# Characterizing slices for proper actions of locally compact groups

## TL;DR

This paper extends key results of proper group actions from compact to locally compact groups, providing new theorems on slices, openness of action maps, and an application to the Banach-Mazur compacta.

## Contribution

It generalizes important properties of slices and proper actions from compact to locally compact groups, including openness and continuity results, and offers a new proof related to Banach-Mazur compacta.

## Key findings

- Openness of the action map for small slices in proper G-spaces.
- Continuity and openness of the slicing map $f_S:G(S)\to G/H$.
- A new proof of the compactness of the Banach-Mazur compacta.

## Abstract

In his seminal work \cite{pal:61}, 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 $G\times S\to G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)\to G/H$ is continuos 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.