Seminari del Dipartimento di Matematica

Seminario del 2023

Explicit measurable cocycles for actions at infinity of some semisimple Lie groups

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Since the formulation of Dupont's conjecture, it has been evident the importance to understand the boundedness of characteristic classes appearing in the cohomology ring of a semisimple Lie group. This problem is deeply related to Monod's conjecture, which relates the continuous bounded cohomology of a semisimple Lie group with its continuous variant. An important step towards a possible proof of those conjectures was the isometric realization of the continuous bounded cohomology of a semisimple Lie group G as the cohomology of the complex of essentially bounded functions on the Furstenberg-Poisson boundary (and more generally for any regular amenable G-space). Surprisingly, Monod has recently proved that the complex of measurable unbounded functions on the same boundary does not compute the continuous cohomology of G unless the rank of the group is not one, but an additional term appears. Nevertheless, there is a way to characterize explicitly the defect in terms of the invariant cohomology of a maximal split torus. In this seminar we will exhibit two main examples of such phenomenon: the product of isometry groups of real hyperbolic spaces and the group SL3. The first part of the seminar will be devoted to an overview about the state of art. Then we will move to examples and we will give a characterization of Monod's Kernel in low degree. Finally we will show that Monod's conjecture is true in those cases. In the second part of the seminar we will discuss in details the main results and the techniques we used, such as the explicit computation on Bloch-Monod spectral sequence. If time allows we will show how we can implement all this stuff using a software like Sagemath.