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The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum-Connes conjecture holds for (X, Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above, namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory.

Let (1 → Nn → Gn → Qn → 1) n∈N be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence (Nn)n∈N with the induced metric from the word metrics of (Gn) n∈N has property A, and the sequence (Qn) n∈N with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence (Gn) n∈N , which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera [3]. Contents

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Let 1 → N → G → G N → 1 be a short exact sequence of countable discrete groups and let B be any G-C * -algebra. In this paper, we show that the strong Novikov conjecture with coefficients in B holds for such a group G when the normal subgroup N and the quotient group G N are coarsely embeddable into Hilbert spaces. As a result, the group G satisfies the Novikov conjecture under the same hypothesis on N and G N .