elementary sum set estimates (sometimes known as “Ruzsa calculus”) that relate the cardinality of various sum sets
and difference sets
as well as iterated sumsets such as , , and so forth. Here, are finite non-empty subsets of some additive group (classically one took or , but nowadays one usually considers more general additive groups). Some basic estimates in this vein are the following:
Lemma 1 (Ruzsa covering lemma) Let be finite non-empty subsets of . Then may be covered by at most translates of .
Proof: Consider a maximal set of disjoint translates of by elements . These translates have cardinality , are disjoint, and lie in , so there are at most of them. By maximality, for any , must intersect at least one of the selected , thus , and the claim follows.
Lemma 2 (Ruzsa triangle inequality) Let be finite non-empty subsets of . Then .
Proof: Consider the addition map from to . Every element of has a preimage of this map of cardinality at least , thanks to the obvious identity for each . Since has cardinality , the claim follows.
Such estimates (which are covered, incidentally, in Section 2 of my book with Van Vu) are particularly useful for controlling finite sets of small doubling, in the sense that for some bounded . (There are deeper theorems, most notably Freiman’s theorem, which give more control than what elementary Ruzsa calculus does, however the known bounds in the latter theorem are worse than polynomial in (although it is conjectured otherwise), whereas the elementary estimates are almost all polynomial in .)
However, there are some settings in which the standard sum set estimates are not quite applicable. One such setting is the continuous setting, where one is dealing with bounded open sets in an additive Lie group (e.g. or a torus ) rather than a finite setting. Here, one can largely replicate the discrete sum set estimates by working with a Haar measure in place of cardinality; this is the approach taken for instance in this paper of mine. However, there is another setting, which one might dub the “discretised” setting (as opposed to the “discrete” setting or “continuous” setting), in which the sets remain finite (or at least discretisable to be finite), but for which there is a certain amount of “roundoff error” coming from the discretisation. As a typical example (working now in a non-commutative multiplicative setting rather than an additive one), consider the orthogonal group of orthogonal matrices, and let be the matrices obtained by starting with all of the orthogonal matrice in and rounding each coefficient of each matrix in this set to the nearest multiple of , for some small . This forms a finite set (whose cardinality grows as like a certain negative power of ). In the limit , the set is not a set of small doubling in the discrete sense. However, is still close to in a metric sense, being contained in the -neighbourhood of . Another key example comes from graphs of maps from a subset of one additive group to another . If is “approximately additive” in the sense that for all , is close to in some metric, then might not have small doubling in the discrete sense (because could take a large number of values), but could be considered a set of small doubling in a discretised sense.
One would like to have a sum set (or product set) theory that can handle these cases, particularly in “high-dimensional” settings in which the standard methods of passing back and forth between continuous, discrete, or discretised settings behave poorly from a quantitative point of view due to the exponentially large doubling constant of balls. One way to do this is to impose a translation invariant metric on the underlying group (reverting back to additive notation), and replace the notion of cardinality by that of metric entropy. There are a number of almost equivalent ways to define this concept:
Definition 3 Let be a metric space, let be a subset of , and let be a radius.
- The packing number is the largest number of points one can pack inside such that the balls are disjoint.
- The internal covering number is the fewest number of points such that the balls cover .
- The external covering number is the fewest number of points such that the balls cover .
- The metric entropy is the largest number of points one can find in that are -separated, thus for all .
It is an easy exercise to verify the inequalities
for any , and that is non-increasing in and non-decreasing in for the three choices (but monotonicity in can fail for !). It turns out that the external covering number is slightly more convenient than the other notions of metric entropy, so we will abbreviate . The cardinality can be viewed as the limit of the entropies as .
If we have the bounded doubling property that is covered by translates of for each , and one has a Haar measure on which assigns a positive finite mass to each ball, then any of the above entropies is comparable to , as can be seen by simple volume packing arguments. Thus in the bounded doubling setting one can usually use the measure-theoretic sum set theory to derive entropy-theoretic sumset bounds (see e.g. this paper of mine for an example of this). However, it turns out that even in the absence of bounded doubling, one still has an entropy analogue of most of the elementary sum set theory, except that one has to accept some degradation in the radius parameter by some absolute constant. Such losses can be acceptable in applications in which the underlying sets are largely “transverse” to the balls , so that the -entropy of is largely independent of ; this is a situation which arises in particular in the case of graphs discussed above, if one works with “vertical” metrics whose balls extend primarily in the vertical direction. (I hope to present a specific application of this type here in the near future.)
Henceforth we work in an additive group equipped with a translation-invariant metric . (One can also generalise things slightly by allowing the metric to attain the values or , without changing much of the analysis below.) By the Heine-Borel theorem, any precompact set will have finite entropy for any . We now have analogues of the two basic Ruzsa lemmas above:
Proof: Let be a maximal set of points such that the sets are all disjoint. Then the sets are disjoint in and have entropy , and furthermore any ball of radius can intersect at most one of the . We conclude that , so . If , then must intersect one of the , so , and the claim follows.
Proof: Consider the addition map from to . The domain may be covered by product balls . Every element of has a preimage of this map which projects to a translate of , and thus must meet at least of these product balls. However, if two elements of are separated by a distance of at least , then no product ball can intersect both preimages. We thus see that , and the claim follows.
Below the fold we will record some further metric entropy analogues of sum set estimates (basically redoing much of Chapter 2 of my book with Van Vu). Unfortunately there does not seem to be a direct way to abstractly deduce metric entropy results from their sum set analogues (basically due to the failure of a certain strong version of Freiman’s theorem, as discussed in this previous post); nevertheless, the proofs of the discrete arguments are elementary enough that they can be modified with a small amount of effort to handle the entropy case. (In fact, there should be a very general model-theoretic framework in which both the discrete and entropy arguments can be processed in a unified manner; see this paper of Hrushovski for one such framework.)
It is also likely that many of the arguments here extend to the non-commutative setting, but for simplicity we will not pursue such generalisations here.
Let be a natural number. We consider the question of how many “almost orthogonal” unit vectors one can place in the Euclidean space . Of course, if we insist on being exactly orthogonal, so that for all distinct , then we can only pack at most unit vectors into this space. However, if one is willing to relax the orthogonality condition a little, so that is small rather than zero, then one can pack a lot more unit vectors into , due to the important fact that pairs of vectors in high dimensions are typically almost orthogonal to each other. For instance, if one chooses uniformly and independently at random on the unit sphere, then a standard computation (based on viewing the as gaussian vectors projected onto the unit sphere) shows that each inner product concentrates around the origin with standard deviation and with gaussian tails, and a simple application of the union bound then shows that for any fixed , one can pack unit vectors into whose inner products are all of size .
One can remove the logarithm by using some number theoretic constructions. For instance, if is twice a prime , one can identify with the space of complex-valued functions , whee is the field of elements, and if one then considers the different quadratic phases for , where is the standard character on , then a standard application of Gauss sum estimates reveals that these unit vectors in all have inner products of magnitude at most with each other. More generally, if we take and consider the different polynomial phases for , then an application of the Weil conjectures for curves, proven by Weil, shows that the inner products of the associated unit vectors with each other have magnitude at most .
As it turns out, this construction is close to optimal, in that there is a polynomial limit to how many unit vectors one can pack into with an inner product of :
In particular, for fixed and large , the number of unit vectors one can pack in whose inner products all have magnitude at most will be . This doesn’t quite match the construction coming from the Weil conjectures, although it is worth noting that the upper bound of for the inner product is usually not sharp (the inner product is actually times the sum of unit phases which one expects (cf. the Sato-Tate conjecture) to be uniformly distributed on the unit circle, and so the typical inner product is actually closer to ).
Note that for , the case of the above theorem (or more precisely, Lemma 2 below) gives the bound , which is essentially optimal as the example of an orthonormal basis shows. For , the condition is trivially true from Cauchy-Schwarz, and can be arbitrariy large. Finally, in the range , we can use a volume packing argument: we have , so of we set , then the open balls of radius around each are disjoint, while all lying in a ball of radius , giving rise to the bound for some absolute constant .
As I learned recently from Philippe Michel, a more precise version of this theorem is due to Kabatjanskii and Levenstein, who studied the closely related problem of sphere packing (or more precisely, cap packing) in the unit sphere of . However, I found a short proof of the above theorem which relies on one of my favorite tricks – the tensor power trick – so I thought I would give it here.
We begin with an easy case, basically the case of the above theorem:
Proof: Suppose for contradiction that . We consider the Gram matrix . This matrix is real symmetric with rank at most , thus if one subtracts off the identity matrix, it has an eigenvalue of with multiplicity at least . Taking Hilbert-Schmidt norms, we conclude that
But by hypothesis, the left-hand side is at most , giving the desired contradiction.
To amplify the above lemma to cover larger values of , we apply the tensor power trick. A direct application of the tensor power trick does not gain very much; however one can do a lot better by using the symmetric tensor power rather than the raw tensor power. This gives
Corollary 3 Let be a natural number, and let be unit vectors in such that for all distinct . Then .
Proof: We work in the symmetric component of the tensor power , which has dimension . Applying the previous lemma to the tensor powers , we obtain the claim.
Using the trivial bound , we can lower bound
We can thus prove Theorem 1 by setting for some sufficiently large absolute constant .
In the last set of notes, we obtained the following structural theorem concerning approximate groups:
Theorem 1 Let be a finite -approximate group. Then there exists a coset nilprogression of rank and step contained in , such that is covered by left-translates of (and hence also by right-translates of ).
