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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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This week there is a conference here at IPAM on expanders in pure and applied mathematics. I was an invited speaker, but I don’t actually work in expanders per se (though I am certainly interested in them). So I spoke instead about the recent simplified proof by Kleiner of the celebrated theorem of Gromov on groups of polynomial growth. (This proof does not directly mention expanders, but the argument nevertheless hinges on the absence of expansion in the Cayley graph of a group of polynomial growth, which is exhibited through the smoothness properties of harmonic functions on such graphs.)

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