Growth and isoperimetric profile of planar graphs

Benjamini, Itai; Papasoglu, Panos

doi:10.1090/S0002-9939-2011-10810-4

Growth and isoperimetric profile of planar graphs
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by Itai Benjamini and Panos Papasoglu PDF

Proc. Amer. Math. Soc. 139 (2011), 4105-4111 Request permission

Abstract:

Let $\Gamma$ be a planar graph such that the volume function of $\Gamma$ satisfies $V(2n)\leq CV(n)$ for some constant $C>0$. Then for every vertex $v$ of $\Gamma$ and $n\in \mathbb N$, there is a domain $\Omega$ such that $B(v,n)\subset \Omega$, $\partial \Omega \subset B(v, 6n)$ and $|\partial \Omega | \precsim n$.

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