# zbMATH — the first resource for mathematics

Geometrically finite Kleinian groups: The completeness of Nayatani’s metric. (English. Abridged French version) Zbl 0895.30030
A Kleinian group $$\Gamma$$ is a group of Möbius transformations acting on the $$n$$-sphere $$S^n$$, with non-empty set of discontinuity $$\Omega(\Gamma) \subset s^n$$. Using Patterson-Sullivan measures, Nayatani constructed a metric $$g$$ on $$\Omega(\Gamma)$$ invariant under $$\Gamma$$ and conformal to the standard metric on $$S^n$$. If $$\Gamma$$ acts freely on $$\Omega(\Gamma)$$ and $$M = \Omega(\Gamma)/ \Gamma$$ is compact the metric $$g$$ is automatically complete on $$\Omega(\Gamma)$$. The main result of the present paper says that, for a torsion-free geometrically finite Kleinian group $$\Gamma$$, the Nayatani metric $$g$$ is complete on $$\Omega(\Gamma)$$ if and only if the limit set of $$\Gamma$$ does not contain any parabolic fixed points of $$\text{rank }< k< \delta(\Gamma)$$ where $$\delta(\Gamma)$$ denotes the critical exponent of $$\Gamma$$.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57S30 Discontinuous groups of transformations
##### Keywords:
Möbius transformations; Kleinian group; Nayatani metric
Full Text: