Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Abstract

Let G be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if G has polynomial growth, then (U^n) is a Folner sequence: that is, the volume of the boundary of U^n divided by U^n goes to zero. Moreover, we give a polynomial estimate of this ratio. Our proof is based on doubling property. As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a balls averages L^p-pointwise ergodic theorem for probability G-spaces, with G of polynomial growth and for all p greater than 1.