The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

Abstract

Corollary 1.2. Let Γ be a finitely generated group. If Γ, as a metric space with a word-length metric, admits a uniform embedding into Hilbert space, and its classifying space BΓ has the homotopy type of a finite CW complex, then the strong Novikov conjecture holds for Γ, i.e. the index map from K∗(BΓ) to K∗(C∗ r (Γ)) is injective. Corollary 1.2 follows from Theorem 1.1 and the descent principle [23]. By index theory, the strong Novikov conjecture implies the Novikov conjecture on the homotopy invariance of higher signatures (cf. [8] for an excellent