5,885 research outputs found
Straightening warped cones
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 -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
We introduce the notions of almost Lipschitz embeddability and nearly
isometric embeddability. We prove that for , every proper
subset of is almost Lipschitzly embeddable into a Banach space if and
only if contains uniformly the '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
The Haagerup property, which is a strong converse of Kazhdan's property
, 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
-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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