Strongly non embeddable metric spaces

TL;DR

This paper constructs a specific metric space that cannot be embedded into any space with positive generalized roundness, highlighting fundamental limitations in embedding theory and implications for universal metric spaces and groups.

Contribution

It introduces a simpler construction of a strongly non-embeddable metric space combining previous examples, and explores its implications for universal spaces and group Cayley graphs.

Findings

Constructed a locally finite metric space that is strongly non-embeddable.

Showed that universal metric spaces may not embed into spaces of positive generalized roundness.

Proved existence of Lipschitz injections into l_p spaces for all locally finite metric spaces.

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 $f : Z \to \ell_{p}$.