Abstract

A quasi-Fuchsian representation of a surface group in PSL is a discrete and faithful representation that preserves a Jordan curve on the Riemann sphere. These classical objects have a very rich structure as they lie at the crossroad of several areas of mathematics such as complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. I will discuss joint work with James Farre and Gabriele Viaggi in which we investigate similar phenomena for a class of representations of surface groups in PSL , hyperconvex representations, and discuss geometric properties of the image subgroups and their parameter space. Among other things we show that the groups admitting hyperconvex  representations are virtually isomorphic to convex-cocompact subgroups of PSL , and more generally they exhibit striking analogies with such groups, such as suitable Ahlfors-Bers parameters.