Geodesic
A product theorem in free groups
Alexander A. Razborov1
1 Steklov Mathematical Institute, Moscow, Russia and Institute for Advanced Study, Princeton, NJ
Annals of mathematics, Tome 179 (2014) no. 2, pp. 405-429.

Voir la notice de l'article provenant de la source Annals of Mathematics website

If $A$ is a finite subset of a free group with at least two noncommuting elements, then $|A\cdot A\cdot A|\geq\frac{|A|^2}{(\log |A|)^{O(1)}}$. More generally, the same conclusion holds in an arbitrary virtually free group, unless $A$ 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.
DOI : 10.4007/annals.2014.179.2.1

Alexander A. Razborov 1

1 Steklov Mathematical Institute, Moscow, Russia and Institute for Advanced Study, Princeton, NJ 
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Alexander A. Razborov. A product theorem in free groups. Annals of mathematics, Tome 179 (2014) no. 2, pp. 405-429. doi : 10.4007/annals.2014.179.2.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2014.179.2.1/

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