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The overall aim of this note is to initiate a "manifold" theory for metricDiophantine approximation on the limit sets of Kleinian groups. We investigatethe notions of singular and extremal limit points within the geometricallyfinite Kleinian group framework. Also, we consider the natural analogue ofDavenport's problem regarding badly approximable limit points in a given subsetof the limit set. Beyond extremality, we discuss potential Khintchine-typestatements for subsets of the limit set. These can be interpreted as theconjectural "manifold" strengthening of Sullivan's logarithmic law forgeodesics.

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points