We consider transition amplitudes in the coloured simplicial Boulatov model
for three-dimensional Riemannian quantum gravity. First, we discuss aspects of
the topology of coloured graphs with non-empty boundaries. Using a modification
of the standard rooting procedure of coloured tensor models, we then write
transition amplitudes systematically as topological expansions. We analyse the
transition amplitudes for the simplest boundary topology, the 2-sphere, and
prove that they factorize into a sum entirely given by the combinatorics of the
boundary spin network state and that the leading order is given by graphs
representing the closed 3-ball in the large N limit. This is the first step
towards a more detailed study of the holographic nature of coloured
Boulatov-type GFT models for topological field theories and quantum gravity.
