TY - JOUR

T1 - Strongly non-embeddable metric spaces

AU - Kelleher, Casey

AU - Miller, Daniel

AU - Osborn, Trenton

AU - Weston, Anthony

N1 - Funding Information:
The research presented in this paper was undertaken and completed at the 2011 Cornell University Summer Mathematics Institute (SMI). The authors would like to thank the Department of Mathematics and the Center for Applied Mathematics at Cornell University for supporting this project, and the National Science Foundation for its financial support of the SMI through NSF grant DMS-0739338. The last named author thanks Canisius College for a faculty summer research fellowship. We would also like to thank the referee for their helpful comments on our paper.

PY - 2012/2/15

Y1 - 2012/2/15

N2 - 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.

AB - 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.

KW - Coarse embedding

KW - Locally finite metric space

KW - Uniform embedding

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U2 - 10.1016/j.topol.2011.11.041

DO - 10.1016/j.topol.2011.11.041

M3 - Article

AN - SCOPUS:84455208186

SN - 0166-8641

VL - 159

SP - 749

EP - 755

JO - Topology and its Applications

JF - Topology and its Applications

IS - 3

ER -