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In mathematics, one frequently starts with some space {X} and wishes to extend it to a larger space {Y}. Generally speaking, there are two ways in which one can extend a space {X}:

  • By embedding {X} into a space {Y} that has {X} (or at least an isomorphic copy of {X}) as a subspace.
  • By covering {X} by a space {Y} that has {X} (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,

\displaystyle  0 \rightarrow X \rightarrow Y

and the latter type of extension be represented by the exact sequence

\displaystyle  Y \rightarrow X \rightarrow 0.

In some cases, {X} can be both embedded in, and covered by, {Y}, 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 {200 \times 200} pixel image of person’s face into a {800 \times 800} 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 {200 \times 200} pixel picture of a face to a {800 \times 800} 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 {k} by a field {l}, the kernel of the covering map {\pi: l \rightarrow k} is necessarily trivial and so {k, l} 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 {G} of a base group {H},

\displaystyle  G \rightarrow H \rightarrow 0,

one can form the kernel {K = \hbox{ker}(\phi)} of the covering map {\pi: G \rightarrow H}, which is a normal subgroup of {G}, and we thus can extend the above sequence canonically to a short exact sequence

\displaystyle  0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0.

On the other hand, an embedding of {K} into {G},

\displaystyle  0 \rightarrow K \rightarrow G

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 {V} into another {W} induces a covering of the dual space {V^*} by the dual space {W^*}, and conversely; similarly, an embedding of a locally compact abelian group {H} in another {G} induces a covering of the Pontryagin dual {\hat H} by the Pontryagin dual {\hat G}. 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 {X} and then considering a space {C(X)} of functions on that space (e.g. continuous real-valued functions, if {X} was a topological space, or in more algebraic settings one could consider homomorphisms from {X} to some fixed space). Embedding {X} into {Y} then induces a covering of {C(X)} by {C(Y)}, and conversely, a covering of {X} by {Y} induces an embedding of {C(X)} into {C(Y)}. 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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I’m continuing the stream of uploaded papers this week with my paper “Freiman’s theorem for solvable groups“, submitted to Contrib. Disc. Math..  This paper concerns the problem (discussed in this earlier blog post) of determining the correct analogue of Freiman’s theorem in a general non-abelian group G = (G,\cdot).  Specifically, if A \subset G is a finite set that obeys the doubling condition |A \cdot A| \leq K|A| for some bounded K, what does this tell us about A?  Heuristically, we expect A to behave like a finite subgroup of G (or perhaps a coset of such a subgroup).

When G is the integers (with the additive group operation), Freiman’s theorem then tells us that A is controlled by a generalised arithmetic progression P, where I say that one set A is controlled by another P if they have comparable size, and the former can be covered by a finite number of translates of the latter.  (One can view generalised arithmetic progressions as an approximate version of a subgroup, in which one only uses the generators of the progression for a finite amount of time before stopping, as opposed to groups which allow words of unbounded length in the generators.) For more general abelian groups, the Freiman theorem of Green and Ruzsa tells us that a set of bounded doubling is controlled by a generalised coset progression P+H, i.e. the sum of a generalised arithmetic progression P and a finite subgroup H of G.  (Of course, if G is torsion-free, the finite subgroup H must be trivial.)

In this paper we address the case when G is a solvable group of bounded derived length.  The main result is that if a subset of G has small doubing, then it is controlled by an object which I call a “coset nilprogression”, which is a certain technical generalisation of a coset progression, in which the generators do not quite commute, but have commutator expressible in terms of “higher order” generators.  This is essentially a sharp characterisation of such sets, except for the fact that one would like a more explicit description of these coset nilprogressions.   In the torsion-free case, a more explicit description (analogous to the Mal’cev basis description of nilpotent groups) has appeared in a very recent paper of Breulliard and Green; in the case of monomial groups (a class of groups that overlaps to a large extent with solvable groups), and assuming a polynomial growth condition rather than a doubling condition, a related result controlling A by balls in a suitable type of metric has appeared in very recent work of Sanders.  In the nilpotent case there is also a nice recent argument of Fisher, Peng, and Katz which shows that sets of small doubling remain of small doubling with respect to the Lie algebra operations of addition and Lie bracket, and thus are amenable to the abelian Freiman theorems.

