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Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group acting in a measure-preserving fashion on a probability space , thus each group element gives rise to a measure-preserving map . Define the *third Gowers-Host-Kra seminorm* of a function via the formula

*Conze-Lesigne factor*or the

*second Host-Kra-Ziegler factor*of the system, and this factor controls all the complexity two recurrence properties of the system.

The analogous theory in complexity one is well understood. Here, one replaces the norm by the norm

and the ergodic systems for which is a norm are called*Kronecker systems*. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group equipped with Haar probability measure and a translation action for some homomorphism with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).

We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) *nilsystems*, in which the space is a quotient of a two-step nilpotent Lie group by a lattice (equipped with Haar probability measure), and the action is given by a translation for some group homomorphism . For instance, the Heisenberg -nilsystem

Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which is a locally compact Polish group rather than a Lie group:

Theorem 1 (Classification of Conze-Lesigne systems)Let be a countable abelian group, and an ergodic measure-preserving -system. Then is a Conze-Lesigne system if and only if it is the inverse limit of translational systems , where is a nilpotent locally compact Polish group of nilpotency class two, and is a lattice in (and also a lattice in the commutator group ), with equipped with the Haar probability measure and a translation action for some homomorphism .

In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm for an arbitrary finite abelian group (with no restrictions on the order of , in particular our result handles the case of even and odd in a unified fashion). In principle, having a higher order version of this theorem will similarly allow us to derive inverse theorems for norms for arbitrary and finite abelian ; we hope to investigate this further in future work.

We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension , where is a Kronecker system (a rotational system on a compact abelian group and translation action ), is another compact abelian group, and the cocycle is a collection of measurable maps obeying the cocycle equation

for almost all . Furthermore, is of “type two”, which means in this concrete setting that it obeys an additional equation for all and almost all , and some measurable function ; roughly speaking this asserts that is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group is a circle , one can use some Fourier analysis to convert (2) into the more tractable*Conze-Lesigne equation*for all , all , and almost all , where for each , is a measurable function, and is a homomorphism. (For technical reasons it is often also convenient to enforce that depend in a measurable fashion on ; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group (as long as one has the measurability in mentioned previously), but the converse turns out to fail for some groups , such as solenoid groups (e.g., inverse limits of as ), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group . Putting together the and cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian

*Lie*groups (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system as a translational system , where the

*Host-Kra group*is the set of all pairs that solve an equation of the form (3) (with these pairs acting on by the law ), and is the stabiliser of a point in this system. This then establishes the theorem in the case when is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)

There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of on is technically only a *near-action* rather than a genuine action, and as such one cannot directly define to be the stabiliser of a point without running into multiple problems. To fix this, one has to pass to a *topological model* of in which the action becomes continuous, and the stabilizer becomes well defined, although one then has to work a little more to check that the action is still transitive. This can be done via Gelfand duality; we proceed using a mixture of a construction from this book of Host and Kra, and the machinery in this recent paper of Asgar and myself.

Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case . As this group embeds into the torus , it is easy to use existing methods obtain (3) but with the homomorphism and the function taking values in rather than in . The main task is then to fix up the homomorphism so that it takes values in , that is to say that vanishes. This only needs to be done locally near the origin, because the claim is easy when lies in the dense subgroup of , and also because the claim can be shown to be additive in . Near the origin one can leverage the Steinhaus lemma to make depend linearly (or more precisely, homomorphically) on , and because the cocycle already takes values in , vanishes and must be an eigenvalue of the system . But as was assumed to be separable, there are only countably many eigenvalues, and by another application of Steinhaus and linearity one can then make vanish on an open neighborhood of the identity, giving the claim.

Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Mackey-Zimmer theorem“. This paper is part of our longer term project to develop “uncountable” versions of various theorems in ergodic theory; see this previous paper of Asgar and myself for the first paper in this series (and another paper will appear shortly).

