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 Ann. of Math. (2), 2014, Volume 179, Issue 2, Pages 405–429 (Mi aom4)

A product theorem in free groups

A. A. Razborovabc

a Steklov Mathematical Institute, Moscow, Russia
b Institute for Advanced Study, Princeton, NJ
c University of Chicago, Chicago, IL

Abstract: If $A$ is a finite subset of a free group with at least two noncommuting elements, then $|A\cdot A\cdot A|\geqslant\frac{|A|^2}{(\log|A|)^{O(1)}}$. More generally, the same conclusion holds in an arbitrary virtually free group, unless AA generates a virtually cyclic subgroup. The central part of the proof of this result is carried on by estimating the number of collisions in multiple products $A_1\cdot\ldots\cdot A_k$. We include a few simple observations showing that in this “statistical” context the analogue of the fundamental Plünnecke–Ruzsa theory looks particularly simple and appealing.

 Funding Agency Grant Number Russian Foundation for Basic Research National Science Foundation ITR-0324906 Supported by the NSF grant ITR-0324906 and by the Russian Foundation for Basic Research.

DOI: https://doi.org/10.4007/annals.2014.179.2.1

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Revised: 20.09.2013
Accepted:25.09.2013
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• http://mi.mathnet.ru/eng/aom4

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