Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh (Author), Dušan Repovš (Author)

Abstract

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 ▫${\rm cov}^\sharp(X)={\rm cov}^\flat(X)$▫.

Keywords

ultrametric space;isometrically homogeneous metric space;coarse equivalence;

Data

Language:	English
Year of publishing:	2016
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.124
COBISS:	17652057
ISSN:	0010-1354
Views:	497
Downloads:	313
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	str. 189-202
Volume:	ǂVol. ǂ144
Issue:	ǂno. ǂ2
Chronology:	2016
DOI:	10.4064/cm6697-9-2015
ID:	11231230