Abstract

Maximal representations of surface groups into symplectic real groups have been extensively studied in the last years. Beautiful results have been obtained using either the algebraic approach offered by the theory of Higgs bundles or a geometric approach based on a formula coming from bounded cohomology. After having recalled those results, we will construct, for a maximal representation $ ho: pi_1(Sigma_g) o mathrm{Sp}(2n, mathbf{R})$, an open subset $Omega ubset mathbf{R} mathbb{P}^{$ where $pi_1( Sigma_g)$ acts properly with compact quotient. The topology of the quotient will then be determined. Finally we shall consider the problem of giving an interpretation of the moduli of maximal symplectic representations as a moduli space of $mathbf{R} mathbb{P}}{2n-1}$-structures and what are the questions that remain to give a complete answer to that problem.