Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh ; Dušan Repovš

Colloquium Mathematicae, Tome 144 (2016), p. 189-202 / Harvested from The Polish Digital Mathematics Library

Résumé

For every metric space X we introduce two cardinal characteristics $cov^{♭}(X)$ and $cov^{♯}(X)$ 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 $cov^{♭}(X) = cov^{♯}(X) = cov^{♭}(Y) = cov^{♯}(Y)$. This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $cov^{♭}(X) = cov^{♯}(X)$. Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and only if $cov^{♯}(X) = cov^{♯}(Y)$ 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 $cov^{♯}(X) = cov^{♭}(X)$.

Détail

Publié le : 2016-01-01

Zbl 06574999

EUDML-ID : urn:eudml:doc:283912