Improved and perfect actions in discrete gravity

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they-although being defined on a triangulation-reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of general relativity. We construct the perfect action in three dimensions with a cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge calculus with curved simplices, which arises naturally in this context.