After a series of partial results, Ilya Kapovich proved in 2009 that if G is a non-elementary geometrically finite Kleinian group acting on a finitely-dimensional unit ball by hyperbolic isometries, then either the Hausdorff dimension of the limit set L(G ) of G is strictly larger than its topological dimension k , or else, L(G ) is a geometric k -sphere.
Using the methods, different than Kapovich's ones, stemming from the theory of conformal iterated function systems and geometric measure theory (recifiability), we shall formulate and discuss the proof of an extension of Kapovich result to the case, where geometric finiteness is replaced by the weaker requirement that the Hausdorff dimension of the limit points that are not radial, is smaller than the Hausdorff dimension of the set of radial points.
A counterpart of this theorem for rational functions of the Riemann sphere will be also discussed.
Finally, the case of Kleinian groups acting on the unit ball of infinitely dimensional separable Hilbert space will be considered.