Strongly non-embeddable metric spaces

Casey Kelleher, Daniel Miller, Trenton Osborn, Anthony Weston

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

Enflo (1969) [4] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu (2002) [3] 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 (Z,ζ) 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 Zω=⊕א0Z admits a Cayley graph which may not be coarsely embedded into any metric space of non-zero generalized roundness. Finally, for each p≥0 and each locally finite metric space (Z, d), we prove the existence of a Lipschitz injection f:Z→ℓp.

Original languageEnglish (US)
Pages (from-to)749-755
Number of pages7
JournalTopology and its Applications
Volume159
Issue number3
DOIs
StatePublished - Feb 15 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Keywords

  • Coarse embedding
  • Locally finite metric space
  • Uniform embedding

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