  In this paper we analyze the functional renormalization group flow of quantum
gravity on the Einstein-Cartan theory space. The latter consists of all action
functionals depending on the spin connection and the vielbein field (co-frame)
which are invariant under both spacetime diffeomorphisms and local frame
rotations. In the first part of the paper we develop a general methodology and
corresponding calculational tools which can be used to analyze the flow
equation for the pertinent effective average action for any truncation of this
theory space. In the second part we apply it to a specific three-dimensional
truncated theory space which is parametrized by Newton's constant, the
cosmological constant, and the Immirzi parameter. A comprehensive analysis of
their scale dependences is performed, and the possibility of defining an
asymptotically safe theory on this hitherto unexplored theory space is
investigated. In principle Asymptotic Safety of metric gravity (at least at the
level of the effective average action) is neither necessary nor sufficient for
Asymptotic Safety on the Einstein-Cartan theory space which might accommodate
different "universality classes" of microscopic quantum gravity theories.
Nevertheless, we do find evidence for the existence of at least one
non-Gaussian renormalization group fixed point which seems suitable for the
Asymptotic Safety construction in a setting where the spin connection and the
vielbein are the fundamental field variables.
