Sur les actions affines des groupes discrets. (On affine actions of discrete groups.). (French) Zbl 0865.57038

Zeghib, Abdelghani

Ann. Inst. Fourier 47, No. 2, 641-685 (1997).

Summary: One would hope that, for lattices in \(\text{SL}(n,{\mathbb R})\), \(n \geq 3\), differentiable, volume preserving actions on compact manifolds might be “classifiable”. However, we are far from realizing this goal, and so many authors have recently been considering actions of lattices in \(\text{SL}(n,{\mathbb R})\) on manifolds of relatively low dimension, precisely, of dimension \(\leq n\), and which, in addition, satisfy some extra dynamical or geometrical conditions. It has been shown, for example, that there is essentially no new action, other than the standard one of \(\text{SL}(n,{\mathbb Z})\) on the \(n\)-torus. Here we generalize this fact to connection preserving actions of lattices in \(\text{SL}(n,{\mathbb R})\) on manifolds of dimension \(n+1\).

MSC:

57S20	Noncompact Lie groups of transformations
53C05	Connections (general theory)
37-XX	Dynamical systems and ergodic theory

Keywords:

lattice; connection; affine; parallel; weakly stable; stable; geodesic foliation; rigidity

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