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

TL;DR

This paper proves the equivariant coarse Baum-Connes conjecture for actions by a-T-menable groups on certain metric spaces, expanding the class of spaces where the conjecture holds.

Contribution

It establishes the conjecture for actions with controlled distortion on spaces with coarsely embeddable quotients, answering an open question.

Findings

The conjecture holds for a-T-menable group actions under specified conditions.

The result applies to spaces with bounded geometry and coarsely embeddable quotients.

Provides new evidence supporting the conjecture in broader settings.

Abstract

The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe [29] as a unified way to approach both the Baum-Connes conjecture and its coarse counterpart. In this paper, we prove that if an a-T-menable group $\Gamma$ acts properly and isometrically on a bounded geometry metric space $X$ with controlled distortion such that the quotient space $X/\Gamma$ is coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds for this action. This answers a question posed in [8] affirmatively.