In this case the theorem in question is the Mackey-Zimmer theorem, previously discussed in this blog post. This theorem gives an important classification of group and homogeneous extensions of measure-preserving systems. Let us first work in the (classical) setting of concrete measure-preserving systems. Let be a measure-preserving system for some group , thus is a (concrete) probability space and is a group homomorphism from to the automorphism group of the probability space. (Here we are abusing notation by using to refer both to the measure-preserving system and to the underlying set. In the notation of the paper we would instead distinguish these two objects as and respectively, reflecting two of the (many) categories one might wish to view as a member of, but for sake of this informal overview we will not maintain such precise distinctions.) If is a compact group, we define a *(concrete) cocycle* to be a collection of measurable functions for that obey the *cocycle equation*

*group skew-product*of , which is another measure-preserving system where

- is the Cartesian product of and ;
- is the product measure of and Haar probability measure on ; and
- The action is given by the formula

*homogeneous skew-product*in which the group is replaced by the homogeneous space for some closed subgroup of , noting that still comes with a left-action of and a Haar measure. Group skew-products are very “explicit” ways to extend a system , as everything is described by the cocycle which is a relatively tractable object to manipulate. (This is not to say that the cohomology of measure-preserving systems is trivial, but at least there are many tools one can use to study them, such as the Moore-Schmidt theorem discussed in this previous post.)

This group skew-product comes with a factor map and a coordinate map , which by (2) are related to the action via the identities

and where in (4) we are implicitly working in the group of (concretely) measurable functions from to . Furthermore, the combined map is measure-preserving (using the product measure on ), indeed the way we have constructed things this map is just the identity map.
We can now generalize the notion of group skew-product by just working with the maps , and weakening the requirement that be measure-preserving. Namely, define a *group extension* of by to be a measure-preserving system equipped with a measure-preserving map obeying (3) and a measurable map obeying (4) for some cocycle , such that the -algebra of is generated by . There is also a more general notion of a *homogeneous extension* in which takes values in rather than . Then every group skew-product is a group extension of by , but not conversely. Here are some key counterexamples:

- (i) If is a closed subgroup of , and is a cocycle taking values in , then can be viewed as a group extension of by , taking to be the vertical coordinate (viewing now as an element of ). This will not be a skew-product by because pushes forward to the wrong measure on : it pushes forward to rather than .
- (ii) If one takes the same example as (i), but twists the vertical coordinate to another vertical coordinate for some measurable “gauge function” , then is still a group extension by , but now with the cocycle replaced by the cohomologous cocycle Again, this will not be a skew product by , because pushes forward to a twisted version of that is supported (at least in the case where is compact and the cocycle is continuous) on the -bundle .
- (iii) With the situation as in (i), take to be the union for some outside of , where we continue to use the action (2) and the standard vertical coordinate but now use the measure .

As it turns out, group extensions and homogeneous extensions arise naturally in the Furstenberg-Zimmer structural theory of measure-preserving systems; roughly speaking, every compact extension of is an inverse limit of group extensions. It is then of interest to classify such extensions.

Examples such as (iii) are annoying, but they can be excluded by imposing the additional condition that the system is ergodic – all invariant (or essentially invariant) sets are of measure zero or measure one. (An essentially invariant set is a measurable subset of such that is equal modulo null sets to for all .) For instance, the system in (iii) is non-ergodic because the set (or ) is invariant but has measure . We then have the following fundamental result of Mackey and Zimmer:

Theorem 1 (Countable Mackey Zimmer theorem)Let be a group, be a concrete measure-preserving system, and be a compact Hausdorff group. Assume that is at most countable, is a standard Borel space, and is metrizable. Then every (concrete) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (concrete) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We will not define precisely what “abstractly isomorphic” means here, but it roughly speaking means “isomorphic after quotienting out the null sets”. A proof of this theorem can be found for instance in .

The main result of this paper is to remove the “countability” hypotheses from the above theorem, at the cost of working with opposite probability algebra systems rather than concrete systems. (We will discuss opposite probability algebras in a subsequent blog post relating to another paper in this series.)

