Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh ¹   ; Dušan Repovš ²

¹ Ivan Franko National University of Lviv Universytetska 1 Lviv, 79000, Ukraine and Jan Kochanowski University in Kielce Świętokrzyska 15 25-406 Kielce, Poland
² Faculty of Education and Faculty of Mathematics and Physics University of Ljubljana Kardeljeva pl. 16 1000 Ljubljana, Slovenia

Colloquium Mathematicum, Tome 144 (2016) no. 2, pp. 189-202

Résumé

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

Détail

DOI : 10.4064/cm6697-9-2015

Keywords: every metric space introduce cardinal characteristics mathrm cov flat mathrm cov sharp describing capacity balls prove these cardinal characteristics invariant under coarse equivalence ultrametric spaces coarsely equivalent mathrm cov flat mathrm cov sharp mathrm cov flat mathrm cov sharp implies ultrametric space coarsely equivalent isometrically homogeneous ultrametric space only mathrm cov flat mathrm cov sharp moreover isometrically homogeneous ultrametric spaces coarsely equivalent only mathrm cov sharp mathrm cov sharp only each coarsely embeds other means coarse structure isometrically homogeneous ultrametric space completely determined value cardinal mathrm cov sharp mathrm cov flat