Numerical invariants for measurable cocycles

The theory of numerical invariants for representations can be generalized to measurable cocycles. This provides a natural notion of maximality for cocycles associated to complex hyperbolic lattices with values in groups of Hermitian type. Among maximal cocycles, the class of Zariski dense ones turns out to have a rigid behavior. An alternative implementation of numerical invariants can be given by using equivariant maps at the level of boundaries and by exploiting the Burger-Monod approach to bounded cohomology. Due to their crucial role in this theory, we prove existence results in two different contexts. Precisely, we construct boundary maps for non-elementary cocycles into the isometry group of CAT(0)-spaces of finite telescopic dimension and for Zariski dense cocycles into simple Lie groups. Then we approach numerical invariants. Our first goal is to study cocycles from complex hyperbolic lattices into the Hermitian group SU(p,q). Following the theory recently developed by Moraschini and Savini, we define the Toledo invariant by using the pullback along cocycles, also by involving boundary maps. For cocycles Γ × X → SU(p,q) with 1<p<q<+∞, we prove that maximality and Zariski density imply superrigidity in the sense of Zimmer, namely such cocycles come from representations PU(1,n) → SU(p,q) of the ambient group. As a consequence, there is no Zariski dense such cocycle when 1<p<q. Then we move to cocycles Γ × X → PU(p,∞) where PU(p,∞) is the infinite dimensional version of PU(p,q). We show that maximal cocycles are reducible, namely that, modulo cohomology, their image is contained in a finite dimensional algebraic subgroup of PU(p,∞). Finally, we classify Zariski dense measurable cocycles Γ × X → G from finitely generated groups into Hermitian groups not of tube-type. Precisely, we show that the pullback of the Kahler class completely determines the cohomology class of such cocycles.