Classifying homogeneous ultrametric spaces up to coarse equivalence

Banakh, Taras (author), Repovš, Dušan (author)

For every metric space ▫$X$▫ we introduce two cardinal characteristics ▫${\rm cov}^\flat(X)$▫ and ▫${\rm cov}^\sharp(X)$▫ describing the capacity of balls in ▫$X$▫. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces ▫$X,Y$▫ are coarsely equivalent if ▫${\rm cov}^\flat(X)={\rm cov}^\sharp(X)={\rm cov}^\flat(Y)={\rm cov}^\sharp(Y)$▫. This result implies that an ultrametric space ▫$X$▫ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if ▫${\rm cov}^\flat(X)={\rm cov}^\sharp(X)$▫. Moreover, two isometrically homogeneous ultrametric spaces ▫$X,Y$▫ are coarsely equivalent if and only if ▫${\rm cov}^\sharp(X)={\rm cov}^\sharp(Y)$▫ if and only if each of these spaces coarsely embeds into the other space. This means that the coarse structure of an isometrically homogeneous ultrametric space ▫$X$▫ is completely determined by the value of the cardinal ▫${\rm cov}^\sharp(X)={\rm cov}^\flat(X)$▫.

• Language: English
• Keywords: ultrametric space, isometrically homogeneous metric space, coarse equivalence
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2016
• Number of pages: Str. 189-202
• Numbering: Vol. 144, no. 2
• PID: 20.500.12556/RUL-111081
• UDC: 515.124
• ISSN on article: 0010-1354
• DOI: 10.4064/cm6697-9-2015
• COBISS.SI-ID: 17652057
• Publication date in RUL: 23.09.2019

Record is a part of a journal

• Title: Colloquium mathematicum
• Shortened title: Colloq. math.
• Publisher: Éditions Scientifiques de Pologne
• ISSN: 0010-1354
• COBISS.SI-ID: 25242624