Bounded cohomology classes from differential forms

Abstract: Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$) and by Battista et al. (for $n>2$) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed differential $2$-forms on $M$ into $H^2_b(M)$. We extend this result to the case when the fundamental group of $M$ is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when $M$ has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an $L^\infty$ function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane.

• The paper proves that integration over geodesic simplices yields an injective map from bounded differential 2-forms to bounded cohomology classes on hyperbolic manifolds.
• It uses advanced Fourier analysis and integral geometry to show that bounded functions are uniquely determined by their integrals over ideal triangles.
• The work extends classical compact manifold results to non-compact cases by linking injectivity to the dynamics of the fundamental group at infinity.

Bounded Cohomology Classes from Differential Forms

Introduction and Context

The paper "Bounded cohomology classes from differential forms" (2604.16289) addresses the interplay between bounded cohomology and differential geometry, focusing on the construction of bounded cohomology classes from bounded, closed differential forms on complete hyperbolic $n$-manifolds. Bounded cohomology, originally introduced by Johnson and Trauber for Banach algebras, and generalized by Gromov to spaces and groups, diverges notably from classical (singular) cohomology, especially regarding excision and dimension. In contrast to its vanishing on amenable or non-negatively curved spaces, bounded cohomology is typically infinite-dimensional for negatively curved manifolds, particularly in degree 2.

For compact hyperbolic manifolds, previous work by Barge-Ghys ($n=2$) and Battista et al. ($n>2$) established that the integration of closed bounded differential 2-forms over geodesic triangles yields an injective map into bounded cohomology, embedding an infinite-dimensional subspace. However, these results were confined to the compact (closed) case. The present paper extends the injectivity criterion to non-compact manifolds, providing a precise characterization in terms of the dynamics of the fundamental group on the boundary at infinity, and establishes important new results in integral geometry on the hyperbolic plane as technical tools.

Main Results and Theoretical Advances

The core contribution is the following sharp theorem: Let $M$ be a complete hyperbolic $n$-manifold ($n\geq 2$). The map

$\psi: Z\Omega^2_b(M) \to H_b^2(M)$

which sends a closed bounded differential 2-form to its associated bounded cohomology class via integration over geodesic simplices, is injective if and only if $M$ is of the first kind (i.e., its fundamental group $\Gamma$ has limit set equal to the entire boundary at infinity $\partial\mathbb{H}^n$).

An immediate corollary is that if $n=2$0 has finite volume, then $n=2$1 is injective, since finite volume implies the first kind. The paper also provides a complete proof of failure of injectivity in the "second kind" case, exploiting the topology of the convex core and the existence of forms supported outside it.

A critical innovation is the reduction of the injectivity question to a problem in integral geometry via amenable boundary actions and the explicit use of the Burger-Monod resolution for bounded group cohomology. This approach leads to the formulation and proof of the following striking result in integral geometry:

Theorem: A bounded function on the hyperbolic plane is uniquely determined by its integrals over all ideal triangles.

This is proved using Fourier analysis on $n=2$2, specifically leveraging the Helgason-Fourier transform and a hyperbolic analog of Wiener's Tauberian Theorem, as established by Mohanty et al.

Technical Approach and Novelty

1. Dynamics on the Boundary at Infinity:

The characterization of injectivity of $n=2$3 is achieved by connecting it to the dynamics of $n=2$4 on $n=2$5. The manifold is of the first kind if the limit set is full, a condition closely related to (but weaker than) ergodicity of the geodesic flow. The paper demonstrates that injectivity holds precisely in this regime, using deep results on the dynamics of Kleinian groups as well as properties of the convex core.

2. Amenable Boundary Actions and Bounded Resolutions:

The authors exploit the amenability of the action of $n=2$6 on $n=2$7, facilitating an explicit computation of bounded group cohomology via invariant alternating $n=2$8-functions. This construction translates the differential form integration map into an explicit cocycle on the boundary, providing a genuinely cochain-level representative.

3. Integral Geometry and Functional Analysis:

Injectivity is ultimately reduced to the surjectivity of a certain Radon-type transform on $n=2$9: the ideal triangle transform. The authors show, by Fourier-analytic means, that a bounded function is uniquely determined by its integrals over the family of all ideal triangles. This transfer to integral geometry is both technically advantageous and conceptually powerful.

4. Use of the Hyperbolic Wiener's Tauberian Theorem:

The technical heart is a careful application of a hyperbolic analog of Wiener's theorem, coupled with explicit computations detailing the absence of non-trivial solutions to a functional equation involving the Helgason-Fourier transform of the ideal triangle indicator function.

5. Generalizations and Constraints:

The paper also discusses limitations when considering higher degree forms ($n>2$0) and non-smooth coefficients, and relates the result to established constructions of bounded cohomology classes in geometric topology (e.g., using volume forms on geometrically infinite manifolds).

Numerical and Structural Highlights

• The ideal triangle transform $n>2$1, defined by $n>2$2, is shown to be injective.
• The crucial equations in the Fourier analysis—specifically, the vanishing of Gamma function ratios—are shown to admit no non-trivial solutions in the analytic strip relevant for bounded functions.
• The proof does not rely on ergodicity of the geodesic flow, which would be a strictly stronger dynamical assumption, but only on the group being of first kind.

Implications and Future Directions

Theoretical Implications:

The injectivity result provides a robust new tool for constructing and analyzing bounded cohomology classes on a large class of hyperbolic manifolds, broadening the landscape previously accessible only in the compact setting. The reduction to integral geometry signals an appealing interplay between rigidity theory, dynamics, and analysis on symmetric spaces.

Practical and Computational Consequences:

Having an explicit, injective embedding permits effective calculations of bounded cohomology classes arising from differential-geometric data, with implications for identifying invariants that distinguish infinite-volume or finite-volume hyperbolic manifolds beyond the reach of classical topology.

Extension to Other Geometries and Generalized Cohomology:

While the present work focuses on hyperbolic manifolds, the techniques suggest applicability to broader contexts—particularly to symmetric spaces of noncompact type and to settings in higher rank, where integral geometry and harmonic analysis play central roles.

Potential in the Study of Rigidity and Invariants:

Applications likely extend to rigidity phenomena, the analysis of measure equivalence, and the study of secondary characteristic classes associated with non-amenable or dynamically rich groups.

Future Research Questions:

• Generalization to non-smooth, $n>2$3-coefficient forms in higher degree
• Explicit inversion formulas for the ideal triangle transform
• Precise characterizations of the image and kernel of such transforms for broader classes of functions or groups
• Analysis of bounded cohomology in higher rank symmetric spaces via similar analytic-geometric techniques

Conclusion

This paper rigorously establishes the precise circumstances under which bounded closed differential 2-forms on complete hyperbolic manifolds yield injective elements in second bounded cohomology, extending foundational results to the non-compact, finite-volume setting and beyond. The reduction to and solution of a previously unexamined problem in hyperbolic integral geometry constitutes a significant methodological advance. By integrating geometric, analytic, and cohomological perspectives, this work opens new avenues for the study of invariants in geometric topology and bounded cohomology theory.