Universal properties of group actions on locally compact spaces

TL;DR

This paper investigates the universal properties of locally compact G-spaces for countable infinite groups G, focusing on their minimal and free actions, and demonstrates that every such group admits a free minimal action on a locally compact non-compact Cantor set.

Contribution

It introduces a framework for understanding universal properties of G-spaces and proves that all countable infinite groups have free minimal actions on a locally compact non-compact Cantor set.

Findings

Every countable infinite group admits a free minimal action on the locally compact non-compact Cantor set.

Analysis of open invariant subsets of the eta-compactification of G.

Characterization of minimal closed invariant subspaces as locally compact free G-spaces.

Abstract

We study universal properties of locally compact G-spaces for countable infinite groups G. In particular we consider open invariant subsets of the \beta-compactification of G (which is a G-space in a natural way), and their minimal closed invariant subspaces. These are locally compact free G-spaces, and the latter are also minimal. We examine the properies of these G-spaces with emphasis on their universal properties. As an example of our resuts, we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non-compact Cantor set.