×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI