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Lattices in semisimple groups and invariant geometric structures on compact manifolds. (English) Zbl 0663.22008
Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 152-210 (1987).
[For the entire collection see Zbl 0632.00015.]
The author describes a geometrization of the Mostow-Margulis theory of rigidity and representations of discrete subgroups of semisimple groups. Namely, let H be a connected semisimple Lie group and \(\Gamma\) \(\subset H\) a lattice subgroup, let G be another Lie group, and \(\pi\) : \(\Gamma\) \(\to G\) a homomorphism. Then the general Mostow-Margulis theory studies such homomorphisms when G is semisimple and \(\pi\) (\(\Gamma)\) is a lattice in G, or when \(\pi\) (\(\Gamma)\) is assumed to be Zariski dense in G. A geometric generalization of the notion of homomorphism is the notion of an action of \(\Gamma\) by automorphisms of a principal G-bundle. Thus by the geometrization of the Mostow-Margulis theory the author means the following general:
Problem 1. With H,\(\Gamma\),G as above, understand the actions of \(\Gamma\) by automorphisms of a principal G-bundle \(P\to M\), where M is some (say compact) \(\Gamma\)-space. In special cases this problem reduces to: Problem 2. Understand the homomorphisms \(\Gamma\) \(\to Diff(M)\) where M is a compact manifold. Problem 3. Understand the actions of \(\Gamma\) on a compact manifold that preserve a G-structure. Problem 4. Can every smooth volume preserving action of \(\Gamma\) on a compact manifold be described in algebraic terms, or do there exist genuinely (differential) topological actions ? The author describes known results concerning these problems, and states some conjectures.
Reviewer: V.Mavenich

22E40 Discrete subgroups of Lie groups
53C10 \(G\)-structures