  A higher-derivative, interacting, scalar field theory in curved spacetime
with the most general action of sigma-model type is studied. The one-loop
counterterms of the general theory are found. The renormalization group
equations corresponding to two different, multiplicatively renormalizable
variants of the same are derived. The analysis of their asymptotic solutions
shows that, depending on the sign of one of the coupling constants, we can
construct an asymptotically free theory which is also asymptotically conformal
invariant at strong (or small) curvature. The connection that can be
established between one of the multiplicatively renormalizable variants of the
theory and the effective theory of the conformal factor, aiming at the
description of quantum gravity at large distances, is investigated.
