Abstract
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.
Highlights
In what follows G denotes a nonelementary, geometrically finite Kleinian group acting on the unit ball model (Bd+1, ρ) of (d + 1)–dimensional hyperbolic space with metric ρ derived from the differential dρ = 2|dx|/(1−|x|2)
The following two Dirichlet-type theorems were first established by Patterson [33, Section 7: Theorems 1 & 2] for finitely generated Fuchsian groups, i.e. Kleinian groups acting on the unit disc model of 2–dimensional hyperbolic space
Motivated by the above classical “singular” theory we introduce the notion of singular limit points within the hyperbolic space setup
Summary
The classical results of Diophantine approximation, in particular those from the onedimensional theory, have natural counterparts and extensions in the hyperbolic space setting. In this setting, instead of approximating real numbers by rationals, one approximates the limit points of a fixed Kleinian group G by points in the orbit (under the group) of a distinguished limit point y. In what follows G denotes a nonelementary, geometrically finite Kleinian group acting on the unit ball model (Bd+1, ρ) of (d + 1)–dimensional hyperbolic space with metric ρ derived from the differential dρ = 2|dx|/(1−|x|2). G is a discrete subgroup of Mob(Bd+1), the group of orientation-preserving Mobius
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