  We consider the quantization of the midi-superspace associated with a class
of spacetimes with toroidal isometries, but without the compact spatial
hypersurfaces of the well-known Gowdy models. By a symmetry reduction, the
phase space for the system at the classical level can be identified with that
of a free massless scalar field on a fixed background spacetime, thereby
providing a simple route to quantization. We are then able to study certain
non-perturbative features of the quantum gravitational system. In particular,
we examine the quantum geometry of the asymptotic regions of the spacetimes
involved and find some surprisingly large dispersive effects of quantum
gravity.
