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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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