Remark 1 Under some mild additional hypotheses (e.g. if the dimensions of are sufficiently large, or if is placed in a certain “normal form”, details of which may be found in this paper), a coset nilprogression of rank and step will be an -approximate group, thus giving a partial converse to Theorem 1. (It is not quite a full converse though, even if one works qualitatively and forgets how the constants depend on : if is covered by a bounded number of left- and right-translates of , one needs the group elements to “approximately normalise” in some sense if one wants to then conclude that is an approximate group.) The mild hypotheses alluded to above can be enforced in the statement of the theorem, but we will not discuss this technicality here, and refer the reader to the above-mentioned paper for details.
By placing the coset nilprogression in a virtually nilpotent group, we have the following corollary in the global case:
In this final set of notes, we give some applications of the above results. The first application is to replace “-approximate group” by “sets of bounded doubling”:
Proposition 3 Let be a finite non-empty subset of a (global) group such that . Then there exists a coset nilprogression of rank and step and cardinality such that can be covered by left-translates of , and also by right-translates of .
We will also establish (a strengthening of) a well-known theorem of Gromov on groups of polynomial growth, as promised back in Notes 0, as well as a variant result (of a type known as a “generalised Margulis lemma”) controlling the almost stabilisers of discrete actions of isometries.
The material here is largely drawn from my recent paper with Emmanuel Breuillard and Ben Green.
In the previous set of notes, we introduced the notion of an ultra approximate group – an ultraproduct of finite -approximate groups for some independent of , where each -approximate group may lie in a distinct ambient group . Although these objects arise initially from the “finitary” objects , it turns out that ultra approximate groups can be profitably analysed by means of infinitary groups (and in particular, locally compact groups or Lie groups ), by means of certain models of (or of the group generated by ). We will define precisely what we mean by a model later, but as a first approximation one can view a model as a representation of the ultra approximate group (or of ) that is “macroscopically faithful” in that it accurately describes the “large scale” behaviour of (or equivalently, that the kernel of the representation is “microscopic” in some sense). In the next section we will see how one can use “Gleason lemma” technology to convert this macroscopic control of an ultra approximate group into microscopic control, which will be the key to classifying approximate groups.
Models of ultra approximate groups can be viewed as the multiplicative combinatorics analogue of the more well known concept of an ultralimit of metric spaces, which we briefly review below the fold as motivation.
The crucial observation is that ultra approximate groups enjoy a local compactness property which allows them to be usefully modeled by locally compact groups (and hence, through the Gleason-Yamabe theorem from previous notes, by Lie groups also). As per the Heine-Borel theorem, the local compactness will come from a combination of a completeness property and a local total boundedness property. The completeness property turns out to be a direct consequence of the countable saturation property of ultraproducts, thus illustrating one of the key advantages of the ultraproduct setting. The local total boundedness property is more interesting. Roughly speaking, it asserts that “large bounded sets” (such as or ) can be covered by finitely many translates of “small bounded sets” , where “small” is a topological group sense, implying in particular that large powers of lie inside a set such as or . The easiest way to obtain such a property comes from the following lemma of Sanders:
This lemma has an elementary combinatorial proof, and is the key to endowing an ultra approximate group with locally compact structure. There is also a closely related lemma of Croot and Sisask which can achieve similar results, and which will also be discussed below. (The locally compact structure can also be established more abstractly using the much more general methods of definability theory, as was first done by Hrushovski, but we will not discuss this approach here.)
By combining the locally compact structure of ultra approximate groups with the Gleason-Yamabe theorem, one ends up being able to model a large “ultra approximate subgroup” of by a Lie group . Such Lie models serve a number of important purposes in the structure theory of approximate groups. Firstly, as all Lie groups have a dimension which is a natural number, they allow one to assign a natural number “dimension” to ultra approximate groups, which opens up the ability to perform “induction on dimension” arguments. Secondly, Lie groups have an escape property (which is in fact equivalent to no small subgroups property): if a group element lies outside of a very small ball , then some power of it will escape a somewhat larger ball . Or equivalently: if a long orbit lies inside the larger ball , one can deduce that the original element lies inside the small ball . Because all Lie groups have this property, we will be able to show that all ultra approximate groups “essentially” have a similar property, in that they are “controlled” by a nearby ultra approximate group which obeys a number of escape-type properties analogous to those enjoyed by small balls in a Lie group, and which we will call a strong ultra approximate group. This will be discussed in the next set of notes, where we will also see how these escape-type properties can be exploited to create a metric structure on strong approximate groups analogous to the Gleason metrics studied in previous notes, which can in turn be exploited (together with an induction on dimension argument) to fully classify such approximate groups (in the finite case, at least).
There are some cases where the analysis is particularly simple. For instance, in the bounded torsion case, one can show that the associated Lie model is necessarily zero-dimensional, which allows for a easy classification of approximate groups of bounded torsion.
