  The structure of the divergences for transverse theories of gravity is
studied to one-loop order. These theories are invariant only under those
diffeomorphisms that enjoy unit Jacobian determinant (TDiff), so that the
determinant of the metric transforms as a true scalar instead of a density.
Generically, the models include an additional scalar degree of freedom
contained in the metric besides the usual spin two component. When the
cosmological constant is fine tuned to zero, there are only two theories which
are on shell finite, namely the one in which the symmetry is enhanced to the
full group of diffeomorphisms, i.e. Einstein's gravity, and another one denoted
by WTDiff which enjoys local Weyl invariance. Both of them are free from the
additional scalar.
