Asymptotic group theory, 1

Budapest bird

The conference opened with a talk by Yoav Segev on his construction, with Eliahu Rips and Katrin Tent, of infinite non-split sharply 2-transitive groups.

A permutation group is sharply 2-transitive if any pair of distinct elements of the domain can be mapped to any other such pair by a unique element of the group. All such finite groups are known, and indeed it is easy to prove that such a finite group is “split”, that is, has a transitive abelian normal subgroup (and hence is the one-dimensional affine group over a finite nearfield – all finite nearfields were determined by Zassenhaus in the 1930s). It is a long-standing open problem whether there exist non-split infinite sharply 2-transitive groups.

Sharply 2-transitive groups give rise to a special low-rank class of independence algebras, a topic that Csaba Szabó and I worked on during my first visit to Budapest more than 22 years ago (at a time when the third Macdonalds had only just opened in Budapest). So it was very interesting to me to hear about this new construction.

The construction is remarkably easy. Yoav took us through the entire thing, apart from some calculations with normal forms in HNN extensions and free products. But the examples seem to be very diverse. In fact they show that any group whatever can be embedded as a subgroup in a sharply 2-transitive group.

In a sharply 2-transitive group, any two points are interchanged by a unique element, which is an involution; all involutions are conjugate, and so all fix the same number of points, which is either 0 or 1. Their construction deals with the former case (which they call “characteristic 2”). It can be formulated in group-theoretic terms. The final result G is a group with a subgroup A having the properties that any two conjugates of A intersect in the identity, that there are only two AA double cosets in G, and that A contains no involutions. (A is the point stabiliser.) So start with any pair G0, A0 having the first and third properties; if there are more than two double cosets, add a new generator to unify two of them. This may create new double cosets, but “in the limit” (repeating the construction enough times) the tortoise catches up with the hare and all the double cosets outside A are pulled into one. Now to get the announced result, we can take G0 to be any group and A0 to be the trivial group.

Balázs Szegedy gave us a very interesting talk on how nilpotent groups force themselves into additive combinatorics (e.g. Szemerédi’s theorem on arithmetic progressions in dense sets of integers) whether we like it or not. Roth proved the theorem for 3-term arithmetic progressions using Fourier analysis; a similar proof of Szemerédi’s theorem involves “higher order Fourier analysis”. Balázs has axiomatised the appropriate objects (which he calls nilspaces) in terms of “cubes”. The axioms are simple enough but the objects captured include nilmanifolds. I cannot really do justice to the talk in a single paragraph; but the claim is that shadows of these objects already appear in Szemerédi’s proof (although it appears to be just complicated combinatorics).

The day ended with Evgeny Vdovin talking about the conjecture that, if a transitive finite permutation group with no soluble normal subgroup has the property that the point stabiliser is soluble, then the group has a base of bounded size (that is, there is a set of points of bounded size whose pointwise stabiliser is the identity). The bound has been variously conjectured to be 7 or 5. This is almost certain to be true. Evgeny’s method shows the importance of choosing the right induction hypothesis. He works down a specially chosen composition series for the group, but the induction hypothesis “there is a base of size k cannot be made to work (there are counterexmples). Instead, he has to take the hypothesis “there are at least 5 regular orbits on k-tuples”. Why 5? I don’t know, but it seems to work!

About Peter Cameron

I count all the things that need to be counted.
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