Some of the material here is drawn from my recent paper with Ben Green and Emmanuel Breuillard, which is in turn inspired by a previous paper of Hrushovski.
Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “The structure of approximate groups“, submitted to Pub. IHES. We had announced the main results of this paper in various forums (including this blog) for a few months now, but it had taken some time to fully write up the paper and put in various refinements and applications.
As announced previously, the main result of this paper is what is a (virtually, qualitatively) complete description of finite approximate groups in an arbitrary (local or global) group . For simplicity let us work in the much more familiar setting of global groups, although our results also apply (but are a bit more technical to state) in the local group setting.
Recall that in a global group , a -approximate group is a symmetric subset of containing the origin, with the property that the product set is covered by left-translates of . Examples of -approximate groups include genuine groups, convex bodies in a bounded dimensional vector space, small balls in a bounded dimensional Lie group, large balls in a discrete nilpotent group of bounded rank or step, or generalised arithmetic progressions (or more generally, coset progressions) of bounded rank in an abelian group. Specialising now to finite approximate groups, a key example of such a group is what we call a coset nilprogression: a set of the form , where is a homomorphism with finite kernel from a subgroup of to a nilpotent group of bounded step, and is a nilprogression with a bounded number of generators in and some lengths , where consists of all the words involving at most copies of , copies of , and so forth up to copies of . One can show (by some nilpotent algebra) that all such coset nilprogressions are -approximate groups so long as the step and the rank are bounded (and if are sufficiently large).
Our main theorem (which was essentially conjectured independently by Helfgott and by Lindenstrauss) asserts, roughly speaking, that coset nilprogressions are essentially the only examples of approximate groups.
Theorem 1 Let be a -approximate group. Then contains a coset nilprogression of rank and step , such that can be covered by left-translates of .
In the torsion-free abelian case, this result is essentially Freiman’s theorem (with an alternate proof by Ruzsa); for general abelian case, it is due to Green and Ruzsa. Various partial results in this direction for some other groups (e.g. free groups, nilpotent groups, solvable groups, or simple groups of Lie type) are also known; see these previous blog posts for a summary of several of these results.
This result has a number of applications to geometric growth theory, and in particular to variants of Gromov’s theorem of groups of polynomial growth, which asserts that a finitely generated group is of polynomial growth if and only if it is virtually nilpotent. The connection lies in the fact that if the balls associated to a finite set of generators has polynomial growth, then some simple volume-packing arguments combined with the pigeonhole principle will show that will end up being a -approximate group for many radii . In fact, since our theorem only needs a single approximate group to obtain virtually nilpotent structure, we are able to obtain some new strengthenings of Gromov’s theorem. For instance, if is any -approximate group in a finitely generated group that contains for some set of generators and some that is sufficiently large depending on , our theorem implies that is virtually nilpotent, answering a question of Petrunin. Among other things, this gives an alternate proof of a recent result of Kapovitch and Wilking (see also this previous paper of Cheeger and Colding) that a compact manifold of bounded diameter and Ricci curvature at least necessarily has a virtually nilpotent fundamental group if is sufficiently small (depending only on dimension). The main point here is that no lower bound on the injectivity radius is required. Another application is a “Margulis-type lemma”, which asserts that if a metric space has “bounded packing” (in the sense that any ball of radius (say) is covered by a bounded number of balls of radius ), and is a group of isometries on that acts discretely (i.e. every orbit has only finitely many elements (counting multiplicity) in each bounded set), then the near-stabiliser of a point is virtually nilpotent if is small enough depending on the packing constant.
There are also some variants and refinements to the main theorem proved in the paper, such as an extension to local groups, and also an improvement on the bound on the rank and step from to (but at the cost of replacing in the theorem with ).
I’ll be discussing the proof of the main theorem in detail in the next few lecture notes of my current graduate course. The full proof is somewhat lengthy (occupying about 50 pages of the 90-page paper), but can be summarised in the following steps:
- (Hrushovski) Take an arbitrary sequence of finite -approximate groups, and show that an appropriate limit of such groups can be “modeled” in some sense by an open bounded subset of a locally compact group. (The precise definition of “model” is technical, but “macroscopically faithful representation” is a good first approximation.) As discussed in the previous lecture notes, we use an ultralimit for this purpose; the paper of Hrushovski where this strategy was first employed also considered more sophisticated model-theoretic limits. To build a locally compact topology, Hrushovski used some tools from definability theory; in our paper, we instead use a combinatorial lemma of Sanders (closely related to a similar result of Croot and Sisask.)
- (Gleason-Yamabe) The locally compact group can in turn be “modeled” by a Lie group (possibly after shrinking the group, and thus the ultralimit , slightly). (This result arose from the solution to Hilbert’s fifth problem, as discussed here. For our extension to local groups, we use a recent local version of the Gleason-Yamabe theorem, due to Goldbring.)
