Download e-book for kindle: Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

ISBN-10: 0821808958

ISBN-13: 9780821808955

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Xi jyi /b D . /. xi jyi /b C c exists, and . xi jyi /b 0 D . 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 !

X; x 0 /j C 6ı Ä jxx 0 j C 6ı for every o 2 X . Now the claim follows easily from the definition of Busemann functions. 4. jyi /o is uniformly bounded, since ¤ !. yi / ! C1 together with joyi j because fyi g converges to infinity. ;o . Since b! jxi /o ! jxi /o ! 1. Hence fxi g 2 !. 6. xi / ! / and a sequence fxi g 2 !. i; exp. xi / D i ! 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 !.

These cross-ratios may depend on the choice of o or b respectively. 1, such a dependence is completely controlled by the hyperbolicity constant ı. Z; / is a quasi-metric space with infinitely remote set Z1 Z, jZ1 j Ä 1. 1. Given a quadruple Q of distinct points a; b; c; d 2 Z, we call the triple M D . Q/. In the case when one of the quadruple points is at infinity, every member of M contains an infinite factor. , if d 2 Z1 then we put M D . Q/ as above. The cross-ratio then becomes the ordinary ratio.