Theorem 2 (Uncountable Mackey Zimmer theorem)Let be a group, be an opposite probability algebra measure-preserving system, and be a compact Hausdorff group. Then every (abstract) ergodic group extension of is abstractly isomorphic to a group skew-product (by some closed subgroup of ), and every (abstract) ergodic homogeneous extension of is similarly abstractly isomorphic to a homogeneous skew-product.

We plan to use this result in future work to obtain uncountable versions of the Furstenberg-Zimmer and Host-Kra structure theorems.

As one might expect, one locates a proof of Theorem 2 by finding a proof of Theorem 1 that does not rely too strongly on “countable” tools, such as disintegration or measurable selection, so that all of those tools can be replaced by “uncountable” counterparts. The proof we use is based on the one given in this previous post, and begins by comparing the system with the group extension . As the examples (i), (ii) show, these two systems need not be isomorphic even in the ergodic case, due to the different probability measures employed. However one can relate the two after performing an additional averaging in . More precisely, there is a canonical factor map given by the formula

This is a factor map not only of -systems, but actually of -systems, where the opposite group to acts (on the left) by right-multiplication of the second coordinate (this reversal of order is why we need to use the opposite group here). The key point is that the ergodicity properties of the system are closely tied the group that is “secretly” controlling the group extension. Indeed, in example (i), the invariant functions on take the form for some measurable , while in example (ii), the invariant functions on take the form . In either case, the invariant factor is isomorphic to , and can be viewed as a factor of the invariant factor of , which is isomorphic to . Pursuing this reasoning (using an abstract ergodic theorem of Alaoglu and Birkhoff, as discussed in the previous post) one obtains the*Mackey range*, and also obtains the quotient of to in this process. The main remaining task is to lift the quotient back up to a map that stays measurable, in order to “untwist” a system that looks like (ii) to make it into one that looks like (i). In countable settings this is where a “measurable selection theorem” would ordinarily be invoked, but in the uncountable setting such theorems are not available for concrete maps. However it turns out that they still remain available for abstract maps: any abstractly measurable map from to has an abstractly measurable lift from to . To prove this we first use a canonical model for opposite probability algebras (which we will discuss in a companion post to this one, to appear shortly) to work with continuous maps (on a Stone space) rather than abstractly measurable maps. The measurable map then induces a probability measure on , formed by pushing forward by the graphing map . This measure in turn has several lifts up to a probability measure on ; for instance, one can construct such a measure via the Riesz representation theorem by demanding for all continuous functions . This measure does not come from a graph of any single lift , but is in some sense an “average” of the entire ensemble of these lifts. But it turns out one can invoke the Krein-Milman theorem to pass to an extremal lifting measure which

*does*come from an (abstract) lift , and this can be used as a substitute for a measurable selection theorem. A variant of this Krein-Milman argument can also be used to express any homogeneous extension as a quotient of a group extension, giving the second part of the Mackey-Zimmer theorem.

Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Moore-Schmidt theorem“. This paper revisits a classical theorem of Moore and Schmidt in measurable cohomology of measure-preserving systems. To state the theorem, let be a probability space, and be the group of measure-preserving automorphisms of this space, that is to say the invertible bimeasurable maps that preserve the measure : . To avoid some ambiguity later in this post when we introduce abstract analogues of measure theory, we will refer to measurable maps as *concrete measurable maps*, and measurable spaces as *concrete measurable spaces*. (One could also call a concrete probability space, but we will not need to do so here as we will not be working explicitly with abstract probability spaces.)

Let be a discrete group. A *(concrete) measure-preserving action* of on is a group homomorphism from to , thus is the identity map and for all . A large portion of ergodic theory is concerned with the study of such measure-preserving actions, especially in the classical case when is the integers (with the additive group law).

Let be a compact Hausdorff abelian group, which we can endow with the Borel -algebra . A *(concrete measurable) –cocycle* is a collection of concrete measurable maps obeying the *cocycle equation*

for -almost every . (Here we are glossing over a measure-theoretic subtlety that we will return to later in this post – see if you can spot it before then!) Cocycles arise naturally in the theory of group extensions of dynamical systems; in particular (and ignoring the aforementioned subtlety), each cocycle induces a measure-preserving action on (which we endow with the product of with Haar probability measure on ), defined by

This connection with group extensions was the original motivation for our study of measurable cohomology, but is not the focus of the current paper.