- (Gleason) Using the escape properties of the Lie model, construct a norm (and thus a left-invariant metric ) on the ultralimit approximate group (and also on the finitary groups ) that obeys a number of good properties, such as a commutator estimate . (This is modeled on an analogous construction used in the theory of Hilbert’s fifth problem, as discussed in this previous set of lecture notes.) This norm is essentially an escape norm associated to (a slight modification) of or .
- (Jordan-Bieberbach-Frobenius) We now take advantage of the finite nature of the by locating the non-trivial element of with minimal escape norm (but one has to first quotient out the elements of zero escape norm first). The commutator estimate mentioned previously ensures that this element is essentially “central” in . One can then quotient out a progression generated by this central element (reducing the dimension of the Lie model by one in the process) and iterates the process until the dimension of the model drops to zero. Reversing the process, this constructs a coset nilprogression inside . This argument is based on the classic proof of Jordan’s theorem due to Bieberbach and Frobenius, as discussed in this blog post.
One quirk of the argument is that it requires one to work in the category of local groups rather than global groups. (This is somewhat analogous to how, in the standard proofs of Freiman’s theorem, one needs to work with the category of Freiman homomorphisms, rather than group homomorphisms.) The reason for this arises when performing the quotienting step in the Jordan-Bieberbach-Frobenius leg of the argument. The obvious way to perform this step (and the thing that we tried first) would be to quotient out by the entire cyclic group generated by the element of minimal escape norm. However, it turns out that this doesn’t work too well, because the group quotiented out is so “large” that it can create a lot of torsion in the quotient. In particular, elements which used to have positive escape norm, can now become trapped in the quotient of , thus sending their escape norm to zero. This leads to an inferior conclusion (in which a coset nilprogression is replaced by a more complicated tower of alternating extensions between central progressions and finite groups, similar to the towers encountered in my previous paper on this topic). To prevent this unwanted creation of torsion, one has to truncate the cyclic group before it escapes , so that one quotients out by a geometric progression rather than the cyclic group. But the operation of quotienting out by a , which is a local group rather than a global one, cannot be formalised in the category of global groups, but only in the category of local groups. Because of this, we were forced to carry out the entire argument using the language of local groups. As it turns out, the arguments are ultimately more natural in this setting, although there is an initial investment of notation required, given that global group theory is much more familiar and well-developed than local group theory.
One interesting feature of the argument is that it does not use much of the existing theory of Freiman-type theorems, instead building the coset nilprogression directly from the geometric properties of the approximate group. In particular, our argument gives a new proof of Freiman’s theorem in the abelian case, which largely avoids Fourier analysis (except through the use of the theory of Hilbert’s fifth problem, which uses the Peter-Weyl theorem (or, in the abelian case, Pontryagin duality), which is basically a version of Fourier analysis).
In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:
Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.
In the next set of notes, we will combine the Gleason-Yamabe theorem with some topological analysis (and in particular, using the invariance of domain theorem) to establish some further control on locally compact groups, and in particular obtaining a solution to Hilbert’s fifth problem.
To prove the Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a Gleason metric:
Definition 2 Let be a topological group. A Gleason metric on is a left-invariant metric which generates the topology on and obeys the following properties for some constant , writing for :
- (Escape property) If and is such that , then .
- (Commutator estimate) If are such that , then
where is the commutator of and .
The second tool is the existence of a left-invariant Haar measure on any locally compact group; see Theorem 3 from Notes 3. Finally, we will also need the compact case of the Gleason-Yamabe theorem (Theorem 8 from Notes 3), which was proven via the Peter-Weyl theorem:
Theorem 4 (Gleason-Yamabe theorem for compact groups) Let be a compact Hausdorff group, and let be a neighbourhood of the identity. Then there exists a compact normal subgroup of contained in such that is isomorphic to a linear group (i.e. a closed subgroup of a general linear group ).
To finish the proof of the Gleason-Yamabe theorem, we have to somehow use the available structures on locally compact groups (such as Haar measure) to build good metrics on those groups (or on suitable subgroups or quotient groups). The basic construction is as follows:
Definition 5 (Building metrics out of test functions) Let be a topological group, and let be a bounded non-negative function. Then we define the pseudometric by the formula
and the semi-norm by the formula
Note that one can also write
where is the “derivative” of in the direction .
Exercise 1 Let the notation and assumptions be as in the above definition. For any , establish the metric-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
- (Left-invariance) . In particular, .
In particular, we have the norm-like properties
- (Identity) , with equality when .
- (Symmetry) .
- (Triangle inequality) .
- (Continuity) If , then the map is continuous.
- (Boundedness) One has . If is supported in a set , then equality occurs unless .
