Rigid Geometric Structures and Representations of Fundamental Groups
Let G be a simple Lie group, ℝ-rank(G) ≥ 2, and F < G a lattice. Assume that Γ acts analytically and ergodically on a compact manifold M preserving a volume and an analytic rigid geometric structure. In , we establish that either the Γ-action is isometric and π1(M) is finite or π1(M) admits a “large image” linear representation. We discuss the proof of this result. We also present related results which use similar techniques to show that under slightly stronger hypotheses the Γ-action is a 0-entropy extension of a standard arithmetic example. We give one new result in which this extension can be shown to be continuous rather than measurable.
KeywordsEntropy Manifold Hull Univer
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- 2.Benoist, Y., Orbites des structures rigides (d’après M. Gromov), Integrable systems and foliations, Progr. Math. 145, Birkhäuser, Boston, 1997.Google Scholar
- 3.Feres, R., Rigid geometric structures and actions of semisimple Lie groups, Proceedings of Strasbourg Conference, (P. Foulon, ed.), to appear.Google Scholar
- 4.Fisher, D., On the arithmetic structure of lattice actions on compact spaces, preprint.Google Scholar
- 5.Fisher, D., Whyte, K., Continuous quotients for lattice actions on compact spaces, Geom. Dedicata, to appear.Google Scholar
- 6.Fisher, D., Zimmer, R. J., Geometric lattice actions, entropy and fundamental groups, preprint.Google Scholar
- 8.Gromov, M., Rigid transformation groups, Géométrie Différentielle (D. Bernard and Y. Choquet-Bruhat, eds.) Hermann, Paris 1988.Google Scholar
- 9.Lubotzky, A., Zimmer, R. J., Arithmetic structure of fundamental groups and actions of semisimple groups, Topology, to appear.Google Scholar
- 12.Shah, N., Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie Groups and Ergodic Theory, Tata Inst. Fund. Res., Bombay, 1998, 220-271.Google Scholar
- 16.Zimmer, R. J., Automorphism groups and fundamental groups of geometric manifolds, Differential Geometry: Riemannian Geometry (Los Angeles, CA 1990), Amer. Math. Soc, Providence, RI (1993), 693-710.Google Scholar
- 17.Zimmer, R. J., Entropy and arithmetic quotients for simple automorphism groups of compact manifolds, Geom. Dedicata, to appear.Google Scholar