Sur les actions affines des groupes discrets

Zeghib, Abdelghani

Annales de l'Institut Fourier, Tome 47 (1997), p. 641-685 / Harvested from Numdam

Résumé

On pourrait espérer “classifier” les actions différentiables en préservant le volume des réseaux de ${\rm SL}(n,{\Bbb R})$ sur les variétés compactes. On en est cependant loin. Ainsi, plusieurs auteurs ont récemment étudié les actions des réseaux de ${\rm SL}(n,{\Bbb R})$ sur des variétés de dimension relativement basse, précisément, $\le n$, et vérifiant en plus certaines conditions géométriques ou dynamiques. On montre alors qu’il s’agit essentiellement de l’action usuelle de ${\rm SL}(n,{\Bbb Z})$ sur un tore de dimension $n$. Ici, on généralise ce fait aux actions des réseaux de ${\rm SL}(n,{\Bbb R})$ sur des variétés de dimension $n+1$, et qui préservent une connexion.

Abstract

One would hope that, for lattices in ${\rm SL}(n, {\Bbb R})$, $n \ge 3$, 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 ${\rm SL}(n, {\Bbb R})$ on manifolds of relatively low dimension, precisely, of dimension $\le n$, 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 ${\rm SL}(n, {\Bbb Z})$ on the $n$-torus. Here we generalize this fact to connection preserving actions of lattices in ${\rm SL}(n, {\Bbb R})$ on manifolds of dimension $n+1$.

Détail

Publié le : 1997-01-01

MR 98d:57068 | Zbl 0865.57038 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1577

Bibliographie

[BL] Y. Benoist, F. Labourie, Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables, Invent. Math., 111 (1993), 285-308. | MR 94d:58114 | Zbl 0777.58029

[Car] Y. Carrière, Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., 95 (1989), 615-628. | MR 89m:53116 | Zbl 0682.53051

[Fer] R. Feres, Connections preserving actions of lattices in SL(n,R), Israel. J. Math., 2, 135 (1992), 1-21. | Zbl 0786.57016

[Fer2] R. Feres, Actions of discrete linear groups and Zimmer's conjecture, J. Diff. Geom., 42 (1995), 554-576. | MR 97a:22016 | Zbl 0858.22011

[Fri] D. Fried, Closed similarity manifolds, Comment. Math. Helv., 55 (1988), 555-565. | Zbl 0455.57005

[Goe] E. Goetze, Connection preserving actions of connected and discrete Lie groups, J. Diff. Geom., 40 (1994), 595-620. | MR 95m:57052 | Zbl 0846.57030

[God] C. Godbillon, Feuilletages, Études géométriques, Birkhäuser, 1991.

[Gro] M. Gromov, Rigid transformation groups, Géometrie différentielle, D. Bernard et Choquet-Bruhat. éd., Travaux en cours 33, Hermann, Paris, 1988. | MR 90d:58173 | Zbl 0652.53023

[Kob] S. Kobayashi, Transformation groups in differential geometry, Springer-Verlag, 1972. | MR 50 #8360 | Zbl 0246.53031

[Lew] J. Lewis, The algebraic hull of a measurable cocycle, preprint.

[Pes] Y.B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Uspekhi Mat. Nauk., 32, 4 (1997), 55-114; english transl. Russian Math. Surveys 32, 4 (1977), 55-112. | Zbl 0383.58011

[P-R] G. Prasad, M.S. Raghunathan, Cartan subgroups and lattices in semisimple groups, Ann. Math., 96 (1972), 296-317. | MR 46 #1965 | Zbl 0245.22013

[Sol] B. Solomon, On foliations of Rn+1 by minimal hypersurfaces, Comment. Math. Helvetici, 61 (1986), 67-83. | MR 87k:49048 | Zbl 0601.53025

[Thu] W. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1978.

[Zeg1] A. Zeghib, Feuilletages géodésiques appliqués, Math. Annalen, 298 (1994), 729-759. | MR 95f:53110 | Zbl 0794.53020

[Zeg2] A. Zeghib, Geodesic foliations in Lorentz 3-manifolds, preprint, 1994.

[Zeg3] A. Zeghib, Le groupe affine d'une variété riemannienne compacte, Comm. Ana. Geom., 5 (1997), 123-135. | MR 98g:53065 | Zbl 0912.53025

[Zim1] R. Zimmer, On connection-preserving actions of discrete linear groups, Erg. Th. Dynam. Systems, 6 (1986), 639-644. | MR 88g:57045 | Zbl 0623.22007

[Zim2] R. Zimmer, Ergodic theory and semisimple Lie groups, Birkhäuser, Boston, 1984. | MR 86j:22014 | Zbl 0571.58015

[Zim3] R. Zimmer, Lattices in semisimple groups and invariant geometric structures on compact manifolds, Discrete Groups in Geometry and Analysis, R. Howe, ed., Birkhäuser, Boston, 1987, 152-210. | MR 88i:22025 | Zbl 0663.22008