In this chapter, we continue the development of the global theory, wherein we emphasize volume and integration. The major theme is the study of the volume growth of Riemannian manifolds, and the fundamental approach is to reduce the study of the volume growth to a corresponding discrete problem.

Such a study presupposes that the Riemannian manifold has sufficient local uniformity to allow us to disregard local fluctuations of the geometric data. The primary example is that of a noncompact manifold covering a compact Riemannian manifold, where the Riemannian metric on the cover is the lift of the Riemannian metric on the compact via the covering (this example was first considered in this context by V. Efremovič (1953), A. S. Svarc (1955), and J. Milnor (1968)). The covering determines a discrete group of isometries of the cover, which, in turn, induces a tiling of the cover by relatively compact fundamental domains – each isometric to the other. Thus, the estimate of the volume of a metric disk in the cover is reduced to counting the number of fundamental domains contained in the disk, and the smallest number of fundamental domains containing the disk. Since the action of the group is free, counting fundamental domains is the same as counting elements of the group. The quantitative estimates used here are the Bishop theorems of §III.4.

More generally, one may relax the degree of local uniformity and nevertheless obtain similar discretizations of Riemannian manifolds. Here, for the calibration of discrete to the continuous, one must use Gromov's refinement of the Bishop theorems (see §III.4). Our treatment, in §IV.4, follows that of M. Kanai (1985).

We consider a number of other matters along the way.