## Parabolic fixed points of Kleinian groups and the horospherical foliation on hyperbolic manifolds.(English)Zbl 0874.57027

From the paper: “We use a dynamical approach to study parabolic fixed points for Kleinian groups $$\Gamma\subset\text{Iso}(\mathbb{H}^n)$$. Let $$\mathcal H$$ be the horospherical foliation on the unit tangent bundle $$SM$$ of a manifold $$M=\Gamma/\mathbb{H}^n$$ of constant negative curvature. We construct examples $$\Gamma\subset \text{Iso}(\mathbb{H}^4)$$ which show that a horosphere based at a parabolic fixed point $$w\in \partial\mathbb{H}^4$$ can project to a leave $${\mathcal H}_x\subset SM$$ of complicated structure: it can be locally closed and not closed; not locally closed and non-dense in the non-wandering set $$\Omega^+\subset SM$$ of $$\mathcal H$$; dense in $$\Omega^+$$ (this is equivalent to $$w$$ being a horospherical limit point). Using the natural duality, one gets the corresponding examples of $$\Gamma$$-orbits on the light cone. We give an elementary proof of the fact that conical limit points $$w\in \partial\mathbb{H}^4$$ cannot be parabolic fixed points”.

### MSC:

 57R30 Foliations in differential topology; geometric theory 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)

### Keywords:

Kleinian groups; horospherical foliation
Full Text: