  In the context of canonical quantum gravity in 3+1 dimensions, we introduce a
new notion of bubble network that represents discrete 3d space geometries.
These are natural extensions of twisted geometries, which represent the
geometrical data underlying loop quantum geometry and are defined as networks
of SU(2) holonomies. In addition to the SU(2) representations encoding the
geometrical flux, the bubble network links carry a compatible SL(2,R)
representation encoding the discretized frame field which composes the flux. In
contrast with twisted geometries, this extra data allows to reconstruct the
frame compatible with the flux unambiguously. At the classical level this data
represents a network of 3d geometrical cells glued together. The SL(2,R) data
contains information about the discretized 2d metrics of the interfaces between
3d cells and SL(2,R) local transformations are understood as the group of
area-preserving diffeomorphisms. We further show that the natural gluing
condition with respect to this extended group structure ensures that the
intrinsic 2d geometry of a boundary surface is the same from the viewpoint of
the two cells sharing it. At the quantum level this gluing corresponds to a
maximal entanglement along the network edges. We emphasize that the nature of
this extension of twisted geometries is compatible with the general analysis of
gauge theories that predicts edge mode degrees of freedom at the interface of
subsystems.
