  We examine the relations between observables in two- and three-dimensional
quantum gravity by studying the coupling of topologically massive gravity to
matter fields in non-trivial representations of the three-dimensional Lorentz
group. We show that the gravitational renormalization of spin up to one-loop
order in these theories reproduces the leading orders of the KPZ scaling
relations for quantum Liouville theory. We demonstrate that the two-dimensional
scaling dimensions can be computed from tree-level Aharonov-Bohm scattering
amplitudes between the charged particles in the limit where the
three-dimensional theory possesses local conformal invariance. We show how the
three-dimensional description defines scale-dependent weights by computing the
one-loop order anomalous magnetic moment of fermions in a background
electromagnetic field due to the renormalization by topologically massive
gravity. We also discuss some aspects concerning the different phases of
three-dimensional quantum gravity and argue that the topological ones may be
related to the branched polymer phase of two-dimensional quantum gravity.
