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Lattice actions on the circle. (Actions de réseaux sur le cercle.) (French) Zbl 0995.57006
A lattice $$\Gamma$$ in a connected semisimple Lie group $$G$$ is called irreducible if there does not exist a nontrivial locally direct decomposition $$G=G_1G_2$$ such that $$(G_1 \cap\Gamma) (G_2\cap \Gamma)$$ is of finite index in $$\Gamma$$. Suppose that the real rank of $$G$$ is $$\geq 2$$ and that $$G$$ has no simple factors isomorphic to $$\text{PSL} (2,\mathbb{R})$$. Let an action of an irreducible lattice $$\Gamma\subset G$$ by orientation-preserving diffeomorphisms of $$\mathbb{S}^1$$ be given. It is proved that the action is reduced to an action of a finite cyclic quotient group of $$\Gamma$$. This main result is actually a consequence of a general theorem concerning actions of an irreducible lattice $$\Gamma\subset G$$ by orientation-preserving homeomorphisms of $$\mathbb{S}^1$$, supposing that the real rank of $$G$$ is $$\geq 2$$. Such an action either preserves a probability measure on $$\mathbb{S}^1$$, or is conjugate (in a certain weak sense) to an action induced by a surjection $$G\to \text{PSL} (2,\mathbb{R})$$ and by the projective action of $$\text{PSL}(2,\mathbb{R})$$ on $$\mathbb{S}^1$$.The main part of the proof is the verification of this property for each non-compact simple group of real rank $$\geq 2$$ and for $$\text{PSL}(2,\mathbb{R}) \times\text{PSL}(2,\mathbb{R})$$.

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 22E40 Discrete subgroups of Lie groups 57S25 Groups acting on specific manifolds 57S30 Discontinuous groups of transformations
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