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

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

2005-06-17

arxiv.org/abs/math/0506362v4

Mathematics

Group Theory

20 pages

Scientific paper

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.

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