Freiman's theorem in an arbitrary nilpotent group
Abstract
We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem.
Keywords
Cite
@article{arxiv.1211.3989,
title = {Freiman's theorem in an arbitrary nilpotent group},
author = {Matthew Tointon},
journal= {arXiv preprint arXiv:1211.3989},
year = {2014}
}
Comments
39 pages. Version 2 contains a corrected proof of Theorem 1.7; improved exposition in Section 7, incorporating suggestions from an anonymous referee; and various minor presentational and organisational changes. Version 3 reflects changes made by a copy editor. Appears in Proc. London Math. Soc. at http://plms.oxfordjournals.org/cgi/content/abstract/pdu005? ijkey=AVizKoIO9JYHczu&keytype=ref