Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality

Let G be a graph which satisfies c−1 ar ≤ |B(v, r)| ≤ c ar, for some constants c, a > 1, every vertex v and every radius r. We prove that this implies the isoperimetric inequality |∂A| ≥ C|A|/ log(2 + |A|) for some constant C = C(a, c) and every finite set of vertices A. A graph G = ( V (G), E(G) ) has pinched growth f(r) if there are two constants 0 < c < C < ∞ so that every ball B(v, r) of radius r centered around a vertex v ∈ V (G) satisfies c f(r) < ∣