Topological and geometric restrictions on hyperconvex representations

We study the geometry of hyperconvex representations of hyperbolic groups in $\mathrm{P}\mathrm{S}\mathrm{L}(d,\mathbb{C})$ and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its hyperconvex limit set in the appropriate flag manifold has Hausdorff dimension strictly smaller than $2$.