By Amnon Neeman
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Extra resources for Algebraic and Analytic Geometry
These expressions can be defined in a multiplicative or in an additive way. Well known and intensively studied is the cross-ratio, which involves the distances between four points and which is written in a multiplicative form. 1. Morphisms of metric spaces and hyperbolicity The correspondence between the multiplicative and the additive version of the expression is more subtle than expected at first glance. zju/o ; where o 2 X is any chosen base point. xjy/o to be the additive counterpart to the multiplicative jxyj (this explains the factor 1=2).
Isometric maps for 1 D 2 D id. On the other hand one can define much wider classes of maps. t / D 1. t / D ct C d , c 1, d 0. t / D ct C d , d 0 and c > 0, c D 1 respectively, play an important role. These characterizations of functions involve just the distance jxyj between two points and how this distance is disturbed by the map f . For the study of hyperbolic spaces it is necessary to consider more complicated expressions which involve the distances between three and four points. This is not surprising since hyperbolicity itself is a condition on quadruples of points.
2. Let X be a geodesic ı-hyperbolic space. t 0 /j Ä C for arbitrarily large t , t 0 . Then j . / 0 . /j Ä ı for all 0; in particular, the rays , 0 are asymptotic. Proof. We have . t 0 /j/ minft; t 0 g C =2: Thus for Ä minft; t 0 g C =2 we have j . / 0 . /j Ä ı by ı-hyperbolicity. Since t , t 0 can be chosen arbitrarily large, this inequality holds for all 0. @1 X . In If a geodesic space X is Gromov hyperbolic, then obviously @g X general, there is no reason that the geodesic boundary of a hyperbolic geodesic space coincides with the boundary at infinity.
Algebraic and Analytic Geometry by Amnon Neeman