5 research outputs found
Freiman's theorem for solvable groups
Freiman's theorem asserts, roughly speaking, if that a finite set in a
torsion-free abelian group has small doubling, then it can be efficiently
contained in (or controlled by) a generalised arithmetic progression. This was
generalised by Green and Ruzsa to arbitrary abelian groups, where the
controlling object is now a coset progression. We extend these results further
to solvable groups of bounded derived length, in which the coset progressions
are replaced by the more complicated notion of a "coset nilprogression". As one
consequence of this result, any subset of such a solvable group of small
doubling is is controlled by a set whose iterated products grow polynomially,
and which are contained inside a virtually nilpotent group. As another
application we establish a strengthening of the Milnor-Wolf theorem that all
solvable groups of polynomial growth are virtually nilpotent, in which only one
large ball needs to be of polynomial size. This result complements recent work
of Breulliard-Green, Fisher-Katz-Peng, and Sanders.Comment: 41 pages, no figures, to appear, Contrib. Disc. Math. More discussion
and examples added, as per referee suggestions; also references to subsequent
wor
Approximate groups and doubling metrics
We develop a version of Freiman'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 Freiman's theorem by Breuillard and Green, Fischer, Katz and Peng,
and Tao.Comment: 21 pp. Corrected typos. Changed title from `From polynomial growth to
metric balls in monomial groups