  Benjamini, Finucane and the first author have shown that if (G_n,S_n) is a
sequence of Cayley graphs such that |S_n^n|=O(n^D|S_n|), then the sequence
(G_n,d_{S_n}/n) is relatively compact for the Gromov-Hausdorff topology and
every cluster point is a connected nilpotent Lie group equipped with a
left-invariant sub-Finsler metric. In this paper we show that the dimension of
such a cluster point is bounded by D, and that, under the stronger bound
|S_n^n|=O(n^D), the homogeneous dimension of a cluster point is bounded by D.
Our approach is roughly to use a well-known structure theorem for approximate
groups due to Breuillard, Green and Tao to replace S_n^n with a coset
nilprogression of bounded rank, and then to use results about nilprogressions
from a previous paper of ours to study the ultralimits of such coset
nilprogressions. As an application we bound the dimension of the scaling limit
of a sequence of vertex-transitive graphs of large diameter. We also recover
and effectivise parts of an argument of Tao concerning the further growth of
single set S satisfying the bound |S^n| < Mn^D|S|.
