  Nonperturbative treatments of the UV limit of pure gravity suggest that it
admits a stable fixed point with positive Newton's constant and cosmological
constant. We prove that this result is stable under the addition of a scalar
field with a generic potential and nonminimal couplings to the scalar
curvature. There is a fixed point where the mass and all nonminimal scalar
interactions vanish while the gravitational couplings have values which are
almost identical to the pure gravity case. We discuss the linearized flow
around this fixed point and find that the critical surface is four-dimensional.
In the presence of other, arbitrary, massless minimally coupled matter fields,
the existence of the fixed point, the sign of the cosmological constant and the
dimension of the critical surface depend on the type and number of fields. In
particular, for some matter content, there exist polynomial asymptotically free
scalar potentials, thus providing a solution to the well-known problem of
triviality.
