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Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let G be a semisimple algebraic R -group such that G = G (R) ∘ is of Hermitian type. If Γ ≤ L is a torsion-free lattice of a finite connected covering of PU (1 , 1) , given a standard Borel probability Γ -space (Ω , μ Ω) , we introduce the notion of Toledo invariant for a measurable cocycle σ : Γ × Ω → G . The Toledo invariant remains unchanged along G-cohomology classes and its absolute value is bounded by the rank of G. This allows to define maximal measurable cocycles. We show that the algebraic hull H of a maximal cocycle σ is reductive and the centralizer of H = H (R) ∘ is compact. If additionally σ admits a boundary map, then H is of tube type and σ is cohomologous to a cocycle stabilizing a unique maximal tube type subdomain. This result is analogous to the one obtained for representations. In the particular case G = PU (n , 1) maximality is sufficient to prove that σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles. [ABSTRACT FROM AUTHOR]

We show that there is a unique functorial way to extend the classical four-point cross ratio on the unit circle to a family of four-point invariants on Shilov boundaries of bounded symmetric domains of tube type. These generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domains. They can also be used to associate with every maximal representation of a surface group with Hermitian target a strict weak cross ratio on the circle in the sense of Labourie. In this context our translation length estimates then imply that maximal representations with Hermitian target are well-displacing, and consequently that the action of the mapping class group on the moduli space of maximal representations into a Hermitian Lie group is proper. [ABSTRACT FROM AUTHOR]