Topological and geometric restrictions on hyperconvex representations

We study the geometry of hyperconvex representations of hyperbolic groups in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>PSL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> PSL(d,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.