  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.
