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**On discrete subgroups of Lie groups and elliptic geometric structures.**
*(English)*
Zbl 0624.22006

The author continues his investigation of actions of lattices in higher rank semisimple Lie groups on compact manifolds.

In more detail, let H be a connected semisimple Lie group with finite center. Suppose the \({\mathbb{R}}\)-rank of every simple factor of H is at least 2. Let \(\Gamma\) be a lattice subgroup of H and M a compact n- manifold with volume density. Let \(P\to M\) be a G-structure on M where G is a real algebraic group. The author has conjectured that if there is a smooth volume preserving action of \(\Gamma\) on M defining a homomorphism \(\Gamma\to Aut(P)\) then the action is of an “algebraic” nature, meaning that either a) there is a nontrivial Lie algebra homomorphism L(H)\(\to L(G)\); or b) there is a \(\Gamma\)-invariant Riemannian metric on M. He has proved this conjecture under the additional assumption that the G- structure is of finite type and the \(\Gamma\)-action is ergodic [Ergodic Theory Dyn. Syst. 5, 301-306 (1985; Zbl 0594.22005)]. In the paper under review he proves it under the assumption that \(\alpha)\) Aut(P) is a Lie group, and \(\beta)\) Aut(P) acts transitively on M.

In more detail, let H be a connected semisimple Lie group with finite center. Suppose the \({\mathbb{R}}\)-rank of every simple factor of H is at least 2. Let \(\Gamma\) be a lattice subgroup of H and M a compact n- manifold with volume density. Let \(P\to M\) be a G-structure on M where G is a real algebraic group. The author has conjectured that if there is a smooth volume preserving action of \(\Gamma\) on M defining a homomorphism \(\Gamma\to Aut(P)\) then the action is of an “algebraic” nature, meaning that either a) there is a nontrivial Lie algebra homomorphism L(H)\(\to L(G)\); or b) there is a \(\Gamma\)-invariant Riemannian metric on M. He has proved this conjecture under the additional assumption that the G- structure is of finite type and the \(\Gamma\)-action is ergodic [Ergodic Theory Dyn. Syst. 5, 301-306 (1985; Zbl 0594.22005)]. In the paper under review he proves it under the assumption that \(\alpha)\) Aut(P) is a Lie group, and \(\beta)\) Aut(P) acts transitively on M.

Reviewer: H.Abel

### MSC:

22E40 | Discrete subgroups of Lie groups |

53C30 | Differential geometry of homogeneous manifolds |

37A99 | Ergodic theory |

57S20 | Noncompact Lie groups of transformations |