This is an announcement for the paper "Strongly non embeddable metric spaces" by Casey Kelleher, Daniel Miller, Trenton Osborn, and Anthony Weston.

Abstract: Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(\mathfrak{Z}, \zeta)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $\mathbb{Z}_\omega=\bigoplus_{\aleph_0}\mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p \geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z \to \ell_{p}$.

Archive classification: math.FA math.GN

Mathematics Subject Classification: 46C05, 46T99

Remarks: 10 pages

Submitted from: westona@canisius.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1109.5300

or

http://arXiv.org/abs/1109.5300