By Sergei Buyalo and Viktor Schroeder
Asymptotic geometry is the examine of metric areas from a wide scale viewpoint, the place the neighborhood geometry doesn't come into play. a major 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 e-book, in analogy with the techniques of classical hyperbolic geometry, the authors supply a scientific account of the fundamental thought of Gromov hyperbolic areas. those areas were studied widely within the final two decades and feature stumbled on purposes in crew idea, geometric topology, Kleinian teams, in addition to dynamics and tension idea. within the moment a part of the publication, a variety of 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 basic case, yet measurement thought proves invaluable for locating fascinating effects and purposes. The textual content leads concisely to a couple primary features of the idea. every one bankruptcy concludes with a separate part containing supplementary effects and bibliographical notes. right here the speculation can also be illustrated with a number of examples in addition to relatives to the neighboring fields of comparability geometry and geometric staff thought. The e-book relies on lectures the authors offered on the Steklov Institute in St. Petersburg and the college of Z??rich. A booklet of the ecu Mathematical Society (EMS). allotted in the Americas via the yankee Mathematical Society.
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Extra resources for Elements of Asymptotic Geometry (EMS Monographs in Mathematics)
Are equivalent to each other in the standard sense. X ! @1 X . g and consider a sequence fxi g 2 . xi jxj /o ! jxj /o are uniformly bounded. xi jxj /b ! C1, and fxi g converges to infinity with respect to !. Similarly, any other fyi g 2 is equivalent to fxi g with respect to !. g ! X . g @1 X . It remains to check that the map f is injective. xi jxj /b ! jxi /o ! 1. jxj /o ! jxi /o C ı for all sufficiently large i , j . We fix such an i and look at the limit as j ! 1. /o C 3ı ! 1 as i ! 1. xi jxj /b !
5. o// because Á Therefore, f does not preserve cross-pairs. ". This phenomenon is related to the fact that R2 is not hyperbolic. 3. 4. Assume that a map f W X ! c; d /-PQ-isometric and the space X 0 is ı 0 -hyperbolic (we do not require that X is hyperbolic). 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 !
1 In particular, Gromov products on Z based at ! 2 @1 X differ from each other by a constant: . jÁ/b . /, some constant c and all ; Á 2 Z. 34 Chapter 3. Busemann functions on hyperbolic spaces Finally, we note that for every ; Á; 2 Z distinct from ! /, the numbers . Áj /b , . 3 because . xi jyi /b for fxi g 2 , fyi g 2 Á. Bibliographical note. /. It is proven in [FS2] that the function b . 1 ; 2 / D e . g, of any CAT. 1/-space X for every ! /. The proof is based on the properties of a Bourdon metric and the Ptolemy inequality.
Elements of Asymptotic Geometry (EMS Monographs in Mathematics) by Sergei Buyalo and Viktor Schroeder