Classifying homogeneous ultrametric spaces up to coarse equivalence

TL;DR

This paper introduces two cardinal invariants for metric spaces, proves their invariance under coarse equivalence, and classifies ultrametric spaces up to coarse equivalence using these invariants, especially focusing on isometrically homogeneous spaces.

Contribution

It establishes a complete classification of ultrametric spaces up to coarse equivalence via new cardinal characteristics and characterizes when such spaces are coarsely equivalent to homogeneous ultrametric spaces.

Findings

Cardinal characteristics are invariant under coarse equivalence.

Ultrametric spaces are classified by these invariants.

Homogeneous ultrametric spaces are characterized by equal invariants.

Abstract

For every metric space $X$ we introduce two cardinal characteristics $\mathrm{cov}^\flat(X)$ and $\mathrm{cov}^\sharp(X)$ describing the capacity of balls in $X$. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces $X,Y$ are coarsely equivalent if $\mathrm{cov}^\flat(X)=\mathrm{cov}^\sharp(X)=\mathrm{cov}^\flat(Y)=\mathrm{cov}^\sharp(Y)$. This result implies that an ultrametric space $X$ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $\mathrm{cov}^\flat(X)=\mathrm{cov}^\sharp(X)$. Moreover, two isometrically homogeneous ultrametric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{cov}^\sharp(X)=\mathrm{cov}^\sharp(Y)$ if and only if each of these spaces coarsely embeds into the other space. This means that the coarse structure of an isometrically homogeneous ultrametric space $X$ is completely determined by the value of the cardinal $\mathrm{cov}^\sharp(X)=\mathrm{cov}^\flat(X)$.