Classifying homogeneous ultrametric spaces up to coarse equivalence
Volume 144 / 2016
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 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 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 them coarsely embeds into the other. This means that the coarse structure of an isometrically homogeneous ultrametric space $X$ is completely determined by the value of the cardinal $\mathrm {cov}^\sharp (X)=\mathrm {cov}^\flat (X)$.
