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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
-
is the Cartesian product of
and
;
-
is the product measure of
and Haar probability measure on
; and
- The action
is given by the formula
This group skew-product comes with a factor map
and a coordinate map
, which by (2) are related to the action via the identities
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) Letbe 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) Letbe 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
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 1 We 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 group
Indeed, 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
.)
In mathematics, one frequently starts with some space and wishes to extend it to a larger space
. Generally speaking, there are two ways in which one can extend a space
:
- By embedding
into a space
that has
(or at least an isomorphic copy of
) as a subspace.
- By covering
by a space
that has
(or an isomorphic copy thereof) as a quotient.
For many important categories of interest (such as abelian categories), the former type of extension can be represented by the exact sequence,
and the latter type of extension be represented by the exact sequence
In some cases, can be both embedded in, and covered by,
, in a consistent fashion; in such cases we sometimes say that the above exact sequences split.
An analogy would be to that of digital images. When a computer represents an image, it is limited both by the scope of the image (what it is picturing), and by the resolution of an image (how much physical space is represented by a given pixel). To make the image “larger”, one could either embed the image in an image of larger scope but equal resolution (e.g. embedding a picture of a pixel image of person’s face into a
pixel image that covers a region of space that is four times larger in both dimensions, e.g. the person’s upper body) or cover the image with an image of higher resolution but of equal scope (e.g. enhancing a
pixel picture of a face to a
pixel of the same face). In the former case, the original image is a sub-image (or cropped image) of the extension, but in the latter case the original image is a quotient (or a pixelation) of the extension. In the former case, each pixel in the original image can be identified with a pixel in the extension, but not every pixel in the extension is covered. In the latter case, every pixel in the original image is covered by several pixels in the extension, but the pixel in the original image is not canonically identified with any particular pixel in the extension that covers it; it “loses its identity” by dispersing into higher resolution pixels.
(Note that “zooming in” the visual representation of an image by making each pixel occupy a larger region of the screen neither increases the scope or the resolution; in this language, a zoomed-in version of an image is merely an isomorphic copy of the original image; it carries the same amount of information as the original image, but has been represented in a new coordinate system which may make it easier to view, especially to the visually impaired.)
In the study of a given category of spaces (e.g. topological spaces, manifolds, groups, fields, etc.), embedding and coverings are both important; this is particularly true in the more topological areas of mathematics, such as manifold theory. But typically, the term extension is reserved for just one of these two operations. For instance, in the category of fields, coverings are quite trivial; if one covers a field by a field
, the kernel of the covering map
is necessarily trivial and so
are in fact isomorphic. So in field theory, a field extension refers to an embedding of a field, rather than a covering of a field. Similarly, in the theory of metric spaces, there are no non-trivial isometric coverings of a metric space, and so the only useful notion of an extension of a metric space is the one given by embedding the original space in the extension.
On the other hand, in group theory (and in group-like theories, such as the theory of dynamical systems, which studies group actions), the term “extension” is reserved for coverings, rather than for embeddings. I think one of the main reasons for this is that coverings of groups automatically generate a special type of embedding (a normal embedding), whereas most embeddings don’t generate coverings. More precisely, given a group extension of a base group
,
one can form the kernel of the covering map
, which is a normal subgroup of
, and we thus can extend the above sequence canonically to a short exact sequence
On the other hand, an embedding of into
,
does not similarly extend to a short exact sequence unless the the embedding is normal.
Another reason for the notion of extension varying between embeddings and coverings from subject to subject is that there are various natural duality operations (and more generally, contravariant functors) which turn embeddings into coverings and vice versa. For instance, an embedding of one vector space into another
induces a covering of the dual space
by the dual space
, and conversely; similarly, an embedding of a locally compact abelian group
in another
induces a covering of the Pontryagin dual
by the Pontryagin dual
. In the language of images, embedding an image in an image of larger scope is largely equivalent to covering the Fourier transform of that image by a transform of higher resolution, and conversely; this is ultimately a manifestation of the basic fact that frequency is inversely proportional to wavelength.
Similarly, a common duality operation arises in many areas of mathematics by starting with a space and then considering a space
of functions on that space (e.g. continuous real-valued functions, if
was a topological space, or in more algebraic settings one could consider homomorphisms from
to some fixed space). Embedding
into
then induces a covering of
by
, and conversely, a covering of
by
induces an embedding of
into
. Returning again to the analogy with images, if one looks at the collection of all images of a fixed scope and resolution, rather than just a single image, then increasing the available resolution causes an embedding of the space of low-resolution images into the space of high-resolution images (since of course every low-resolution image is an example of a high-resolution image), whereas increasing the available scope causes a covering of the space of narrow-scope images by the space of wide-scope images (since every wide-scope image can be cropped into a narrow-scope image). Note in the case of images, that these extensions can be split: not only can a low-resolution image be viewed as a special case of a high-resolution image, but any high-resolution image can be pixelated into a low-resolution one. Similarly, not only can any wide-scope image be cropped into a narrow-scope one, a narrow-scope image can be extended to a wide-scope one simply by filling in all the new areas of scope with black (or by using more advanced image processing tools to create a more visually pleasing extension). (In the category of sets, the statement that every covering can be split is precisely the axiom of choice.)
I’ve recently found myself having to deal quite a bit with group extensions in my research, so I have decided to make some notes on the basic theory of such extensions here. This is utterly elementary material for a group theorist, but I found this task useful for organising my own thoughts on this topic, and also in pinning down some of the jargon in this field.
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