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Isometric actions of Lie subgroups of the Moebius group. (English) Zbl 1073.53016
The following theorem is considered: Let \(G\) be a non-compact connected Lie subgroup of \(\text{Isom}(\mathbb H^n)\) which does not fix any point at infinity (i.e. on \(\partial \mathbb H^n \simeq\mathbb S^{n-1})\). Then, up to conjugacy, \(G\) preserves a hyperbolic subspace \(\mathbb H^d\) with \(1\leq d\leq n\) and contains \(O^0(d,1)\).
The authors give a geometric – essentially dynamic – proof and an application to the holonomy of Lorentzian manifolds.

MSC:
53A30 Conformal differential geometry (MSC2010)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57S20 Noncompact Lie groups of transformations
53C30 Differential geometry of homogeneous manifolds
53C29 Issues of holonomy in differential geometry
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