Topological and geometric restrictions on hyperconvex representations

Abstract

We study the geometry of hyperconvex representations of hyperbolic groups in $\mathrm{PSL}(d,\mathbb{C})$ 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 $2$.

23 pages. Minor revision, corrected a small inaccuracy in the proof of Proposition 4.16