Classifying homogeneous ultrametric spaces up to coarse equivalence

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 $\mathrm{cov}^\sharp(X)=\mathrm{cov}^\flat(X)$.