ON THE BETTI NUMBERS OF FINITE VOLUME REAL- AND COMPLEX-HYPERBOLIC MANIFOLDS

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 [DS22]. 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 L²-cohomology ranks. Finally, we provide bounds on the de Rham cohomology of such spaces, using a combination of our bounds on L²-cohomology, bounds on the number of cusps in terms of the volume, and the topologi- cal interpretation of reduced L²-cohomology on certain rank one locally symmetric spaces.