Homology and homotopy complexity in negative curvature

A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We prove an analog of this theorem for the torsion part of the homology in all dimensions d not equal 3. Thus the total homology is controlled by the volume. This applies in particular to the classical case of hyperbolic manifolds. In dimension 3 the size of torsion homology cannot be bounded in terms of the volume. As a byproduct, in dimension d >= 4 we give a fairly precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d >= 5 up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest.