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

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

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 ) n is a Følner sequence and we give a polynomial estimate of the rate of decay of μ ( U n + 1 U n ) μ ( U n ) . Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). 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 L p -pointwise...

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