Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture

TL;DR

This paper explores the relationship between fibred coarse embeddings, a-T-menability, and the coarse Baum-Connes conjecture, establishing new links and characterizations in coarse geometry and groupoid theory.

Contribution

It connects fibred coarse embeddings to the coarse Baum-Connes conjecture and characterizes a-T-menability for residually finite groups.

Findings

Fibred coarse embeddings imply the coarse Baum-Connes conjecture under certain conditions.

A new characterization of a-T-menability for residually finite groups is provided.

Homological algebra techniques are used to relate fibred embeddings to groupoid properties.

Abstract

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum-Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum-Connes conjecture and in this paper we connect this property to the traditional coarse Baum-Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.