5,885 research outputs found

    Straightening warped cones

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    We provide the converses to two results of J. Roe (Geom. Topol. 2005): first, the warped cone associated to a free action of an a-T-menable group admits a fibred coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for LpL^p-spaces.Comment: Final authors' version of the article published by JTA. Changes since v2: the proof of Lem. 3.8 (now Prop. 3.10) is split between several lemmata, the proof of Thm 4.2 simplified and more detaile

    Tight embeddability of proper and stable metric spaces

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    We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p∈[1,∞]p\in [1,\infty], every proper subset of LpL_p is almost Lipschitzly embeddable into a Banach space XX if and only if XX contains uniformly the ℓpn\ell_p^n's. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.Comment: 19 page

    The Haagerup property is stable under graph products

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    The Haagerup property, which is a strong converse of Kazhdan's property (T)(T), has translations and applications in various fields of mathematics such as representation theory, harmonic analysis, operator K-theory and so on. Moreover, this group property implies the Baum-Connes conjecture and related Novikov conjecture. The Haagerup property is not preserved under arbitrary group extensions and amalgamated free products over infinite groups, but it is preserved under wreath products and amalgamated free products over finite groups. In this paper, we show that it is also preserved under graph products. We moreover give bounds on the equivariant and non-equivariant LpL_p-compressions of a graph product in terms of the corresponding compressions of the vertex groups. Finally, we give an upper bound on the asymptotic dimension in terms of the asymptotic dimensions of the vertex groups. This generalizes a result from Dranishnikov on the asymptotic dimension of right-angled Coxeter groups.Comment: 20 pages, v3 minor change
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