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By American Mathematical Society, János Kollár, Robert Lazarsfeld

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Additional info for Algebraic Geometry Santa Cruz 1995, Part 2: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz

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U; v/ D ln v based at 1. 2 Gromov products based at infinity Let X be a ı-hyperbolic space, ! 2 @1 X . Busemann functions allow to define a Gromov product based at !. We first define it on X . xjy/b might be negative. 2) up to an error Ä 10ı for every x; y 2 X . / does not play an essential role. 2. 1. jy/o for every x; y 2 X . / satisfies the -inequality with 0 depending only on the hyperbolicity constant ı of X . 2. Assume that numbers a; b; c 2 R form a ı-triple, and a0 D a, b 0 D b, : c 0 D c up to an error Ä .

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. However, there are several important cases when @g X D @1 X .

For hyperbolic spaces this new crossdifference contains essentially the same information as the classical one, but is often easier to handle. In the previous section, we have already observed the following. yju/o is independent of the choice of a base point o 2 X , actually, it coincides with the classical cross-difference hx; y; z; ui. This expression has an interpretation in the spirit of the Tetrahedron Lemma. Consider the quadruple Q as an abstract tetrahedron with vertices x, y, z, u. Every edge of Q is labelled by the Gromov product of its vertices with respect to o.

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