A product theorem in free groups

• Autores: Alexander Razborov
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 179, Nº 2, 2014, págs. 405-429
• Idioma: inglés
• DOI: 10.4007/annals.2014.179.2.1
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• Resumen

  • 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 fundamental Plünnecke-Ruzsa theory looks particularly simple and appealing