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.
Reviewer: H.Abel


22E40 Discrete subgroups of Lie groups
53C30 Differential geometry of homogeneous manifolds
37A99 Ergodic theory
57S20 Noncompact Lie groups of transformations


Zbl 0594.22005
Full Text: DOI EuDML