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Local product structure for group actions

Published online by Cambridge University Press:  19 September 2008

Arlan Ramsay
Arlan Ramsay
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado, 80309-0426, USA
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Abstract

A differentiable G-space is introduced, for a Lie group G, into which every countably separated Borel G-space can be imbedded. The imbedding can be a continuous map if the space is a separable metric space. Such a G-space is called a universal G-space. This universal G-space has a local product structure for the action of G. That structure is inherited by invariant subspaces, giving a local product structure on general G-spaces. This information is used to prove that G-spaces are stratified by the subsets consisting of points whose orbits have the same dimension, to prove that G-spaces with stabilizers of constant dimension are foliated, to give a short proof that closed subgroups of Lie groups are Lie groups, to give a new proof and a stronger version of the Ambrose—Kakutani Theorem and to give a new proof of the existence of near-slices at points having compact stabilizers and hence of the existence of slices for Cartan G-spaces.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 1 , March 1991 , pp. 209 - 217
Copyright
Copyright © Cambridge University Press 1991

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References

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