  We discuss the hypothesis of a fixed point for quantum gravity coupled to a
scalar, in the limit where the scalar field goes to infinity, accompanied by a
suitable scaling of the metric. We propose that no scalar potential is present
for the dilatation symmetric quantum effective action at the fixed point.
Dimensional reduction of such a higher dimensional effective action leads to
solutions with a vanishing effective four-dimensional constant. Under rather
general circumstances these are the only quasistatic stable solutions with
finite four-dimensional gravitational constant. If cosmological runaway
solutions approach the fixed point as time goes to infinity, the cosmological
constant vanishes asymptotically. For our old Universe the fixed point is not
yet reached completely, resulting in a tiny amount of dark energy, comparable
to dark matter. We discuss explicitly higher dimensional geometries which
realize such asymptotic solutions for $t\to\infty$. They include Ricci-flat
spaces as well as warped spaces, potentially with singularities.
