Characterizing slices for proper actions of locally compact groups

In his seminal work [13], R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if...

Full description

Saved in:
Bibliographic Details
Published in:Topology and its applications 2018-04, Vol.239, p.152-159
Main Author: Antonyan, Sergey A.
Format: Article
Language:English
Subjects:
Global slice
John ellipsoid
Orbit space
Proper action
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In his seminal work [13], R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if H is a compact subgroup of a locally compact group G and S is a small (in the sense of Palais) H-slice in a proper G-space, then the action map G×S→G(S) is open. This is applied to prove that the slicing map fS:G(S)→G/H is continuous and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach–Mazur compacta.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2018.02.026