TAILIEUCHUNG - Đề tài " Groups acting properly on “bolic” spaces and the Novikov conjecture "

We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a “bolic”, weakly geodesic metric space of bounded geometry. 1. Introduction This work has grown out of an attempt to give a purely KK-theoretic proof of a result of A. Connes and H. Moscovici ([CM], [CGM]) that hyperbolic groups satisfy the Novikov conjecture. . | Annals of Mathematics Groups acting properly on bolic spaces and the Novikov conjecture By Gennadi Kasparov and Georges Skandalis Annals of Mathematics 158 2003 165 206 Groups acting properly on bolic spaces and the Novikov conjecture By Gennadi Kasparov and Georges Skandalis Abstract We introduce a class of metric spaces which we call bolic . They include hyperbolic spaces simply connected complete manifolds of nonpositive curvature euclidean buildings etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a bolic weakly geodesic metric space of bounded geometry. 1. Introduction This work has grown out of an attempt to give a purely KK-theoretic proof of a result of A. Connes and H. Moscovici CM CGM that hyperbolic groups satisfy the Novikov conjecture. However the main result of the present paper appears to be much more general than this. In the process of this work we have found a class of metric spaces which contains hyperbolic spaces in the sense of M. Gromov simply connected complete Riemannian manifolds of nonpositive sectional curvature euclidean buildings and probably a number of other interesting geometric objects. We called these spaces bolic spaces . Our main result is the following Theorem . Novikov s conjecture on higher signatures is true for any discrete group acting properly by isometries on a weakly bolic weakly geodesic metric space of bounded coarse geometry. - The notion of a bolic and weakly bolic space is defined in Section 2 as well as the notion of a weakly geodesic space - bounded coarse geometry . bounded geometry in the sense of P. Fan see HR is discussed in Section 3. All conditions of the theorem are satisfied for example for any discrete group acting properly and isometrically either on a simply connected complete Riemannian manifold of nonpositive bounded sectional curvature or on a euclidean building with uniformly bounded ramification numbers. All condi 166 .

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