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Topology of the isometry group of the Urysohn space

Julien Melleray (2010)

Fundamenta Mathematicae

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space ℓ²(ℕ). The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group...

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 ∈ 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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