A special case of a -valued cocycle is a *(concrete measurable) -valued coboundary*, in which for each takes the special form

for -almost every , where is some measurable function; note that (ignoring the aforementioned subtlety), every function of this form is automatically a concrete measurable -valued cocycle. One of the first basic questions in measurable cohomology is to try to characterize which -valued cocycles are in fact -valued coboundaries. This is a difficult question in general. However, there is a general result of Moore and Schmidt that at least allows one to reduce to the model case when is the unit circle , by taking advantage of the Pontryagin dual group of characters , that is to say the collection of continuous homomorphisms to the unit circle. More precisely, we have

Theorem 1 (Countable Moore-Schmidt theorem)Let be a discrete group acting in a concrete measure-preserving fashion on a probability space . Let be a compact Hausdorff abelian group. Assume the following additional hypotheses:

- (i) is at most countable.
- (ii) is a standard Borel space.
- (iii) is metrisable.
Then a -valued concrete measurable cocycle is a concrete coboundary if and only if for each character , the -valued cocycles are concrete coboundaries.

The hypotheses (i), (ii), (iii) are saying in some sense that the data are not too “large”; in all three cases they are saying in some sense that the data are only “countably complicated”. For instance, (iii) is equivalent to being second countable, and (ii) is equivalent to being modeled by a complete separable metric space. It is because of this restriction that we refer to this result as a “countable” Moore-Schmidt theorem. This theorem is a useful tool in several other applications, such as the Host-Kra structure theorem for ergodic systems; I hope to return to these subsequent applications in a future post.

Let us very briefly sketch the main ideas of the proof of Theorem 1. Ignore for now issues of measurability, and pretend that something that holds almost everywhere in fact holds everywhere. The hard direction is to show that if each is a coboundary, then so is . By hypothesis, we then have an equation of the form

for all and some functions , and our task is then to produce a function for which

for all .

Comparing the two equations, the task would be easy if we could find an for which

for all . However there is an obstruction to this: the left-hand side of (3) is additive in , so the right-hand side would have to be also in order to obtain such a representation. In other words, for this strategy to work, one would have to first establish the identity

for all . On the other hand, the good news is that if we somehow manage to obtain the equation, then we can obtain a function obeying (3), thanks to Pontryagin duality, which gives a one-to-one correspondence between and the homomorphisms of the (discrete) group to .

Now, it turns out that one cannot derive the equation (4) directly from the given information (2). However, the left-hand side of (2) is additive in , so the right-hand side must be also. Manipulating this fact, we eventually arrive at

In other words, we don’t get to show that the left-hand side of (4) vanishes, but we do at least get to show that it is -invariant. Now let us assume for sake of argument that the action of is ergodic, which (ignoring issues about sets of measure zero) basically asserts that the only -invariant functions are constant. So now we get a weaker version of (4), namely

for some constants .

Now we need to eliminate the constants. This can be done by the following group-theoretic projection. Let denote the space of concrete measurable maps from to , up to almost everywhere equivalence; this is an abelian group where the various terms in (5) naturally live. Inside this group we have the subgroup of constant functions (up to almost everywhere equivalence); this is where the right-hand side of (5) lives. Because is a divisible group, there is an application of Zorn’s lemma (a good exercise for those who are not acquainted with these things) to show that there exists a retraction , that is to say a group homomorphism that is the identity on the subgroup . We can use this retraction, or more precisely the complement , to eliminate the constant in (5). Indeed, if we set

then from (5) we see that

while from (2) one has

and now the previous strategy works with replaced by . This concludes the sketch of proof of Theorem 1.

