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Groups acting properly on “bolic” spaces and the Novikov conjecture. (English) Zbl 1029.19003
The paper under reviewing is a fundamental one in which we can follow some new progress in proving the Novikov conjecture and its relation with the Baum-Connes conjecture.
In §5 the authors state their main result (Theorem 5.2) that for any discrete group \(\Gamma\) acting properly by isometries on a weakly bolic (Definition 2.2), weakly geodesic metric space of bounded coarse geometry (Definition 3.1) and for every \(\Gamma\)-algebra \(A\) the Baum-Connes map (Definition 5.1) \[ \beta^A_{\text{red}} : RK^\Gamma_*(\underline{E}\Gamma; A)= \varinjlim KK^i_\Gamma(C_0(Y),A) \to K_*(C^*_{\text{red}} (\Gamma),A), \] where the limit is taken on cocompact \(\Gamma\)-invariant closed subsets Y in \(\underline{E}\Gamma\), and \(C^*_{\text{red}}(\Gamma)\) is the reduced group C*-algebra of \(\Gamma\), \(\underline{E}\Gamma\) the universal classifying space for free proper actions of \(\Gamma\). From this, the authors then deduced the Novikov conjecture on “homotopy invariance of higher signatures” for this kind of discrete groups.

MSC:
19K35 Kasparov theory (\(KK\)-theory)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
57R99 Differential topology
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