De Rham quasimorphisms, de Rham classes, and continuous functions on S¹

Thanks to a result by Fournier-Facio and Wade, we now know that for nonelementary hyperbolic groups, there exists an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. However, their construction is not very explicit. Therefore, it makes sense to restrict ourselves to specific Aut-invariant subspaces of quasimorphisms, derived, for example, from geometric or combinatorial constructions. During this talk, our focus is on de Rham quasimorphisms, which are defined for fundamental groups of hyperbolic surfaces. In this context, the action of the automorphism group can be translated into two different frameworks: bounded cohomology and the set of continuous functions on the circle. We delve into how these interpretations provide a more geometric perspective, facilitating a deeper understanding of the action of the automorphism group. By the end of the talk, after translating the results back into the original context, we will be able to describe all the finite-dimensional invariant subspaces of the action of the automorphism group on the space of de Rham quasimorphisms.