We remark that the first three properties of in the above exercise ensure that is indeed a pseudometric.
To get good metrics (such as Gleason metrics) on groups , it thus suffices to obtain test functions that obey suitably good “regularity” properties. We will achieve this primarily by means of two tricks. The first trick is to obtain high-regularity test functions by convolving together two low-regularity test functions, taking advantage of the existence of a left-invariant Haar measure on . The second trick is to obtain low-regularity test functions by means of a metric-like object on . This latter trick may seem circular, as our whole objective is to get a metric on in the first place, but the key point is that the metric one starts with does not need to have as many “good properties” as the metric one ends up with, thanks to the regularity-improving properties of convolution. As such, one can use a “bootstrap argument” (or induction argument) to create a good metric out of almost nothing. It is this bootstrap miracle which is at the heart of the proof of the Gleason-Yamabe theorem (and hence to the solution of Hilbert’s fifth problem).
The arguments here are based on the nonstandard analysis arguments used to establish Hilbert’s fifth problem by Hirschfeld and by Goldbring (and also some unpublished lecture notes of Goldbring and van den Dries). However, we will not explicitly use any nonstandard analysis in this post.
One of the fundamental inequalities in convex geometry is the Brunn-Minkowski inequality, which asserts that if are two non-empty bounded open subsets of , then
is the sumset of and , and denotes Lebesgue measure. The estimate is sharp, as can be seen by considering the case when are convex bodies that are dilates of each other, thus for some , since in this case one has , , and .
The Brunn-Minkowski inequality has many applications in convex geometry. To give just one example, if we assume that has a smooth boundary , and set equal to a small ball , then , and in the limit one has
where is the surface measure of ; applying the Brunn-Minkowski inequality and performing a Taylor expansion, one soon arrives at the isoperimetric inequality
Thus one can view the isoperimetric inequality as an infinitesimal limit of the Brunn-Minkowski inequality.
There are many proofs known of the Brunn-Minkowski inequality. Firstly, the inequality is trivial in one dimension:
Proof: By inner regularity we may assume that are compact. The claim then follows since contains the sets and , which meet only at a single point .
For the higher dimensional case, the inequality can be established from the Prékopa-Leindler inequality:
This inequality is usually stated using instead of in order to eliminate the ungainly factor . However, we formulate the inequality in this fashion in order to avoid any reference to the dilation maps ; the reason for this will become clearer later.
for any . We can now optimise in ; the optimal value turns out to be
which yields (1).
To prove the Prékopa-Leindler inequality, we first observe that the inequality tensorises in the sense that if it is true in dimensions and , then it is automatically true in dimension . Indeed, if are measurable functions obeying (2) in dimension , then for any , the functions obey (2) in dimension . Applying the Prékopa-Leindler inequality in dimension , we conclude that
for all , where and similarly for . But then if we apply the Prékopa-Leindler inequality again, this time in dimension and to the functions , , and , and then use the Fubini-Tonelli theorem, we obtain (3).
From tensorisation, we see that to prove the Prékopa-Leindler inequality it suffices to do so in the one-dimensional case. We can derive this from Lemma 1 by reversing the “Prékopa-Leindler implies Brunn-Minkowski” argument given earlier, as follows. We can normalise to have sup norm . If (2) holds (in one dimension), then the super-level sets are related by the set-theoretic inclusion
and thus by Lemma 1
whenever . On the other hand, from the Fubini-Tonelli theorem one has the distributional identity
(and similarly for , but with restricted to ), and thus
The claim then follows from the weighted arithmetic mean-geometric mean inequality .
In this post, I wanted to record the simple observation (which appears in this paper of Leonardi and Mansou in the case of the Heisenberg group, but may have also been stated elsewhere in the literature) that the above argument carries through without much difficulty to the nilpotent setting, to give a nilpotent Brunn-Minkowski inequality:
Theorem 3 (Nilpotent Brunn-Minkowski) Let be a connected, simply connected nilpotent Lie group of (topological) dimension , and let be bounded open subsets of . Let be a Haar measure on (note that nilpotent groups are unimodular, so there is no distinction between left and right Haar measure). Then
Here of course is the product set of and .
Indeed, by repeating the previous arguments, the nilpotent Brunn-Minkowski inequality will follow from
Theorem 4 (Nilpotent Prékopa-Leindler inequality) Let be a connected, simply connected nilpotent Lie group of topological dimension with a Haar measure . Let , and let be non-negative measurable functions obeying the inequality
To prove the nilpotent Prékopa-Leindler inequality, the key observation is that this inequality not only tensorises; it splits with respect to short exact sequences. Indeed, suppose one has a short exact sequence
of connected, simply connected nilpotent Lie groups. The adjoint action of the connected group on acts nilpotently on the Lie algebra of and is thus unimodular. Because of this, we can split a Haar measure on into Haar measures on respectively so that we have the Fubini-Tonelli formula
for any measurable , where is defined by the formula
for any coset representative of (the choice of is not important, thanks to unimodularity of the conjugation action). It is then not difficult to repeat the proof of tensorisation (relying heavily on the unimodularity of conjugation) to conclude that the nilpotent Prékopa-Leindler inequality for and implies the Prékopa-Leindler inequality for ; we leave this as an exercise to the interested reader.
