We study the geometry of hyperconvex representations of surface groups in ${ \rm PSL}(d,\mathbb{ C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller 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 ${ \rm PSL}(d,\mathbb{ R})$.
Categories: math.GT math.DG math.GR