UPDATE (Jan 4 2013): See also this: Approximate groupoids again.
At the end of his talk Harald stated as a conclusion that (if I well understood) the approximate groups field should be regarded as the study of
“stable configurations under the action of a group”
where “stable configuration” means a set (or a family…) with some controlled behaviour of its growth under the “action of a group”, understood maybe in a lax, approximate sense.
This applies to approximate groups, he said, where the action is by left translations, but it could apply to the conjugate action as well, and evenin more general settings, where one has a space where a group acts.
My understanding of this bold claim is that Harald suggests that the approximate groups theory is a kind of geometry in the sense of Felix Klein: the (approximate, in this case) study of stable configuratiuons under the action of a group.
Very glad about hearing this!
Today I just remembered a comment that I have made last november on the blog of Tao, where I proposed to study “approximate groupoids”.
Why is this related?
Because a group acting on a set is just a particular type of groupoid, an action groupoid.
Here is the comment (link), reproduced further.
“Question (please erase if not appropriate): as a metric space is just a particular example of a normed groupoid, could you point me to papers where “approximate groupoids” are studied? For starters, something like an extension of your paper “Product set estimates for non-commutative groups”, along the lines in the introduction “one also consider partial sum sets …”, could be relevant, but I am unable to locate any. Following your proofs could be straightforward, but lengthy. Has this been done?
For example, a approximate groupoid of a groupoid (where denotes the set of arrows) could be a
(i)- symmetric subset of : , for any and
(ii)- there is another, symmetric subset such that for any there are such that ,
(iii)- for any we have .
One may replace the cardinal, as you did, by a measure (or random walk kernel, etc), or even by a norm.”
UPDATE (28.10.2012): Apparently unaware about the classical proof of Ronald Brown, by way of groupoids, of the Van Kampen theorem on the fundamental group of a union of spaces, Terence Tao has a post about this subject. I wonder if he is after some applications of his results on approximate groups in the realm of groupoids.