For negatively curved manifolds, a way of building classes in bounded cohomology is to consider cocycles defined by integrating differential forms over straight simplices. A classical result of Barge and Ghys shows that for closed, negatively curved surfaces this procedure defines an injective map from the space of differential 2-forms to the second bounded cohomology of the surface. Recently, Battista, Francaviglia, Moraschini, Sartini and Savini showed that a similar result holds for 2-forms on closed negatively curved manifolds in any dimension.
In this talk, I will present a generalisation of these results for non-compact hyperbolic manifolds, characterising the injectivity of the Barge-Ghys map in terms of the action of the fundamental group on hyperbolic space.
This is (ongoing) joint work with Gian Maria dall'Ara and Roberto Frigerio.
