A product theorem in free groups

Alexander Razborov
2014 Annals of Mathematics  
If A is a finite subset of a free group with at least two noncommuting elements then |A · A · A| ≥ |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 · . . . · A k . We include a few simple observations showing that in this "statistical" context the analogue of the
more » ... Plünnecke-Ruzsa theory looks particularly simple and appealing.
doi:10.4007/annals.2014.179.2.1 fatcat:h3yfxnbfr5fuvopccnioslbffy