Alessio Savini

Starting from the pioneering work by Mostow and Margulis, rigidity of lattices has been widely studied. When the lattice has rank greater or equal than 2, Zimmer generalized Margulis rigidity to the context of measurable cocycles (or virtual actions in the sense of Mackey). The importance of such generalization relies on the surprising applications to orbit equivalence, measure equivalence and foliations by symmetric spaces.

When the lattice has rank equal to one, one can still introduce additional conditions to obtain similar rigidity results. More precisely, given a lattice inside the group SU(n,1), for any measurable cocycle with target SU(m,n) one can define a numerical invariant, called Toledo invariant. Such invariant has bounded absolute value, so it makes sense to speak about maximal cocycles. In this talk we are going to show that maximal Zariski dense cocycles are superrigidity, generalizing a previous result by Pozzetti for representations. As a consequence, we will see that there are no such cocycles if n is not equal to m. 

This is a joint work with Filippo Sarti.