On the Betti Numbers of Finite Volume Hyperbolic Manifolds

Abstract

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.

arXiv commentSome changes and references updated following the comments of the referees. 56 pages, no figures. To appear in J. Differential Geom

References in corpus (2)

• Effective results for complex hyperbolic manifolds
• The Kodaira dimension of complex hyperbolic manifolds with cusps

Cited by in corpus (1)

• Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic $3$-Manifolds