Bounded cohomology classes from differential forms

TL;DR

This paper extends the known injective embedding of bounded differential 2-forms into bounded cohomology for hyperbolic manifolds to cases where the fundamental group is of the first kind, using Fourier analysis on the hyperbolic plane.

Contribution

It generalizes previous results by establishing the embedding for a broader class of hyperbolic manifolds with first kind fundamental groups, employing a new Fourier analysis approach.

Findings

Bounded differential 2-forms induce bounded cohomology classes via geodesic simplices.

The embedding is injective for manifolds with fundamental groups of the first kind.

An $L^ abla$ function on the hyperbolic plane is uniquely determined by its integrals over all ideal triangles.

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.