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Classifying homogeneous ultrametric spaces up to coarse equivalence

Volume 144 / 2016

Taras Banakh, Dušan Repovš Colloquium Mathematicum 144 (2016), 189-202 MSC: Primary 54E35; Secondary 51F99. DOI: 10.4064/cm6697-9-2015 Published online: 4 March 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 $\mathrm {cov}^\sharp (X)=\mathrm {cov}^\flat (X)$.

Authors

  • Taras BanakhIvan Franko National University of Lviv
    Universytetska 1
    Lviv, 79000, Ukraine
    and
    Jan Kochanowski University in Kielce
    Świętokrzyska 15
    25-406 Kielce, Poland
    e-mail
  • Dušan RepovšFaculty of Education
    and
    Faculty of Mathematics and Physics
    University of Ljubljana
    Kardeljeva pl. 16
    1000 Ljubljana, Slovenia
    e-mail

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