A product theorem in free groups

• 成果类型: Article
• 署名作者: Razborov, Alexander A.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2014.179.2.1
• 发表日期: 2014
• 页码: 405-429
• 关键词: elementary theory

摘要

If A is a finite subset of a free group with at least two noncommuting elements, then vertical bar A.A.A vertical bar >= vertical bar A vertical bar(2)/(log vertical bar A vertical bar)(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 Pliinnecke-Ruzsa theory looks particularly simple and appealing.