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Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality

Itai Benjamini·Oded Schramm
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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 $A$.

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Itai Benjamini, Oded Schramm. 2003-03-11. Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. https://arxiv.org/abs/math/0303127

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