Abstract
The aim of this paper is to describe a geometrization of the Mostow-Margulis theory of rigidity and representations of discrete subgroups of semisimple groups. More precisely let H be a connected semisimple Lie group and Γ ⊂ Halattice subgroup. Let G be another Lie group. The general problem considered by Mostow and Margulis was to study the homomorphisms π: Γ → G. The Mostow rigidity theorem of course deals with the case in which G is semisimple and π(Γ) is a lattice in G, and the Margulis superrigidity theorem deals with the more general case in which π(Γ) is merely assumed to be Zariski dense in G. While we shall recall the precise results later, we simply remark here that the ultimate conclusion is that one can essentially understand all such homomorphisms. Roughly speaking, π either extends to a smooth homomorphism of H or π(Γ) has compact closure (in which case one also has information on the closure), or is a combination of these cases. A geometric generalization of the notion of a homomorphism Γ → G is of course the notion of an action of Γ by automorphisms of a principal G-bundle.
Research partially supported by NSF Grant DMS-8301882.
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Zimmer, R.J. (1987). Lattices in Semisimple Groups and Invariant Geometric Structures on Compact Manifolds. In: Howe, R. (eds) Discrete Groups in Geometry and Analysis. Progress in Mathematics, vol 67. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6664-3_6
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DOI: https://doi.org/10.1007/978-1-4899-6664-3_6
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