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On the Betti Numbers of Finite Volume Hyperbolic Manifolds

Luca F. Di Cerbo, Mark Stern

TL;DR

This work develops sharp upper bounds for Betti numbers of complete finite-volume rank-one locally symmetric spaces, notably complex- and real-hyperbolic manifolds, by extending Price inequalities with holonomy-aware refinements and introducing peaking techniques to bound $L^2$-cohomology. It treats both compact and cusped quotients, delivering effective bounds for complex-, quaternionic-, and Cayley-hyperbolic cases, and extends the framework to towers of coverings, where normalized cusp counts and $L^2$-cohomology decay govern asymptotics. The authors lever combinatorial-geometric analyses of cusps, Fourier primitives in critical degrees, and lattice invariants to bound middle-degree and critical-degree Betti numbers, including noncompact congruence subgroups. They also connect $L^2$-cohomology to de Rham cohomology in towers and provide appendices comparing their geometric approach with trace-formula methods and supplying explicit real/complex-rank-one calculations. Collectively, the results imply sublinear growth of Betti numbers in towers and yield quantitative control on cusp numbers and cohomology in rank-one locally symmetric spaces, with potential implications for topology and arithmetic of 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.

On the Betti Numbers of Finite Volume Hyperbolic Manifolds

TL;DR

This work develops sharp upper bounds for Betti numbers of complete finite-volume rank-one locally symmetric spaces, notably complex- and real-hyperbolic manifolds, by extending Price inequalities with holonomy-aware refinements and introducing peaking techniques to bound -cohomology. It treats both compact and cusped quotients, delivering effective bounds for complex-, quaternionic-, and Cayley-hyperbolic cases, and extends the framework to towers of coverings, where normalized cusp counts and -cohomology decay govern asymptotics. The authors lever combinatorial-geometric analyses of cusps, Fourier primitives in critical degrees, and lattice invariants to bound middle-degree and critical-degree Betti numbers, including noncompact congruence subgroups. They also connect -cohomology to de Rham cohomology in towers and provide appendices comparing their geometric approach with trace-formula methods and supplying explicit real/complex-rank-one calculations. Collectively, the results imply sublinear growth of Betti numbers in towers and yield quantitative control on cusp numbers and cohomology in rank-one locally symmetric spaces, with potential implications for topology and arithmetic of 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 -cohomology ranks. Finally, we provide bounds on the de Rham cohomology of such spaces, using a combination of our bounds on -cohomology, bounds on the number of cusps in terms of the volume, and the topological interpretation of reduced -cohomology on certain rank one locally symmetric spaces.

Paper Structure

This paper contains 23 sections, 42 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. The Geometry of Geodesic Balls in Rank One Symmetric Spaces
  3. On the Betti Numbers of Compact Quotients of $\textbf{H}^{n}_{\mathbb{K}}$
  4. Refined Estimates for Compact Quotients of $\textbf{H}^{n}_{\mathbb{C}}$
  5. Peaking Revisited
  6. $L^{2}$-Harmonic Forms on Real-Hyperbolic Cusps
  7. $L^2$-Harmonic Forms on Complex-Hyperbolic Cusps
  8. $L^2$-Betti Number Estimates away from Middle Degree
  9. Critical Degree for Real-Hyperbolic Manifolds
  10. Fourier Primitives
  11. Cusps and Lattices
  12. $L^2$-Cohomology on a Tower of Coverings
  13. Fattening the Thick Part
  14. Asymptotic Behavior of $L^2$-Cohomology
  15. Noncompact congruence subgroup quotients
  16. ...and 8 more sections

Key Result

Theorem 2

Let $(M^{n}=\Gamma\backslash\textbf{H}^{n}_{\mathbb{C}}, g_{\mathbb{C}})$ be a compact complex-hyperbolic manifold, with $-4\leq sec_{g_{\mathbb{C}}}\leq -1$. For $k<n$, there exists a positive constant $d(n, k)$ depending only on the dimension and the degree $k$ such that

Theorems & Definitions (77)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Proposition 13
  • proof
  • Corollary 14
  • ...and 67 more