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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 \(\Gamma\).

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57S30 Discontinuous groups of transformations
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