  We study the Einstein gravity and dust system in three spacetime dimensions
as an example of a non-perturbative quantum gravity model with local degrees of
freedom. We derive the Hamiltonian theory in the dust time gauge and show that
it has a rich class of exact solutions. These include the
Ba\~nados-Teitelboim-Zanelli black hole, static solutions with naked
singularities and travelling wave solutions with dynamical horizons. We give a
complete quantization of the wave sector of the theory, including a definition
of a self-adjoint spacetime metric operator. This operator is used to
demonstrate the quantization of deficit angle and the fluctuation of dynamical
horizons.
