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Geometry And Topology

Ergodic theory, groups, and geometry by Zimmer R.J., Morris D.W.

By Zimmer R.J., Morris D.W.

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In the course of the proof, we assumed Λ is discrete in GL(n, R). In general, some standard algebraic methods are used to convert Λ to a discrete group. Filling a gap in the proof. 2(2) is incomplete, because we did not show that ρ is nontrivial. ) This is where we use the engaging hypothesis. Suppose ρ is trivial. Recall that this implies we have a section s : M → P with g · s(m) = s(gm) · c(g, m), so we have a G-equivariant map M → P/C. This means we have a G-invariant reduction of P to C, so there is a G-invariant H-equivariant map P → H/C.

1, 35–58. MR 0924701 (89c:22019) [8] R. J. Zimmer: Superrigidity, Ratner’s theorem, and fundamental groups, Israel J. Math. 74 (1991), no. 2-3, 199–207. MR 1135234 (93b:22019) LECTURE 8 Locally Homogeneous Spaces This lecture is not intended to be a survey of locally homogeneous spaces, but just an indication of how two techniques of ergodic theory can be applied to handle a number of particular directions in the subject. 8A. 1) Definition. Suppose M and X are spaces, and G is a pseudogroup of local homeomorphisms of X.

Then g is the space of rightinvariant vector fields on G, so V is the space of left-invariant vector fields on G. 4). • Conclusion (1) is obvious because Λ centralizes G and the connection is Λ-invariant (being lifted from a connection on M ). ” It follows from the fact that any connection defines a Riemannian metric on the frame bundle, and the fact that the isometry group of a Riemannian manifold is finite dimensional. • Conclusion (3) is immediate from the definition of V . • Conclusion (4) is not at all obvious; it is not even clear that there is a single nonzero vector field that commutes with G.

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