In making the above argument rigorous, the hypotheses (i)-(iii) are used in several places. For instance, to reduce to the ergodic case one relies on the ergodic decomposition, which requires the hypothesis (ii). Also, most of the above equations only hold outside of a set of measure zero, and the hypothesis (i) and the hypothesis (iii) (which is equivalent to being at most countable) to avoid the problem that an uncountable union of sets of measure zero could have positive measure (or fail to be measurable at all).

My co-author Asgar Jamneshan and I are working on a long-term project to extend many results in ergodic theory (such as the aforementioned Host-Kra structure theorem) to “uncountable” settings in which hypotheses analogous to (i)-(iii) are omitted; thus we wish to consider actions on uncountable groups, on spaces that are not standard Borel, and cocycles taking values in groups that are not metrisable. Such uncountable contexts naturally arise when trying to apply ergodic theory techniques to combinatorial problems (such as the inverse conjecture for the Gowers norms), as one often relies on the ultraproduct construction (or something similar) to generate an ergodic theory translation of these problems, and these constructions usually give “uncountable” objects rather than “countable” ones. (For instance, the ultraproduct of finite groups is a hyperfinite group, which is usually uncountable.). This paper marks the first step in this project by extending the Moore-Schmidt theorem to the uncountable setting.

If one simply drops the hypotheses (i)-(iii) and tries to prove the Moore-Schmidt theorem, several serious difficulties arise. We have already mentioned the loss of the ergodic decomposition and the possibility that one has to control an uncountable union of null sets. But there is in fact a more basic problem when one deletes (iii): the addition operation , while still continuous, can fail to be measurable as a map from to ! Thus for instance the sum of two measurable functions need not remain measurable, which makes even the very definition of a measurable cocycle or measurable coboundary problematic (or at least unnatural). This phenomenon is known as the *Nedoma pathology*. A standard example arises when is the uncountable torus , endowed with the product topology. Crucially, the Borel -algebra generated by this uncountable product is *not* the product of the factor Borel -algebras (the discrepancy ultimately arises from the fact that topologies permit uncountable unions, but -algebras do not); relating to this, the product -algebra is *not* the same as the Borel -algebra , but is instead a strict sub-algebra. If the group operations on were measurable, then the diagonal set

would be measurable in . But it is an easy exercise in manipulation of -algebras to show that if are any two measurable spaces and is measurable in , then the fibres of are contained in some countably generated subalgebra of . Thus if were -measurable, then all the points of would lie in a single countably generated -algebra. But the cardinality of such an algebra is at most while the cardinality of is , and Cantor’s theorem then gives a contradiction.

To resolve this problem, we give a coarser -algebra than the Borel -algebra, namely the *Baire -algebra* , thus coarsening the measurable space structure on to a new measurable space . In the case of compact Hausdorff abelian groups, can be defined as the -algebra generated by the characters ; for more general compact abelian groups, one can define as the -algebra generated by all continuous maps into metric spaces. This -algebra is equal to when is metrisable but can be smaller for other . With this measurable structure, becomes a measurable group; it seems that once one leaves the metrisable world that is a superior (or at least equally good) space to work with than for analysis, as it avoids the Nedoma pathology. (For instance, from Plancherel’s theorem, we see that if is the Haar probability measure on , then (thus, every -measurable set is equivalent modulo -null sets to a -measurable set), so there is no damage to Plancherel caused by passing to the Baire -algebra.

Passing to the Baire -algebra fixes the most severe problems with an uncountable Moore-Schmidt theorem, but one is still faced with an issue of having to potentially take an uncountable union of null sets. To avoid this sort of problem, we pass to the framework of *abstract measure theory*, in which we remove explicit mention of “points” and can easily delete all null sets at a very early stage of the formalism. In this setup, the category of concrete measurable spaces is replaced with the larger category of *abstract measurable spaces*, which we formally define as the opposite category of the category of -algebras (with Boolean algebra homomorphisms). Thus, we define an *abstract measurable space* to be an object of the form , where is an (abstract) -algebra and is a formal placeholder symbol that signifies use of the opposite category, and an *abstract measurable map* is an object of the form , where is a Boolean algebra homomorphism and is again used as a formal placeholder; we call the *pullback map* associated to . [UPDATE: It turns out that this definition of a measurable map led to technical issues. In a forthcoming revision of the paper we also impose the requirement that the abstract measurable map be -complete (i.e., it respects countable joins).] The composition of two abstract measurable maps , is defined by the formula , or equivalently .

