About: Quasi-isometry

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In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.

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rdfs:label	Quasi-Isometrie (de) 준등거리사상 (ko) Quasi-isometry (en) Квазиизометрия (ru) 擬等距同構 (zh) Квазіізометрія (uk)
rdfs:comment	Der Begriff der Quasi-Isometrie dient in der Mathematik dazu, die „grobe“ globale Geometrie metrischer Räume zu untersuchen. Er spielt in zahlreichen Gebieten der Geometrie, Analysis und geometrischen Gruppentheorie eine wichtige Rolle, etwa in der Theorie der hyperbolischen Gruppen oder in Beweisen von Starrheitssätzen. (de) In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov. (en) 수학에서 준등거리사상(영어: quasi-isometry), 준등거리동형사상, 준등거리변환, 준거리동형사상 혹은 준등장사상은 거리 공간의 일정한 집합 위에 줄 수 있는 동치관계로서, (coarse structure)를 탐구하기 위해 일반적인 등거리사상에서 약간의 세부사항을 무시하는 것이다. 미하일 그로모프의 에서 중요한 역할을 한다. (ko) 擬等距同構是數學上度量空間之間的等價關係，著重在度量空間上的，而忽略掉小尺寸上的細節。這樣有如從遠處觀看度量空間，看到其大概，而察看不出細處的分別。 (zh) Квазиизометрия — обобщение понятия изометрии на метрических пространствах, игнорирующая конечные отклонения, как абсолютные, так и относительные. Это понятие особенно важно в геометрической теории групп. Введено Михаилом Громовым. (ru)
foaf:depiction
dcterms:subject	Equivalence (mathematics) Geometric group theory Metric geometry
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Link from a Wikipage to another Wikipage	Properly discontinuous action Rounding Mostow rigidity theorem Euclidean metric Biautomatic group John von Neumann Index of a subgroup Invariant (mathematics) Equivalence (mathematics) Geometric group theory Connected component (topology) Mathematics Geometric group theory Class (set theory) Function (mathematics) Generating set of a group German language Lipschitz continuity Locally compact Compactification (mathematics) Stallings theorem about ends of groups Subgroup Banach–Tarski paradox Topological group Følner sequence Amenable group Equivalence relation Nilpotent group Cayley graph Group (mathematics) Isometry Hyperbolic manifold Relatively hyperbolic group Metric geometry Coarse structure Homeomorphism Discrete topology Word problem for groups Automatic group Gromov's theorem on groups of polynomial growth Identity function Infinity Metric space Mikhail Leonidovich Gromov Natural numbers Manhattan distance Švarc–Milnor lemma Euclidean plane Finitely generated group Topological space Δ-hyperbolic space Morse lemma Functional composition Discrete group theory Virtually nilpotent
sameAs	Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry Quasi-isometry
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has abstract	Der Begriff der Quasi-Isometrie dient in der Mathematik dazu, die „grobe“ globale Geometrie metrischer Räume zu untersuchen. Er spielt in zahlreichen Gebieten der Geometrie, Analysis und geometrischen Gruppentheorie eine wichtige Rolle, etwa in der Theorie der hyperbolischen Gruppen oder in Beweisen von Starrheitssätzen. (de) In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov. (en) 수학에서 준등거리사상(영어: quasi-isometry), 준등거리동형사상, 준등거리변환, 준거리동형사상 혹은 준등장사상은 거리 공간의 일정한 집합 위에 줄 수 있는 동치관계로서, (coarse structure)를 탐구하기 위해 일반적인 등거리사상에서 약간의 세부사항을 무시하는 것이다. 미하일 그로모프의 에서 중요한 역할을 한다. (ko) 擬等距同構是數學上度量空間之間的等價關係，著重在度量空間上的，而忽略掉小尺寸上的細節。這樣有如從遠處觀看度量空間，看到其大概，而察看不出細處的分別。 (zh) Квазиизометрия — обобщение понятия изометрии на метрических пространствах, игнорирующая конечные отклонения, как абсолютные, так и относительные. Это понятие особенно важно в геометрической теории групп. Введено Михаилом Громовым. (ru)
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