A discrete and faithful representation of a surface group in PSL(2,C) is said to be quasi-Fuchsian when it preserves a Jordan curve on the Riemann sphere. Classically these objects lie at the intersection of several areas of mathematics and have been studied (for example) using complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. From a dynamical perspective, an important invariant of such representations is the Hausdorff dimension of the invariant Jordan curves (typically a very fractal object). It is elementary to see that this number is always at least 1. A celebrated result of Bowen establishes it is equal to 1 if and only if the quasi-Fuchsian representation is Fuchsian, that is, it is conjugate in PSL(2,R). I will first describe this classical picture and then report on recent joint work with James Farre and Beatrice Pozzetti where we prove a generalization of Bowen's result for the much larger class of hyperconvex representations of surface groups in PSL(d,C) (where d is arbitrary).