Now if is a connected simply connected Lie group, then the abeliansation is connected and simply connected and thus isomorphic to a vector space. This implies that is a retract of and is thus also connected and simply connected. From this and an induction of the step of the nilpotent group, we see that the nilpotent Prékopa-Leindler inequality follows from the abelian case, which we have already established in Theorem 2.
Remark 1 Some connected, simply connected nilpotent groups (and specifically, the Carnot groups) can be equipped with a one-parameter family of dilations , which are a family of automorphisms on , which dilate the Haar measure by the formula
for an integer , called the homogeneous dimension of , which is typically larger than the topological dimension. For instance, in the case of the Heisenberg group
which has topological dimension , the natural family of dilations is given by
with homogeneous dimension . Because the two notions of dimension are usually distinct in the nilpotent case, it is no longer helpful to try to use these dilations to simplify the proof of the Brunn-Minkowski inequality, in contrast to the Euclidean case. This is why we avoided using dilations in the preceding discussion. It is natural to wonder whether one could replace by in (4), but it can be easily shown that the exponent is best possible (an observation that essentially appeared first in this paper of Monti). Indeed, working in the Heisenberg group for sake of concreteness, consider the set
for some large parameter . This set has measure using the standard Haar measure on . The product set is contained in
and thus has measure at most . This already shows that the exponent in (4) cannot be improved beyond ; note that the homogeneous dimension is making its presence known in the term in the measure of , but this is a lower order term only.
It is somewhat unfortunate that the nilpotent Brunn-Minkowski inequality is adapted to the topological dimension rather than the homogeneous one, because it means that some of the applications of the inequality (such as the application to isoperimetric inequalities mentioned at the start of the post) break down. (Indeed, the topic of isoperimetric inequalities for the Heisenberg group is a subtle one, with many naive formulations of the inequality being false. See the paper of Monti for more discussion.)
Remark 2 The inequality can be extended to non-simply-connected connected nilpotent groups , if is now set to the dimension of the largest simply connected quotient of . It seems to me that this is the best one can do in general; for instance, if is a torus, then the inequality fails for any , as can be seen by setting .
Remark 3 Specialising the nilpotent Brunn-Minkowski inequality to the case , we conclude that
This inequality actually has a much simpler proof (attributed to Tsachik Gelander in this paper of Hrushovski, as pointed out to me by Emmanuel Breuillard): one can show that for a connected, simply connected Lie group , the exponential map is a measure-preserving homeomorphism, for some choice of Haar measure on , so it suffices to show that
But contains all the squares of , so contains the isotropic dilation , and the claim follows. Note that if we set to be a small ball around the origin, we can modify this argument to give another demonstration of why the topological dimension cannot be replaced with any larger exponent in (4).
One may tentatively conjecture that the inequality in fact holds in all unimodular connected, simply connected Lie groups , and all bounded open subsets of ; I do not know if this bound is always true, however.
Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group to ensure that it is actually a Lie group. In the previous set of notes, we saw that one could reduce the regularity hypothesis imposed on to a “” condition, namely that there was an open neighbourhood of that was isomorphic (as a local group) to an open subset of a Euclidean space with identity element , and with group operation obeying the asymptotic
for sufficiently small . We will call such local groups local groups.
We now reduce the regularity hypothesis further, to one in which there is no explicit Euclidean space that is initially attached to . Of course, Lie groups are still locally Euclidean, so if the hypotheses on do not involve any explicit Euclidean spaces, then one must somehow build such spaces from other structures. One way to do so is to exploit an ambient space with Euclidean or Lie structure that is embedded or immersed in. A trivial example of this is provided by the following basic fact from linear algebra:
Lemma 1 If is a finite-dimensional vector space (i.e. it is isomorphic to for some ), and is a linear subspace of , then is also a finite-dimensional vector space.
We will establish a non-linear version of this statement, known as Cartan’s theorem. Recall that a subset of a -dimensional smooth manifold is a -dimensional smooth (embedded) submanifold of for some if for every point there is a smooth coordinate chart of a neighbourhood of in that maps to , such that , where we identify with a subspace of . Informally, locally sits inside the same way that sits inside .
Note that the hypothesis that is closed is essential; for instance, the rationals are a subgroup of the (additive) group of reals , but the former is not a Lie group even though the latter is.
Exercise 1 Let be a subgroup of a locally compact group . Show that is closed in if and only if it is locally compact.
