  As it stands, quantum gravity coupled with matter in three spacetime
dimensions is not finite. In this paper I show that an algorithmic procedure
that makes it finite exists, under certain conditions. To achieve this result,
gravity is coupled with an interacting conformal field theory C. The Newton
constant and the marginal parameters of C are taken as independent couplings.
The values of the other irrelevant couplings are determined iteratively in the
loop- and energy-expansions, imposing that their beta functions vanish. The
finiteness equations are solvable thanks to the following properties: the beta
functions of the irrelevant couplings have a simple structure; the irrelevant
terms made with the Riemann tensor can be reabsorbed by means of field
redefinitions; the other irrelevant terms have, generically, non-vanishing
anomalous dimensions. The perturbative expansion is governed by an effective
Planck mass that takes care of the interactions in the matter sector. As an
example, I study gravity coupled with Chern-Simons U(1) gauge theory with
massless fermions, solve the finiteness equations and determine the
four-fermion couplings to two-loop order. The construction of this paper does
not immediately apply to four-dimensional quantum gravity.

High Energy Physics - Theory