  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.
