Diversity of Lipschitz-free spaces over countable complete discrete 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$\}=\{{\omega }^2,{\omega }^3\}$.