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This article is cited in 8 scientific papers (total in 8 papers)
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.
Received: 18.06.2007 Revised: 20.09.2013 Accepted: 25.09.2013
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