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




A product theorem in free groups
A. A. Razborov^{abc} ^{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 $A\cdot A\cdot A\geqslant\frac{A^2}{(\logA)^{O(1)}}$. 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.
DOI:
https://doi.org/10.4007/annals.2014.179.2.1
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Received: 18.06.2007 Revised: 20.09.2013 Accepted:25.09.2013
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