By Buyalo S., Schroeder V.

Asymptotic geometry is the examine of metric areas from a wide scale standpoint, the place the neighborhood geometry doesn't come into play. an immense category of version areas are the hyperbolic areas (in the experience 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 elemental thought of Gromov hyperbolic areas. those areas were studied broadly within the final two decades and feature stumbled on purposes in staff idea, geometric topology, Kleinian teams, in addition to dynamics and tension conception. within the moment a part of the publication, numerous points of the asymptotic geometry of arbitrary metric areas are thought of. It seems that the boundary at infinity technique isn't acceptable within the basic case, yet size thought proves beneficial for locating attention-grabbing effects and functions. The textual content leads concisely to a couple principal elements of the speculation. each one bankruptcy concludes with a separate part containing supplementary effects and bibliographical notes. the following the speculation can be illustrated with various examples in addition to relatives to the neighboring fields of comparability geometry and geometric team idea. The publication is predicated on lectures the authors provided on the Steklov Institute in St. Petersburg and the college of Zürich. A booklet of the eu Mathematical Society (EMS). allotted in the Americas through the yank Mathematical Society.

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5. g. to any closed ball of radius 1. However, Hn is by no means doubling. 6. Let X be a proper geodesic hyperbolic space. Show that @1 X is compact. 7. , X is a hyperbolic group. Then @1 X is doubling with respect to any visual metric. yjz/o / as in the Tetrahedron Lemma and call it the cross-difference triple of Q. Q/. 1. Q/ Ä ı for every quadruple Q X . 0, if and only if Proof. Q/ to be a ı-triple. Note that this property is independent of the choice of o and take as o any point of Q. 20 Chapter 2.

X of X with respect to ! is defined as the set of equivalence classes of sequences converging to infinity with respect to !. 4. 1. For every ! g ! X ! g @1 X . Proof. 1. Also recall that the Gromov product based at o 2 X is nonnegative. xi jxj /b ! xi jxj /o ! , any sequence which converges to infinity with respect to ! converges to infinity in the standard sense. Similarly, two sequences equivalent to each other with respect to ! are equivalent to each other in the standard sense. X ! @1 X . g and consider a sequence fxi g 2 .

1. Defining a Busemann function. All of this motivates the following considerations. Let X be a ı-hyperbolic space. For every ! 2 @1 X , we have a well-defined function b! W X X ! R; b! jx/y : The function b! is obviously skew symmetric in its variables. Busemann functions based at ! associated with different reference points o differ by a constant. We show that this property holds for b! up to an error depending only on the hyperbolicity constant of X . : We extend our agreement about D as follows.

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