Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality

Abstract

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

5 pages