Every concrete measurable space can be identified with an abstract counterpart , and similarly every concrete measurable map can be identified with an abstract counterpart , where is the pullback map . Thus the category of concrete measurable spaces can be viewed as a subcategory of the category of abstract measurable spaces. The advantage of working in the abstract setting is that it gives us access to more spaces that could not be directly defined in the concrete setting. Most importantly for us, we have a new abstract space, the *opposite measure algebra* of , defined as where is the ideal of null sets in . Informally, is the space with all the null sets removed; there is a canonical abstract embedding map , which allows one to convert any concrete measurable map into an abstract one . One can then define the notion of an abstract action, abstract cocycle, and abstract coboundary by replacing every occurrence of the category of concrete measurable spaces with their abstract counterparts, and replacing with the opposite measure algebra ; see the paper for details. Our main theorem is then

Theorem 2 (Uncountable Moore-Schmidt theorem)Let be a discrete group acting abstractly on a -finite measure space . Let be a compact Hausdorff abelian group. Then a -valued abstract measurable cocycle is an abstract coboundary if and only if for each character , the -valued cocycles are abstract coboundaries.

With the abstract formalism, the proof of the uncountable Moore-Schmidt theorem is almost identical to the countable one (in fact we were able to make some simplifications, such as avoiding the use of the ergodic decomposition). A key tool is what we call a “conditional Pontryagin duality” theorem, which asserts that if one has an abstract measurable map for each obeying the identity for all , then there is an abstract measurable map such that for all . This is derived from the usual Pontryagin duality and some other tools, most notably the completeness of the -algebra of , and the Sikorski extension theorem.

We feel that it is natural to stay within the abstract measure theory formalism whenever dealing with uncountable situations. However, it is still an interesting question as to when one can guarantee that the abstract objects constructed in this formalism are representable by concrete analogues. The basic questions in this regard are:

- (i) Suppose one has an abstract measurable map into a concrete measurable space. Does there exist a representation of by a concrete measurable map ? Is it unique up to almost everywhere equivalence?
- (ii) Suppose one has a concrete cocycle that is an abstract coboundary. When can it be represented by a concrete coboundary?

For (i) the answer is somewhat interesting (as I learned after posing this MathOverflow question):

- If does not separate points, or is not compact metrisable or Polish, there can be counterexamples to uniqueness. If is not compact or Polish, there can be counterexamples to existence.
- If is a compact metric space or a Polish space, then one always has existence and uniqueness.
- If is a compact Hausdorff abelian group, one always has existence.
- If is a complete measure space, then one always has existence (from a theorem of Maharam).
- If is the unit interval with the Borel -algebra and Lebesgue measure, then one has existence for all compact Hausdorff assuming the continuum hypothesis (from a theorem of von Neumann) but existence can fail under other extensions of ZFC (from a theorem of Shelah, using the method of forcing).
- For more general , existence for all compact Hausdorff is equivalent to the existence of a lifting from the -algebra to (or, in the language of abstract measurable spaces, the existence of an abstract retraction from to ).
- It is a long-standing open question (posed for instance by Fremlin) whether it is relatively consistent with ZFC that existence holds whenever is compact Hausdorff.

Our understanding of (ii) is much less complete:

- If is metrisable, the answer is “always” (which among other things establishes the countable Moore-Schmidt theorem as a corollary of the uncountable one).
- If is at most countable and is a complete measure space, then the answer is again “always”.

In view of the answers to (i), I would not be surprised if the full answer to (ii) was also sensitive to axioms of set theory. However, such set theoretic issues seem to be almost completely avoided if one sticks with the abstract formalism throughout; they only arise when trying to pass back and forth between the abstract and concrete categories.

