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The main result is a step towards the classification of -approximate groups, in the specific setting of simple and semisimple Lie groups (with some partial results for more general Lie groups). For , define a -approximate group to be a finite subset of a group which is a symmetric neighbourhood of the origin (thus and is equal to ), and such that the product set is covered by left-translates (or equivalently, right-translates) of . For , this is the same concept as a finite subgroup of , but for larger values of , one also gets some interesting objects which are close to, but not exactly groups, such as geometric progressions for some and .
The expectation is that -approximate groups are -controlled by “structured” objects, such as actual groups and progressions, though the precise formulation of this has not yet been finalised. (We say that one finite set -controls another if is at most times larger than in cardinality, and can be covered by at most left translates or right translates of .) The task of stating and proving this statement is the noncommutative Freiman theorem problem, discussed in these earlier blog posts.
While this problem remains unsolved for general groups, significant progress has been made in special groups, notably abelian, nilpotent, and solvable groups. Furthermore, the work of Chang (over ) and Helfgott (over ) has established the important special cases of the special linear groups and :
Theorem 1 (Helfgott’s theorem) Let and let be either or for some prime . Let be a -approximate subgroup of .
- If generates the entire group (which is only possible in the finite case ), then is either controlled by the trivial group or the whole group.
- If , then is -controlled by a solvable -approximate subgroup of , or by itself. If , the latter possibility cannot occur, and must be abelian.
Our main result is an extension of Helfgott’s theorem to for general . In fact, we obtain an analogous result for any simple (or almost simple) Chevalley group over an arbitrary finite field (not necessarily of prime order), or over . (Standard embedding arguments then allow us to in fact handle arbitrary fields.) The results from simple groups can also be extended to (almost) semisimple Lie groups by an approximate version of Goursat’s lemma. Given that general Lie groups are known to split as extensions of (almost) semisimple Lie groups by solvable Lie groups, and Freiman-type theorems are known for solvable groups also, this in principle gives a Freiman-type theorem for arbitrary Lie groups; we have already established this in the characteristic zero case , but there are some technical issues in the finite characteristic case that we are currently in the process of resolving.
We remark that a qualitative version of this result (with the polynomial bounds replaced by an ineffective bound ) was also recently obtained by Hrushovski.
Our arguments are based in part on Helfgott’s arguments, in particular maximal tori play a major role in our arguments for much the same reason they do in Helfgott’s arguments. Our main new ingredient is a surprisingly simple argument, which we call the pivot argument, which is an analogue of a corresponding argument of Konyagin and Bourgain-Glibichuk-Konyagin that was used to prove a sum-product estimate. Indeed, it seems that Helfgott-type results in these groups can be viewed as a manifestation of a product-conjugation phenomenon analogous to the sum-product phenomenon. Namely, the sum-product phenomenon asserts that it is difficult for a subset of a field to be simultaneously approximately closed under sums and products, without being close to an actual field; similarly, the product-conjugation phenomenon asserts that it is difficult for a union of (subsets of) tori to be simultaneously approximately closed under products and conjugations, unless it is coming from a genuine group. In both cases, the key is to exploit a sizeable gap between the behaviour of two types of “pivots” (which are scaling parameters in the sum-product case, and tori in the product-conjugation case): ones which interact strongly with the underlying set , and ones which do not interact at all. The point is that there is no middle ground of pivots which only interact weakly with the set. This separation between interacting (or “involved”) and non-interacting (or “non-involved”) pivots can then be exploited to bootstrap approximate algebraic structure into exact algebraic structure. (Curiously, a similar argument is used all the time in PDE, where it goes under the name of the “bootstrap argument”.)
Below the fold we give more details of this crucial pivot argument.
One piece of trivia about the writing of this paper: this was the first time any of us had used
modern version control software to collaboratively write a paper; specifically, we used Subversion, with the repository being hosted online by xp-dev. (See this post at the Secret Blogging Seminar for how to get started with this software.) There were a certain number of technical glitches in getting everything to install and run smoothly, but once it was set up, it was significantly easier to use than our traditional system of emailing draft versions of the paper back and forth, as one could simply download and upload the most recent versions whenever one wished, with all changes merged successfully. I had a positive impression of this software and am likely to try it again in future collaborations, particularly those involving at least three people. (It would also work well for polymath projects, modulo the technical barrier of every participant having to install some software.)