Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality

Let $G$ be a graph which satisfies $c^{-1}{a}^r\le \mathrm{\mid }B(v,r)\mathrm{\mid }\le c{a}^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$forsomeconstant$C=C(a,c)$andeveryfinitesetofvertices$A$.