Dimension-raising maps in a large scale

Miyata, Takahisa; Virk, Žiga

doi:10.4064/fm223-1-6

Cited by 15 publications

(33 citation statements)

References 5 publications

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“…Corollary 4.6 (Miyata-Virk [11]). Suppose f : X → Y is a surjective function of metric spaces that is coarsely n-to-1 for some n ≥ 1.…”

Section: Definition 43mentioning

confidence: 99%

“…Lemma 4.5 (Lemma 3.6 of [11]). Suppose f : X → Y is coarsely n to 1 with control C. Then for every cover U of X and for every r > 0 we have…”

Section: Definition 43mentioning

confidence: 99%

“…The most important of them is the class of coarsely n-to-1 functions recently introduced by Miyata and Virk [11]. As shown in Section 5 that class is contained in the class of functions of asymptotic dimension 0 introduced in [2].…”

Section: Introductionmentioning

confidence: 99%

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Preserving coarse properties

Dydak

Virk

2015

Rev Mat Complut

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Abstract. The aim of this paper is to investigate properties preserved and co-preserved by coarsely n-to-1 functions, in particular by the quotient maps X → X/ ∼ induced by a finite group G acting by isometries on a metric space X. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.

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“…Corollary 4.6 (Miyata-Virk [11]). Suppose f : X → Y is a surjective function of metric spaces that is coarsely n-to-1 for some n ≥ 1.…”

Section: Definition 43mentioning

confidence: 99%

“…Lemma 4.5 (Lemma 3.6 of [11]). Suppose f : X → Y is coarsely n to 1 with control C. Then for every cover U of X and for every r > 0 we have…”

Section: Definition 43mentioning

confidence: 99%

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Preserving coarse properties

Dydak

Virk

2015

Rev Mat Complut

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“…Such properties include (coarse) surjectivity and (coarse) n-to-1 property. As a consequence, we present an improved dimension raising theorem for radial functions (see [14] for the original theorem). In the penultimate section we provide a natural example of a situation as described in the main theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Coarsely n-to-1 maps were introduced in [14]. They appear naturally as quotient maps via a finite group action by coarse functions on a metric space.…”

Section: Introductionmentioning

confidence: 99%

Inducing Maps Between Gromov Boundaries

Dydak

Virk

2015

Mediterr. J. Math.

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Abstract. It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the class of visual functions f that do induce continuous mapsf between Gromov boundaries. Its subclass, the class of radial functions, induces Hölder maps between Gromov boundaries. Conversely, every Hölder map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale properties of f and small scale properties off , especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension.

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Coarse infinite-dimensionality of hyperspaces of finite subsets

Weighill

Yamauchi

Zava

2021

European Journal of Mathematics

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We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

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Dimension-raising maps in a large scale

Cited by 15 publications

References 5 publications

Preserving coarse properties

Preserving coarse properties

Inducing Maps Between Gromov Boundaries

Coarse infinite-dimensionality of hyperspaces of finite subsets

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