Geometry of hyperconvex representations of surface groups

We study the geometry of hyperconvex representations of surface groups in $\mathrm{P}\mathrm{S}\mathrm{L}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors–Bers map to a product of Teichmüller spaces of a canonical Riemann surface lamination and prove that the limit set of a hyperconvex representation in the full flag space has Hausdorff dimension 1 if and only if the representation is conjugate in $\mathrm{P}\mathrm{S}\mathrm{L}(d,\mathbb{R})$.