Sur les actions affines des groupes discrets
Annales de l'Institut Fourier, Volume 47 (1997) no. 2, p. 641-685
One would hope that, for lattices in SL (n,), n3, 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 SL (n,) on manifolds of relatively low dimension, precisely, of dimension 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 SL (n,) on the n-torus. Here we generalize this fact to connection preserving actions of lattices in SL (n,) on manifolds of dimension n+1.
On pourrait espérer “classifier” les actions différentiables en préservant le volume des réseaux de SL (n,) sur les variétés compactes. On en est cependant loin. Ainsi, plusieurs auteurs ont récemment étudié les actions des réseaux de SL (n,) sur des variétés de dimension relativement basse, précisément, n, et vérifiant en plus certaines conditions géométriques ou dynamiques. On montre alors qu’il s’agit essentiellement de l’action usuelle de SL (n,) sur un tore de dimension n. Ici, on généralise ce fait aux actions des réseaux de SL (n,) sur des variétés de dimension n+1, et qui préservent une connexion.
@article{AIF_1997__47_2_641_0,
     author = {Zeghib, Abdelghani},
     title = {Sur les actions affines des groupes discrets},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     pages = {641-685},
     doi = {10.5802/aif.1577},
     zbl = {0865.57038},
     mrnumber = {98d:57068},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1997__47_2_641_0}
}
Zeghib, Abdelghani. Sur les actions affines des groupes discrets. Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 641-685. doi : 10.5802/aif.1577. http://www.numdam.org/item/AIF_1997__47_2_641_0/

[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