Lou van den Dries (Department of Mathematics, University of Illinois at Urbana-Champaign)
The structure of approximate groups according to Breuillard, Green, Tao.
Abstract: Roughly speaking, an approximate group is a finite symmetric subset A of a group such that AA can be covered by a small number of left-translates of A. Last year the authors mentioned in the title established a conjecture of H. Helfgott and E. Lindenstrauss to the effect that approximate groups are ``finite-by-nilpotent''. This may be viewed as a sweeping generalisation of both the Freiman-Ruzsa theorem on sets of small doubling in the additive group of integers, and of Gromov's characterization of groups of polynomial growth. Among the applications of the main result are a finitary refinement of Gromov's theorem and a generalized Margulis lemma conjectured by Gromov. Prior work by Hrushovski on approximate groups is fundamental in the approach taken by the authors. They were able to reduce the role of logic to elementary arguments with ultra products. The point is that an ultraproduct of approximate groups can be modeled in a useful way by a neighborhood of the identity in a Lie group. This allows arguments by induction on the dimension of the Lie group. I will give two talks: the one on Tuesday (1pm in 345 AH) will describe the main results, and the sequel on Friday (4pm in 347 AH) will try to give a rough idea of the proofs. |
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