On an isoperimetric inequality for infinite finitely generated groups

Let Γ be an infinite, finitely generated group. We prove that for any finite subset A of Γ the following inequality is true:|A|⩽∑γ∈∂A dist (e, γ),where dist (e, γ) is a distance in Γ of γ to the identity element e, and ∂A is a boundary of A. This inequality implies that the volume form on the universal cover of a compact Riemannian manifold with infinite fundamental group has a primitive of at most linear growth.