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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 \(\text{PSL}(2,\mathbb{R}) \times\text{PSL}(2,\mathbb{R})\).

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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