  We obtain strong upper bounds for the Betti numbers of compact
complex-hyperbolic manifolds. We use the unitary holonomy to improve the
results given by the most direct application of the techniques of [DS17]. We
also provide effective upper bounds for Betti numbers of compact quaternionic-
and Cayley-hyperbolic manifolds in most degrees. More importantly, we extend
our techniques to complete finite volume real- and complex-hyperbolic
manifolds. In this setting, we develop new monotonicity inequalities for
strongly harmonic forms on hyperbolic cusps and employ a new peaking argument
to estimate $L^2$-cohomology ranks. Finally, we provide bounds on the de Rham
cohomology of such spaces, using a combination of our bounds on
$L^2$-cohomology, bounds on the number of cusps in terms of the volume, and the
topological interpretation of reduced $L^2$-cohomology on certain rank one
locally symmetric spaces.
