  We introduce a linearized version of group field theory. It can be viewed
either as a group field theory over the additive group of a vector space or as
an asymptotic expansion of any group field theory around the unit group
element. We prove exact power counting theorems for any graph of such models.
For linearized colored models the power counting of any amplitude is further
computed in term of the homology of the graph. An exact power counting theorem
is also established for a particular class of graphs of the nonlinearized
models, which satisfy a planarity condition. Examples and connections with
previous results are discussed.
