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One of my favorite open problems, which I have blogged about in the past, is that of establishing (or even correctly formulating) a non-commutative analogue of Freiman’s theorem. Roughly speaking, the question is this: given a finite set in a non-commutative group which is of small doubling in the sense that the product set is not much larger than (e.g. for some ), what does this say about the structure of ? (For various technical reasons one may wish to replace small doubling by, say, small tripling (i.e. ), and one may also wish to assume that contains the identity and is symmetric, , but these are relatively minor details.)
Sets of small doubling (or tripling), etc. can be thought of as “approximate groups”, since groups themselves have a doubling constant equal to one. Another obvious example of an approximate group is that of an arithmetic progression in an additive group, and more generally of a ball (in the word metric) in a nilpotent group of bounded rank and step. It is tentatively conjectured that in fact all examples can somehow be “generated” out of these basic examples, although it is not fully clear at present what “generated” should mean.
A weaker conjecture along the same lines is that if is a set of small doubling, then there should be some sort of “pseudo-metric” on which is left-invariant, and for which is controlled (in some suitable sense) by the unit ball in this metric. (For instance, if was a subgroup of , one would take the metric which identified all the left cosets of to a point, but was otherwise a discrete metric; if were a ball in a nilpotent group, one would use some rescaled version of the word metric, and so forth.) Actually for technical reasons one would like to work with a slightly weaker notion than a pseudo-metric, namely a Bourgain system, but let us again ignore this technicality here.
Recently, using some powerful tools from model theory combined with the theory of topological groups, Ehud Hrushovski has apparently achieved some breakthroughs on this problem, obtaining new structural control on sets of small doubling in arbitrary groups that was not previously accessible to the known combinatorial methods. The precise results are technical to state, but here are informal versions of two typical theorems. The first applies to sets of small tripling in an arbitrary group:
Theorem 1 (Rough version of Hrushovski Theorem 1.1) Let be a set of small tripling, then one can find a long sequence of nested symmetric sets , all of size comparable to and contained in , which are somewhat closed under multiplication in the sense that for all , and which are fairly well closed under commutation in the sense that . (There are also some additional statements to the effect that the efficiently cover each other, and also cover , but I will omit those here.)
This nested sequence is somewhat analogous to a Bourgain system, though it is not quite the same notion.
If one assumes that is “perfect” in a certain sense, which roughly means that there is no non-trivial abelian quotient, then one can do significantly better:
Theorem 2 (Rough version of Hrushovski Corollary 1.2) Let be a set of small tripling, let , and suppose that for almost all -tuples (where ), the conjugacy classes generate most of in the sense that . Then a large part of is contained in a subgroup of size comparable to .
Note that if one quotiented out by the commutator , then all of the conjugacy classes would collapse to points. So the hypothesis here is basically a strong quantitative assertion to the effect that the commutator is extremely large, and rapidly fills out most of itself.
Here at UCLA, a group of logicians and I (consisting of Matthias Aschenbrenner, Isaac Goldbring, Greg Hjorth, Henry Towsner, Anush Tserunyan, and possibly others) have just started a weekly reading seminar to come to grips with the various combinatorial, logical, and group-theoretic notions in Hrushovski’s paper, of which we only have a partial understanding at present. The seminar is a physical one, rather than an online one, but I am going to try to put some notes on the seminar on this blog as it progresses, as I know that there are a couple of other mathematicians who are interested in these developments.
So far there have been two meetings of the seminar. In the first, I surveyed the state of knowledge of the noncommutative Freiman theorem, covering broadly the material in my previous blog post. In the second meeting, Isaac reviewed some key notions of model theory used in Hrushovski’s paper, in particular the notions of definability and type, which I will review below. It is not yet clear how these are going to be connected with the combinatorial side of things, but this is something which we will hopefully develop in future seminars. The near-term objective is to understand the statement of the main theorem on the model-theoretic side (Theorem 3.4 of Hrushovski), and then understand some of its easier combinatorial consequences, before going back and trying to understand the proof of that theorem.
[Update, Oct 19: Given the level of interest in this paper, readers are encouraged to discuss any aspect of that paper in the comments below, even if they are not currently being covered by the UCLA seminar.]