Growth of fundamental groups and isoembolic volume and diameter

Some properties of fundamental groups of Riemannian manifolds M will be studied without a lower bound assumption on Ricci curvature. The main method is to relate the local packing to global packing instead of using the Bishop-Gromov relative volume comparison. This method allows us to control the volume growth of the universal cover M and yields bounds on the number of generators of π 1 (M) in terms of some isoembolic geometric invariants of M.