Diversity of Lipschitz-free spaces over countable complete discrete metric spaces

TL;DR

This paper demonstrates the vast diversity of Lipschitz-free spaces over countable, complete, discrete metric spaces, revealing uncountably many non-isomorphic spaces and complex behaviors of the dentability index.

Contribution

It establishes the uncountable diversity of Lipschitz-free spaces over certain metric spaces and analyzes the non-binary behavior of the dentability index in this context.

Findings

Uncountably many non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces.

Existence of a space whose free space does not embed into any uniformly discrete space's free space.

Complex, uncountable variation of the dentability index across different classes of metric spaces.

Abstract

We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index $D$ presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of $\{D(\mathcal F(M)): M$ countable, complete, discrete$\}$ is uncountable while $\{D(\mathcal F(M)):M$ infinite, compact, purely 1-unrectifiable$\}=\{\omega,\omega^2\}$. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: $\{D(\mathcal F(M)):M$ infinite, uniformly discrete$\}=\{\omega^2,\omega^3\}$.