  We show how finiteness properties of a group and a subgroup transfer to
finiteness properties of the Schlichting completion relative to this subgroup.
Further, we provide a criterion when the dense embedding of a discrete group
into the Schlichting completion relative to one of its subgroups induces an
isomorphism in (continuous) cohomology. As an application, we show that the
continuous cohomology of the Neretin group vanishes in all positive degrees.