The conclusion of my paper is easiest to state (and easiest to prove) in the model case of the lamplighter group G = {\Bbb Z} \rtimes {\Bbb F}_2^\omega, where {\Bbb F}_2^\omega = \lim_{n \to \infty} {\Bbb F}_2^n is the additive group of doubly infinite sequences in the finite field {\Bbb F}_2 with only finitely many non-zero entries, and {\Bbb Z} acts on this space by translations.  This is a solvable group of derived length two.  The main result here is

Theorem 1. (Freiman’s theorem for the lamplighter group) If A \subset {\Bbb Z} \ltimes {\Bbb F}_2^\omega has bounded doubling, then A is controlled either by a finite subspace of the “vertical” group \{0\} \times {\Bbb F}_2^\omega, or else by a set of the form \{ (n,\phi(n)): n \in P \}, where P \subset {\Bbb Z} is a generalised arithmetic progression, and \phi: P \to {\Bbb F}_2^{\omega} obeys the Freiman isomorphism property (n_1,\phi(n_1)) \cdot (n_2, \phi(n_2)) = (n_3,\phi(n_3)) \cdot (n_4,\phi(n_4)) whenever n_1,n_2,n_3,n_4 \in P and n_1+n_2=n_3+n_4.

This result, incidentally, recovers an earlier result of Lindenstrauss that the lamplighter group does not contain a Følner sequence of sets of uniformly bounded doubling.  It is a good exercise to establish the “exact” version of this theorem, in which one classifies subgroups of the lamplighter group rather than sets of small doubling; indeed, the proof of this the above theorem follows fairly closely the natural proof of the exact version.

One application of the solvable Freiman theorem is the following quantitative version of a classical result of Milnor and of Wolf, which asserts that any solvable group of polynomial growth is virtually nilpotent:

Theorem 2. (Quantitative Milnor-Wolf theorem) Let G be a solvable group of derived length O(1), let S be a set of generators for G, and suppose one has the polynomial growth condition |B_S(R)| \leq R^d for some d = O(1), where B_S(R) is the set of all words generated by S of length at most R.  If R is sufficiently large, then this implies that G is virtually nilpotent; more precisely, G contains a nilpotent subgroup of step O(1) and index O(R^{O(1)}).

The key points here are that one only needs polynomial growth at a single scale R, rather than on many scales, and that the index of the nilpotent subgroup has polynomial size.

The proofs are based on an induction on the derived length.  After some standard manipulations (basically, splitting A by an approximate version of a short exact sequence), the problem boils down to that of understanding the action \rho of some finite set A on a set E in an additive group.  If one assumes that E has small doubling and that the action of A leaves E approximately invariant, then one can show that E is a coset progression, and the action of A can be described efficiently using the generators of that progression (after refining the set A a bit).

In the course of the proof we need two new additive combinatorial results which may be of independent interest.  The first is a variant of a well-known theorem of Sárközy, which asserts that if A is a large subset of an arithmetic progression P, then an iterated sumset kA of A for some k=O(1) itself contains a long arithmetic progression. Here, we need the related fact that if A is a large subset of a coset progression, then an iterated subset kA for k=O(1) contains a large coset progression Q, and furthermore this inclusion is “robust” in the sense that all elements the elements of Q have a large number of representations as sums of elements of A.  We also need a new (non-commutative) variant of the Balog-Szemerédi(-Gowers) lemma, which asserts that if A has small doubling, then A (or more precisely A \cdot A^{-1}) contains a large “core” subset D such that almost all of a large iterated subset kD of D still lies inside A \cdot A^{-1}).  (This may not look like the usual Balog-Szemerédi lemma, but the proof of the lemma is almost identical to the original proof of Balog and Szemerédi, in particular relying on the Szemerédi regularity lemma.