Dynamics and diffeomorphism symmetry in discrete quantum gravity

Diffeomorphism symmetry is the underlying symmetry of general relativity and deeply intertwined with its dynamics. The notion of diffeomorphism symmetry is however often obscured in discrete gravity, which underlies most of the current quantum gravity models. We propose a notion of diffeomorphism symmetry in discrete models and find that such a symmetry is weakly broken in many models. We explain the connection to the problem of finding a consistent canonical dynamics for discrete gravity. To overcome these problems we discuss methods to construct models with exact symmetries via so called perfect actions which involves a renormalization group analysis of the discrete quantum gravity models.