By Sergei Buyalo and Viktor Schroeder

Asymptotic geometry is the examine of metric areas from a wide scale standpoint, the place the neighborhood geometry doesn't come into play. a massive category of version areas are the hyperbolic areas (in the feel of Gromov), for which the asymptotic geometry is well encoded within the boundary at infinity. within the first a part of this ebook, in analogy with the thoughts of classical hyperbolic geometry, the authors offer a scientific account of the elemental conception of Gromov hyperbolic areas. those areas were studied greatly within the final two decades and feature came across functions in team thought, geometric topology, Kleinian teams, in addition to dynamics and tension idea. within the moment a part of the booklet, quite a few points of the asymptotic geometry of arbitrary metric areas are thought of. It seems that the boundary at infinity technique isn't really applicable within the common case, yet measurement idea proves invaluable for locating fascinating effects and purposes. The textual content leads concisely to a couple principal features of the idea. every one bankruptcy concludes with a separate part containing supplementary effects and bibliographical notes. the following the speculation is additionally illustrated with a number of examples in addition to kin to the neighboring fields of comparability geometry and geometric staff concept. The booklet is predicated on lectures the authors awarded on the Steklov Institute in St. Petersburg and the college of Z??rich. A ebook of the ecu Mathematical Society (EMS). allotted in the Americas by means of the yank Mathematical Society.

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Furthermore, b . ; Á/ D 1 if and only if one of the points , Á coincides with !. / and some parameter a > 1: c1 a . jÁ/b Ä d. ; Á/ Ä c2 a . jÁ/b : 30 Chapter 3. Busemann functions on hyperbolic spaces In this case we say that d is visual with respect to the Busemann function b and the parameter a. 3), the property of a metric to be visual is independent of the choice of b. 3. Let X be a hyperbolic space, ! 2 @1 X . g with respect to b and a. 4. 0; 1/, and the parameter a D e. / is unbounded with infinitely remote point !

D C ı 0 /. Proof. 3 that f is PQ-isometric. x; y; z; u/ X . Q/ D hx; y; z; ui. Q/ is a ı 0 -triple. We conclude that f preserves cross-pairs of Q. 5. Every quasi-isometric map f W X ! Q/. 6. Vice versa, one can show that any PQ-isometric map f W X ! Q/, is strongly PQ-isometric. We leave this as an exercise to the reader. Bibliographical note. 3 that every quasi-isometric map f W X ! Y between hyperbolic geodesic spaces naturally induces a map between their boundaries at infinity. This extension property was discovered by V.

Instead, we narrow the class of quasi-isometric maps by putting the stronger condition that a map should have a bilipschitz type control over cross-differences, and we call such a map power quasi-isometric. 10). 36 Chapter 4. Morphisms of hyperbolic spaces From our point of view, power quasi-isometric maps constitute a natural class of morphisms between hyperbolic spaces. 1), and in that way, we recover the extension property of quasi-isometric maps. Moreover, we show in Chapter 7 that any hyperbolic space (with the mild natural restriction to be visual) is roughly isometric to a subspace of a geodesic hyperbolic space with the same boundary at infinity, and hence there is no need for quasification of geodesics.