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Classifying homogeneous ultrametric spaces up to coarse equivalence - MaRDI portal

Classifying homogeneous ultrametric spaces up to coarse equivalence (Q2804285)

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scientific article; zbMATH DE number 6574999
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Classifying homogeneous ultrametric spaces up to coarse equivalence
scientific article; zbMATH DE number 6574999

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    Classifying homogeneous ultrametric spaces up to coarse equivalence (English)
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    28 April 2016
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    ultrametric space
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    isometrically homogeneous metric space
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    coarse equivalence
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    In this paper the authors review the standard definitions of the coarse category and provide a coarse characterization of homogenous ultrametric spaces. It is well known that a space is an ultrametric space if and only if its asymptotic dimension is zero. Hence the paper provides a coarse classification of homogenous metric spaces of asymptotic dimension zero.NEWLINENEWLINEThe approach is based on cardinal characteristics \(\mathrm{cov}^\sharp\) and \(\mathrm{cov}^\flat\) arising from the minimal cardinality of the family of balls of certain diameter covering a ball of larger diameter. These are an adaptation of the characteristics used by the first author and I. Zarichnyi to prove a coarse classification of coarsely homogenous separable ultrametric spaces. It turns out that \(\mathrm{cov}^\sharp (X)\) and \(\mathrm{cov}^\flat(X)\) are coarse invariants and coincide if and only if \(X\) is coarsely equivalent to an isometrically homogeneous ultrametric space. In the later case \(\mathrm{cov}^\sharp (X)=\mathrm{cov}^\flat(X)\) completely determines the coarse type of a homogeneous ultrametric space \(X\).
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