Entropy of random coverings and 4-D quantum gravity

We discuss the counting of minimal geodesic ball coverings of n-dimensional (n≥3) Riemannian manifolds of bounded geometry, fixed Euler characteristic, and Reidemester torsion in a given representation of the fundamental group. This counting bears relevance to the analyis of the continuum limit of dicrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corrisponding Reidemester torsion. We discuss the summ over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponenents are obtained in both cases.