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Expositiones Mathematicae
Article
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Expositiones Mathematicae
Article . 2005
License: Elsevier Non-Commercial
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Expositiones Mathematicae
Article . 2005 . Peer-reviewed
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zbMATH Open
Article . 2005
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Gromov hyperbolic spaces

Authors: Väisälä, Jussi;

Gromov hyperbolic spaces

Abstract

This is a mini monograph on Gromov hyperbolic spaces, which are not necessarily geodesic or proper. As the author notes, the purpose of the article is to give a fairly detailed treatment of the basic theory of hyperbolic spaces more general than proper and geodesic. It is often assumed that the metric of the space is intrinsic, that is, the distance between any two points is equal to the infimum of the lengths of curves joining these points. The main idea is that geodesics are replaced by arcs whose length differs from the distance between the end points by uniformly bounded amount. The geodesic rays are replaced by sequences of such arcs called roads, and geodesic lines between boundary points are replaced by another kind of arc sequences called biroads. The advantage is that contrary to geodesics, each point of an intrinsic hyperbolic space is connected with every boundary point by a road, and each two boundary points are connected by a biroad, while rough versions of usual properties of geodesics in a hyperbolic space are preserved for h-arcs, roads and biroads. Another remarkable feature of the paper is the notion of a metametric. A metametric differs from a metric only by that the distance of a point to itself may be positive; points with this property are called thick. A metametric naturally arises on every hyperbolic space via the Gromov product when all points of the space are thick and all boundary points are ordinary. Notions of quasi-symmetric and quasi-Möbius maps are introduced in terms of metametrics, and appropriate boundary extension theorems of quasi-isometries are proved.

Related Organizations
Keywords

quasi-Möbius map, Mathematics(all), Quasi-möbius map, Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces, Gromov hyperbolic space, Gromov boundary

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
103
Top 1%
Top 10%
Top 10%
hybrid