  We introduce a notion of covolume for point sets in locally compact groups
that simultaneously generalizes the covolume of a lattice and the reciprocal of
the Beurling density for amenable, unimodular groups. This notion of covolume
arises naturally from transverse measure theory applied to the hull dynamical
system associated to a point set. Using groupoid techniques, we prove necessary
conditions for sampling and interpolation in reproducing kernel Hilbert spaces
on unimodular groups in terms of this new notion of covolume. These conditions
generalize previously known density theorems for compactly generated groups of
polynomial growth, while also covering important new examples, in particular
model sets arising from cut-and-project schemes.
