Critical phenomena for Riemannian manifolds: Simple Homotopy and simplicial quantum gravity

Simplicial quantum gravity is an approach to quantizing gravity by using as a regularization scheme dynamical triangulations of riemannian manifolds. In dimension two, such an approach has led to considerable progress in our understanding of 2D-gravity. In dimension three and four, the current state of affairs is less favourable, and dynamically triangulated models of quantum gravity suffer from serious difficulties which can be only partially removed by the use of computer simulations. In this paper, which to a large extent reviews some of our recent work, we show how Gromov’s spaces of bounded geometries provide a general mathematical framework for addressing and solving many of the issues of 3D-simplicial quantum gravity. In particular, we establish entropy estimates characterizing the asymptotic distribution of combinatorially inequivalent triangulated 3-manifolds, as the number of tetrahedra diverges. Moreover, we offer a rather detailed presentation of how spaces of three-dimensional riemannian manifolds with natural bounds on curvatures, diameter, and volume can be used to prove that three-dimensional simplicial quantum gravity is connected to a Gaussian model determined by the simple homotopy types of the underlying manifolds. This connection is determined by a Gaussian measure defined over the general linear group GL(R,∞). By exploiting these results it is shown that the partition function of threedimensional simplicial quantum gravity is well-defined, in the thermodynamic