Bounded cohomology classes from measurable differential forms (Roberto Frigerio, Pisa)

10.03.2026 10:30

Let S be a complete hyperbolic surface. Via integration over geodesic simplices, any bounded differential 2-form on S defines a degree-2 bounded cohomology class in H^2_b(S). It was proved by Barge and Ghys that, if S is closed, this procedure defines an injective embedding of the (infinite-dimensional) space of differential 2-forms on S into H^2_b(S). We extend Barge and Ghys’ Theorem in two directions, showing that the same result holds under the weaker hypothesis that S is of parabolic type (this holds, for example, when it has finite area), and letting forms vary among all measurable bounded 2-forms on S, rather than only among smooth ones.
Our argument is different from Barge and Ghys’ original one, and relies on the proof of a fact of independent interest: ideal triangles in the hyperbolic plane are Pompeiu sets.
This is joint work with Gian Maria Dall'Ara and Ervin Hadziosmanovic.

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