Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - May 2012 - Volume 152, Issue 03  

Research Article

Approximate groups and doubling metrics

TOM SANDERSa1

a1 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail: t.sanders@dpmms.cam.ac.uk

Abstract

We develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.

Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.

(Received December 07 2009)

(Revised May 26 2011)

(Online publication December 13 2011)