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Gromov hyperbolic spaces. (English) Zbl 1087.53039

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.

MathOverflow Questions:

Metric ”in the limit”?

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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