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Benjamini, Finucane and the first author have shown that if (G_n,S_n) is asequence 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 andevery cluster point is a connected nilpotent Lie group equipped with aleft-invariant sub-Finsler metric. In this paper we show that the dimension ofsuch 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 approximategroups due to Breuillard, Green and Tao to replace S_n^n with a cosetnilprogression of bounded rank, and then to use results about nilprogressionsfrom a previous paper of ours to study the ultralimits of such cosetnilprogressions. As an application we bound the dimension of the scaling limitof a sequence of vertex-transitive graphs of large diameter. We also recoverand effectivise parts of an argument of Tao concerning the further growth ofsingle set S satisfying the bound |S^n| < Mn^D|S|.

Scaling limits of Cayley graphs with polynomially growing balls