  We study Hecke pairs using the coarse geometry of their coset space and their
Schlichting completion. We prove new stability results for the Baum-Connes and
the Novikov conjectures in the case where the pair is co-Haagerup. This allows
to generalize previous results, while providing new examples of groups
satisfying the Baum-Connes conjecture with coefficients. For instance, we show
that for some S-arithmetic subgroups of Sp(5,1) and Sp(3,1) the conjecture with
coefficients holds.
