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Urysohn universal spaces as metric groups of exponent 2

Piotr Niemiec (2009)

Fundamenta Mathematicae

The aim of the paper is to prove that the bounded and unbounded Urysohn universal spaces have unique (up to isometric isomorphism) structures of metric groups of exponent 2. An algebraic-geometric characterization of Boolean Urysohn spaces (i.e. metric groups of exponent 2 which are metrically Urysohn spaces) is given.

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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