Sur les actions affines des groupes discrets

Zeghib, Abdelghani

Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 641-685.

On pourrait espérer “classifier” les actions différentiables en préservant le volume des réseaux de $\mathrm{SL}(n,ℝ)$ sur les variétés compactes. On en est cependant loin. Ainsi, plusieurs auteurs ont récemment étudié les actions des réseaux de $\mathrm{SL}(n,ℝ)$ 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 $\mathrm{SL}(n,\mathbb{Z})$ sur un tore de dimension $n$. Ici, on généralise ce fait aux actions des réseaux de $\mathrm{SL}(n,ℝ)$ sur des variétés de dimension $n+1$, et qui préservent une connexion.

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

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

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

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

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

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

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

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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 | Zbl

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

[Zim2] R. Zimmer, Ergodic theory and semisimple Lie groups, Birkhäuser, Boston, 1984. | MR | Zbl

[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 | Zbl