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Let G be a graph which satisfies c-1 ar £ |B(v,r)| £ c arc^{-1} a^r\le \vert B(v,r)\vert\le c a^r , for some constants c, a > 1, every vertex v and every radius r. We prove that this implies the isoperimetric inequality |¶A| ³ C |A|/ log|A|\vert\partial A\vert \ge C \vert A\vert / \log \vert A\vert for some constant C = C( a, c) and every nonempty finite set of vertices A.

Pinched Exponential Volume Growth Implies an Infinite Dimensional Isoperimetric Inequality