Groups acting properly on “bolic” spaces and the Novikov conjecture

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.

The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

“…It is now a direct consequence of the above proposition that if G is a K-exact group possessing a γ-element, and if 0 → I → A → A/I → 0 is a short exact sequence of Galgebras, then G satisfying BC for two of the algebras in this sequence implies that G satisfies BC for all three algebras in the sequence. The same result holds without the assumption on the γ-element (see [10,Proposition 4.2] -which was actually deduced as an easy consequence of a result of Kasparov and Skandalis in [25]). …”

An analogue of the Novikov Conjecture in complex algebraic geometry

“…The Strong Novikov Conjecture is weaker than the Baum-Connes Conjecture and is now known to hold for many (overlapping) classes of groups of geometric interest: discrete subgroups of Lie groups [16], groups which act properly on "bolic" spaces [19], groups with finite asymptotic dimension [31], groups with a uniform embedding into a Hilbert space [32], and hyperbolic groups [21].…”

Banach KK-Theory and the Baum-Connes Conjecture

“…The injectivity of the Baum-Connes map µ red (and therefore of µ L 1 ) is known for the following very large classes of groups : a) groups acting continuously properly isometrically on a complete simply connected riemannian manifold with controlled non-positive sectional curvature, and in particular closed subgroups of reductive Lie groups ( [29,31]), b) groups acting continuously properly isometrically on an affine building and in particular closed subgroups of reductive p-adic groups ( [32]), c) groups acting continuously properly isometrically on a discrete metric space with good properies at infinity (weakly geodesic, uniformly locally finite, and "bolic" [33,34]), and in particular hyperbolic groups (i.e. word-hyperbolic in the sense of Gromov), d) groups acting continuously amenably on a compact space ( [22]).…”