  We analyze the edge mode structure of Euclidean three dimensional gravity
from within the quantum theory as embodied by a Ponzano-Regge-Turaev-Viro
discrete state sum with Gibbons-=-Hawking-York boundary conditions. This
structure is encoded in a pair of dual statistical models of the vertex and
face kind, which for specific choices of boundary conditions turn out to be
integrable. The duality is just the manifestation of a pervasive dual structure
which manifests at different levels of the classical and quantum theories.
Emphasis will be put on the geometrical interpretation of the edge modes which
leads in particular to the identification of the quantum analogue of Carlip's
would-be normal diffeomorphisms. We also provide a reinterpretation of our
construction in terms of a non-Abelian 2+1 topological phase with electric
boundary conditions.
