Geometry of hyperconvex representations of surface groups

Abstract

We study the geometry of hyperconvex representations of surface groups in $\mathrm{PSL}(d,\mathbb{C})$ 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 $\mathrm{PSL}(d,\mathbb{R})$.

70 pages. Comments are welcome!