B.O. Stratmann K. Falk
Remarks on Hausdorff Dimensions for Transient Limit Sets of Kleinian Groups
Preprint series: Mathematica Gottingensis

MSC:

20H10 Fuchsian groups and their generalizations

30F40 Kleinian groups

Abstract: In this paper we study discrepancy groups (d-groups), that are Kleinian
groups whose exponent of convergence is strictly less than the
Hausdorff dimension of their limit set. We show that the limit set
of a d-group always contains continuous families of fractal sets, each of
which contains the set of radial limit points and has Hausdorff
dimension strictly less than the Hausdorff dimension of the whole limit
set. Subsequently, we consider special d-groups which are normal
subgroups of some geometrically finite Kleinian group. For these we
obtain the result that their Poincare exponent is always bounded
from below by half of the Poincare exponent of the associated
geometrically finite group in which they are normal. Finally, we give
a discussion of various examples of d-groups, which in particular
also contains explicit constructions of these groups.