  A Kaluza-Klein like approach for a 4d spin foam model is considered. By
applying this approach to a model based on group field theory in 4d (TOCY
model), and using the Peter-Weyl expansion of the gravitational field,
reconstruction of new non gravitational fields and interactions in the action
are found. The perturbative expansion of the partition function produces graphs
colored with su(2) algebraic data, from which one can reconstruct a 3d
simplicial complex representing space-time and its geometry; (like in the
Ponzano-Regge formulation of pure 3d quantum gravity), as well as the Feynman
graph for typical matter fields. Thus a mechanism for generation of matter and
construction of new dimensions are found from pure gravity.
