Bounded cohomology classes from differential forms
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917416365195264 |
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| author | Dall'Ara, Gian Maria Frigerio, Roberto Hadziosmanovic, Ervin |
| author_facet | Dall'Ara, Gian Maria Frigerio, Roberto Hadziosmanovic, Ervin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounded cohomology classes from differential forms Dall'Ara, Gian Maria Frigerio, Roberto Hadziosmanovic, Ervin Geometric Topology Algebraic Topology Classical Analysis and ODEs Differential Geometry 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. |
| title | Bounded cohomology classes from differential forms |
| topic | Geometric Topology Algebraic Topology Classical Analysis and ODEs Differential Geometry |
| url | https://arxiv.org/abs/2604.16289 |