Fixed point structure of the conformal factor field in quantum gravity

The O(partial derivative(2)) background- independent flow equations for conformally reduced gravity are shown to be equivalent to flow equations naturally adapted to scalar field theory with a wrong-sign kinetic term. This sign change is shown to have a profound effect on the renormalization group properties, broadly resulting in a continuum of fixed points supporting both a discrete and a continuous eigenoperator spectrum, the latter always including relevant directions. The properties at the Gaussian fixed point are understood in particular depth, but also detailed studies of the local potential approximation, and the full O(partial derivative(2)) approximation are given. These results are related to evidence for asymptotic safety found by other authors.