Homology and homotopy complexity in negative curvature

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions d≠3. This extends a classical theorem by Gromov. In dimension 3, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of 3-dimensional hyperbolic manifolds that converges to H3 in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in [0,∞]. In dimension d≥4 a somewhat precise estimate is given 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.