In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report on this connection at some later point on this blog. Among other things, this theory achieves the remarkable feat of creating a smooth Lie group structure out of what is ostensibly a much weaker structure, namely the structure of a locally compact group. The ability of algebraic structure (in this case, group structure) to upgrade weak regularity (in this case, continuous structure) to strong regularity (in this case, smooth and even analytic structure) seems to be a recurring theme in mathematics, and an important part of what I like to call the “dichotomy between structure and randomness”.

The theory of Hilbert’s fifth problem sprawls across many subfields of mathematics: Lie theory, representation theory, group theory, nonabelian Fourier analysis, point-set topology, and even a little bit of group cohomology. The latter aspect of this theory is what I want to focus on today. The general question that comes into play here is the *extension problem*: given two (topological or Lie) groups and , what is the structure of the possible groups that are formed by extending by . In other words, given a short exact sequence

to what extent is the structure of determined by that of and ?

As an example of why understanding the extension problem would help in structural theory, let us consider the task of classifying the structure of a Lie group . Firstly, we factor out the connected component of the identity as

as Lie groups are locally connected, is discrete. Thus, to understand general Lie groups, it suffices to understand the extensions of discrete groups by connected Lie groups.

Next, to study a connected Lie group , we can consider the conjugation action on the Lie algebra , which gives the adjoint representation . The kernel of this representation consists of all the group elements that commute with all elements of the Lie algebra, and thus (by connectedness) is the center of . The adjoint representation is then faithful on the quotient . The short exact sequence

then describes as a central extension (by the abelian Lie group ) of , which is a connected Lie group with a faithful finite-dimensional linear representation.

This suggests a route to Hilbert’s fifth problem, at least in the case of connected groups . Let be a connected locally compact group that we hope to demonstrate is isomorphic to a Lie group. As discussed in a previous post, we first form the space of one-parameter subgroups of (which should, eventually, become the Lie algebra of ). Hopefully, has the structure of a vector space. The group acts on by conjugation; this action should be both continuous and linear, giving an “adjoint representation” . The kernel of this representation should then be the center of . The quotient is locally compact and has a faithful linear representation, and is thus a Lie group by von Neumann’s version of Cartan’s theorem (discussed in this previous post). The group is locally compact abelian, and so it should be a relatively easy task to establish that it is also a Lie group. To finish the job, one needs the following result:

Theorem 1 (Central extensions of Lie are Lie)Let be a locally compact group which is a central extension of a Lie group by an abelian Lie group . Then is also isomorphic to a Lie group.

This result can be obtained by combining a result of Kuranishi with a result of Gleason; I am recording this argument below the fold. The point here is that while is initially only a topological group, the smooth structures of and can be combined (after a little bit of cohomology) to create the smooth structure on required to upgrade from a topological group to a Lie group. One of the main ideas here is to improve the behaviour of a cocycle by averaging it; this basic trick is helpful elsewhere in the theory, resolving a number of cohomological issues in topological group theory. The result can be generalised to show in fact that arbitrary (topological) extensions of Lie groups by Lie groups remain Lie; this was shown by Gleason. However, the above special case of this result is already sufficient (in conjunction with the rest of the theory, of course) to resolve Hilbert’s fifth problem.

Remark 1We have shown in the above discussion that every connected Lie group is a central extension (by an abelian Lie group) of a Lie group with a faithful continuous linear representation. It is natural to ask whether this central extension is necessary. Unfortunately, not every connected Lie group admits a faithful continuous linear representation. An example (due to Birkhoff) is the Heisenberg-Weyl groupIndeed, if we consider the group elements

and

for some prime , then one easily verifies that has order and is central, and that is conjugate to . If we have a faithful linear representation of , then must have at least one eigenvalue that is a primitive root of unity. If is the eigenspace associated to , then must preserve , and be conjugate to on this space. This forces to have at least distinct eigenvalues on , and hence (and thus ) must have dimension at least . Letting we obtain a contradiction. (On the other hand, is certainly isomorphic to the extension of the linear group by the abelian group .)

Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our paper “An inverse theorem for the uniformity seminorms associated with the action of “. This paper establishes the ergodic inverse theorems that are needed in our other recent paper to establish the inverse conjecture for the Gowers norms over finite fields in high characteristic (and to establish a partial result in low characteristic), as follows:

Theorem.Let be a finite field of characteristic p. Suppose that is a probability space with an ergodic measure-preserving action of . Let be such that the Gowers-Host-Kra seminorm (defined in a previous post) is non-zero.

- In the high-characteristic case , there exists a phase polynomial g of degree <k (as defined in the previous post) such that .
- In general characteristic, there exists a phase polynomial of degree <C(k) for some C(k) depending only on k such that .

This theorem is closely analogous to a similar theorem of Host and Kra on ergodic actions of , in which the role of phase polynomials is played by functions that arise from nilsystem factors of X. Indeed, our arguments rely heavily on the machinery of Host and Kra.

The paper is rather technical (60+ pages!) and difficult to describe in detail here, but I will try to sketch out (in very broad brush strokes) what the key steps in the proof of part 2 of the theorem are. (Part 1 is similar but requires a more delicate analysis at various stages, keeping more careful track of the degrees of various polynomials.)

A dynamical system is a space X, together with an action of some group . [In practice, one often places topological or measure-theoretic structure on X or G, but this will not be relevant for the current discussion. In most applications, G is an abelian (additive) group such as the integers or the reals , but I prefer to use multiplicative notation here.] A useful notion in the subject is that of an (abelian) *cocycle*; this is a function taking values in an abelian group that obeys the *cocycle equation*

(1)

for all and . [Again, if one is placing topological or measure-theoretic structure on the system, one would want to be continuous or measurable, but we will ignore these issues.] The significance of cocycles in the subject is that they allow one to construct (abelian) *extensions* or *skew products* of the original dynamical system X, defined as the Cartesian product with the group action . (The cocycle equation (1) is needed to ensure that one indeed has a group action, and in particular that .) This turns out to be a useful means to build complex dynamical systems out of simpler ones. (For instance, one can build nilsystems by starting with a point and taking a finite number of abelian extensions of that point by a certain type of cocycle.)

A special type of cocycle is a *coboundary*; this is a cocycle that takes the form for some function . (Note that the cocycle equation (1) is automaticaly satisfied if is of this form.) An extension of a dynamical system by a coboundary can be conjugated to the trivial extension by the change of variables .

While every coboundary is a cocycle, the converse is not always true. (For instance, if X is a point, the only coboundary is the zero function, whereas a cocycle is essentially the same thing as a homomorphism from G to U, so in many cases there will be more cocycles than coboundaries. For a contrasting example, if X and G are finite (for simplicity) and G acts freely on X, it is not difficult to see that every cocycle is a coboundary.) One can measure the extent to which this converse fails by introducing the *first cohomology group* , where is the space of cocycles and is the space of coboundaries (note that both spaces are abelian groups). In my forthcoming paper with Vitaly Bergelson and Tamar Ziegler on the ergodic inverse Gowers conjecture (which should be available shortly), we make substantial use of some basic facts about this cohomology group (in the category of measure-preserving systems) that were established in a paper of Host and Kra.

The above terminology of cocycles, coboundaries, and cohomology groups of course comes from the theory of cohomology in algebraic topology. Comparing the formal definitions of cohomology groups in that theory with the ones given above, there is certainly quite a bit of similarity, but in the dynamical systems literature the precise connection does not seem to be heavily emphasised. The purpose of this post is to record the precise fashion in which dynamical systems cohomology is a special case of cochain complex cohomology from algebraic topology, and more specifically is analogous to singular cohomology (and can also be viewed as the group cohomology of the space of scalar-valued functions on X, when viewed as a G-module); this is not particularly difficult, but I found it an instructive exercise (especially given that my algebraic topology is extremely rusty), though perhaps this post is more for my own benefit that for anyone else.

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