  We address in this paper the issue of renormalizability for SU(2) Tensorial
Group Field Theories (TGFT) with geometric Boulatov-type conditions in three
dimensions. We prove that tensorial interactions up to degree 6 are just
renormalizable without any anomaly. Our new models define the renormalizable
TGFT version of the Boulatov model and provide therefore a new approach to
quantum gravity in three dimensions. Among the many new technical results
established in this paper are a general classification of just renormalizable
models with gauge invariance condition, and in particular concerning properties
of melonic graphs, the second order expansion of melonic two point subgraphs
needed for wave-function renormalization.
