Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh, and Dušan Repovš. "Classifying homogeneous ultrametric spaces up to coarse equivalence." Colloquium Mathematicae 144.2 (2016): 189-202. <http://eudml.org/doc/283912>.

@article{TarasBanakh2016,
abstract = {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 $cov^\{♯\}(X) = cov^\{♭\}(X)$.},
author = {Taras Banakh, Dušan Repovš},
journal = {Colloquium Mathematicae},
keywords = {ultrametric space; isometrically homogeneous metric space; coarse equivalence},
language = {eng},
number = {2},
pages = {189-202},
title = {Classifying homogeneous ultrametric spaces up to coarse equivalence},
url = {http://eudml.org/doc/283912},
volume = {144},
year = {2016},
}

TY - JOUR
AU - Taras Banakh
AU - Dušan Repovš
TI - Classifying homogeneous ultrametric spaces up to coarse equivalence
JO - Colloquium Mathematicae
PY - 2016
VL - 144
IS - 2
SP - 189
EP - 202
AB - 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 $cov^{♯}(X) = cov^{♭}(X)$.
LA - eng
KW - ultrametric space; isometrically homogeneous metric space; coarse equivalence
UR - http://eudml.org/doc/283912
ER -