  The discrete spatial geometry underlying Loop Quantum Gravity (LQG) is
degenerate almost everywhere. This is at apparent odds with the non- degeneracy
of asymptotically flat metrics near spatial infinity. Koslowski generalised the
LQG representation so as to describe states labelled by smooth non- degenerate
triad fields. His representation was further studied by Sahlmann with a view to
imposing gauge and spatial diffeomorphism invariance through group averaging
methods. Motivated by the desire to model asymptotically flat quantum geometry
by states with triad labels which are non- degenerate at infinity but not
necessarily so in the interior, we initiate a generalisation of Sahlmann's
considerations to triads of varying degeneracy. In doing so, we include
delicate phase contributions to the averaging procedure which are crucial for
the correct implementation of the gauge and diffeomorphism constraints, and
whose existence can be traced to the background exponential functions recently
constructed by one of us. Our treatment emphasizes the role of symmetries of
quantum states in the averaging procedure. Semianalyticity, influential in the
proofs of the beautiful uniqueness results for LQG, plays a key role in our
considerations. As a by product, we re- derive the group averaging map for
standard LQG, highlighting the role of state symmetries and explicitly
exhibiting the essential uniqueness of its specification.
