The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

Abstract

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.

The equivariant coarse Baum-Connes conjecture for actions by a-T-menable groups

“…Along this way, very recently the first author together with Deng and Wang [8] proved that the equivariant coarse Baum-Connes conjecture holds for amenable groups acting on metric spaces such that all orbits are equivariantly uniformly coarsely equivalent and quotient spaces are coarsely embeddable. They also asked a question whether the result remains true when the involved group is a-T-menable, inspired by Higson and Kasparov's significant result that the Baum-Connes conjecture holds for a-T-menable groups [16].…”

“…They also asked a question whether the result remains true when the involved group is a-T-menable, inspired by Higson and Kasparov's significant result that the Baum-Connes conjecture holds for a-T-menable groups [16]. However, the techniques in [8] are no longer applicable for a-T-menable groups due to an obstruction in coarse K-amenability (see [8,Theorem 1.2]).…”

“…In this paper, we use a different approach to bypass the issue of coarse Kamenability and answer the question posed in [8] affirmatively. More precisely, we show that the equivariant coarse Baum-Connes conjecture holds for a-T-menable groups acting on metric spaces with controlled distortion such that quotient spaces are coarsely embeddable.…”

“…We would like to highlight the difference between our approach and the one used in [8]. Although both of them consult Yu's machinery, the constructions of equivariant twisted algebras and index maps are different.…”

“…Although both of them consult Yu's machinery, the constructions of equivariant twisted algebras and index maps are different. More precisely in the approach of [8], index maps (usually called the Bott and Dirac maps) are defined on dense subalgebras in the form of asymptotic morphisms, and therefore one needs a coarse K-amenability result to extend these maps. While in the current paper, we use a coarse Mayer-Vietoris argument to chop the space and decompose the associated algebras at first, which allows us to construct index maps directly on the K-theories of the whole algebras (see Proposition 5.7).…”