LAUSR

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 $$\mathbf {G}$$ G be a semisimple algebraic $${\mathbb {R}}$$ R -group such that $$G=\mathbf {G}({\mathbb {R}})^{\circ }$$ G = G ( R ) ∘ is of Hermitian type. If $$\Gamma \le L$$ Γ ≤ L is a torsion-free lattice of a finite connected covering of $$\mathrm{PU}(1,1)$$ PU ( 1 , 1 ) , given a standard Borel probability $$\Gamma $$ Γ -space $$(\Omega ,\mu _\Omega )$$ ( Ω , μ Ω ) , we introduce the notion of Toledo invariant for a measurable cocycle $$\sigma :\Gamma \times \Omega \rightarrow G$$ 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 $$\mathbf {H}$$ H of a maximal cocycle $$\sigma $$ σ is reductive and the centralizer of $$H=\mathbf {H}({\mathbb {R}})^{\circ }$$ H = H ( R ) ∘ is compact. If additionally $$\sigma $$ σ admits a boundary map, then H is of tube type and $$\sigma $$ 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 $$G=\mathrm{PU}(n,1)$$ G = PU ( n , 1 ) maximality is sufficient to prove that $$\sigma $$ σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.