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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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