Growth and isoperimetric profile of planar graphs

Abstract

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