Journal ArticleDOI
Parabolic Fixed Points of Kleinian Groups and the Horospherical Foliation on Hyperbolic Manifolds
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In this paper, the authors used dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn) with constant negative curvature.Abstract:
We use dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn). Let ℋ be the horospherical foliation on the unit tangent bundle SM of manifold M = Γ\ℍn with constant negative curvature. We construct examples Γ ⊂ Iso(ℍ4) which show that horosphere based at parabolic fixed point w ∈ ∂ℍ4 can project to leaf ℋx ⊂ SM of complicated structure: it can be locally closed and not closed; not locally closed and non-dense in the non-wandering set Ω+ ⊂ SM of ℋ; dense in Ω+ (this is equivalent to w being a horospherical limit point). Using the natural duality, one gets the corresponding examples of Γ-orbits on the light cone. We give an elementary proof of the fact that conical limit point w ∈ ∂ℍn cannot be a parabolic fixed point.read more
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