A variant of the above results is provided by using (faithful) representations instead of embeddings. Again, the linear version is trivial:
Lemma 3 If is a finite-dimensional vector space, and is another vector space with an injective linear transformation from to , then is also a finite-dimensional vector space.
Here is the non-linear version:
Actually, it will suffice for the homomorphism to be locally injective rather than injective; related to this, von Neumann’s theorem localises to the case when is a local group rather a group. The requirement that be locally compact is necessary, for much the same reason that the requirement that be closed was necessary in Cartan’s theorem.
Example 1 Let be the two-dimensional torus, let , and let be the map , where is a fixed real number. Then is a continuous homomorphism which is locally injective, and is even globally injective if is irrational, and so Theorem 4 is consistent with the fact that is a Lie group. On the other hand, note that when is irrational, then is not closed; and so Theorem 4 does not follow immediately from Theorem 2 in this case. (We will see, though, that Theorem 4 follows from a local version of Theorem 2.)
As a corollary of Theorem 4, we observe that any locally compact Hausdorff group with a faithful linear representation, i.e. a continuous injective homomorphism from into a linear group such as or , is necessarily a Lie group. This suggests a representation-theoretic approach to Hilbert’s fifth problem. While this approach does not seem to readily solve the entire problem, it can be used to establish a number of important special cases with a well-understood representation theory, such as the compact case or the abelian case (for which the requisite representation theory is given by the Peter-Weyl theorem and Pontryagin duality respectively). We will discuss these cases further in later notes.
In all of these cases, one is not really building up Euclidean or Lie structure completely from scratch, because there is already a Euclidean or Lie structure present in another object in the hypotheses. Now we turn to results that can create such structure assuming only what is ostensibly a weaker amount of structure. In the linear case, one example of this is is the following classical result in the theory of topological vector spaces.
Remark 1 The Banach-Alaoglu theorem asserts that in a normed vector space , the closed unit ball in the dual space is always compact in the weak-* topology. Of course, this dual space may be infinite-dimensional. This however does not contradict the above theorem, because the closed unit ball is not a neighbourhood of the origin in the weak-* topology (it is only a neighbourhood with respect to the strong topology).
The full non-linear analogue of this theorem would be the Gleason-Yamabe theorem, which we are not yet ready to prove in this set of notes. However, by using methods similar to that used to prove Cartan’s theorem and von Neumann’s theorem, one can obtain a partial non-linear analogue which requires an additional hypothesis of a special type of metric, which we will call a Gleason metric:
Definition 6 Let be a topological group. A Gleason metric on is a left-invariant metric which generates the topology on and obeys the following properties for some constant , writing for :
- (Escape property) If and is such that , then .
- (Commutator estimate) If are such that , then
where is the commutator of and .
Exercise 2 Let be a topological group that contains a neighbourhood of the identity isomorphic to a local group. Show that admits at least one Gleason metric.
We will rely on Theorem 7 to solve Hilbert’s fifth problem; this theorem reduces the task of establishing Lie structure on a locally compact group to that of building a metric with suitable properties. Thus, much of the remainder of the solution of Hilbert’s fifth problem will now be focused on the problem of how to construct good metrics on a locally compact group.
In all of the above results, a key idea is to use one-parameter subgroups to convert from the nonlinear setting to the linear setting. Recall from the previous notes that in a Lie group , the one-parameter subgroups are in one-to-one correspondence with the elements of the Lie algebra , which is a vector space. In a general topological group , the concept of a one-parameter subgroup (i.e. a continuous homomorphism from to ) still makes sense; the main difficulties are then to show that the space of such subgroups continues to form a vector space, and that the associated exponential map is still a local homeomorphism near the origin.
Exercise 3 The purpose of this exercise is to illustrate the perspective that a topological group can be viewed as a non-linear analogue of a vector space. Let be locally compact groups. For technical reasons we assume that are both -compact and metrisable.
- (i) (Open mapping theorem) Show that if is a continuous homomorphism which is surjective, then it is open (i.e. the image of open sets is open). (Hint: mimic the proof of the open mapping theorem for Banach spaces, as discussed for instance in these notes. In particular, take advantage of the Baire category theorem.)
- (ii) (Closed graph theorem) Show that if a homomorphism is closed (i.e. its graph is a closed subset of ), then it is continuous. (Hint: mimic the derivation of the closed graph theorem from the open mapping theorem in the Banach space case, as again discussed in these notes.)
- (iii) Let be a homomorphism, and let be a continuous injective homomorphism into another Hausdorff topological group . Show that is continuous if and only if is continuous.
- (iv) Relax the condition of metrisability to that of being Hausdorff. (Hint: Now one cannot use the Baire category theorem for metric spaces; but there is an analogue of this theorem for locally compact Hausdorff spaces.)