  We show that doubling at some large scale in a Cayley graph implies uniform
doubling at all subsequent scales. The proof is based on the structure theorem
for approximate subgroups proved by Green, Tao and the first author. We also
give a number of applications to the geometry and spectrum of finite Cayley
graphs. For example, we show that a finite group has moderate growth in the
sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a
fixed power of the cardinality of the group. We call such groups almost flat
and show that they have a subgroup of bounded index admitting a cyclic quotient
of comparable diameter. We also give bounds on the Cheeger constant, first
eigenvalue of the Laplacian, and mixing time. This can be seen as a
finite-group version of Gromov's theorem on groups with polynomial growth. It
also improves on a result of Lackenby regarding property (tau) in towers of
coverings. Another consequence is a universal upper bound on the diameter of
all finite simple groups, independent of the CFSG.
