# alpha brooks Book Archive

Geometry And Topology

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

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

Similar geometry and topology books

Zeta Functions, Topology and Quantum Physics

This quantity makes a speciality of numerous elements of zeta services: a number of zeta values, Ohno’s kinfolk, the Riemann speculation, L-functions, polylogarithms, and their interaction with different disciplines. 11 articles on fresh advances are written by means of extraordinary specialists within the above-mentioned fields. each one article starts off with an introductory survey resulting in the intriguing new learn advancements finished via the participants.

Extra resources for Ergodic theory, groups, and geometry

Example text

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.