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”.


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