  The overall aim of this note is to initiate a "manifold" theory for metric
Diophantine approximation on the limit sets of Kleinian groups. We investigate
the notions of singular and extremal limit points within the geometrically
finite Kleinian group framework. Also, we consider the natural analogue of
Davenport's problem regarding badly approximable limit points in a given subset
of the limit set. Beyond extremality, we discuss potential Khintchine-type
statements for subsets of the limit set. These can be interpreted as the
conjectural "manifold" strengthening of Sullivan's logarithmic law for
geodesics.
