  An effective field theory for matter coupled to three-dimensional quantum
gravity was recently derived in the context of spinfoam models in
hep-th/0512113. In this paper, we show how this relates to group field theories
and generalized matrix models. In the first part, we realize that the effective
field theory can be recasted as a matrix model where couplings between matrices
of different sizes can occur. In a second part, we provide a family of
classical solutions to the three-dimensional group field theory. By studying
perturbations around these solutions, we generate the dynamics of the effective
field theory. We identify a particular case which leads to the action of
hep-th/0512113 for a massive field living in a flat non-commutative space-time.
The most general solutions lead to field theories with non-linear redefinitions
of the momentum which we propose to interpret as living on curved space-times.
We conclude by discussing the possible extension to four-dimensional spinfoam
models.
