On actions of abelian Cantor groups

TL;DR

This paper investigates actions of abelian Cantor groups on compact metric spaces, demonstrating that such groups can be assumed to be abelian in certain orbit space constructions for dimensions greater than one.

Contribution

It shows that in the context of orbit spaces of compact metric spaces, the acting Cantor group can be taken as abelian when the dimension exceeds one.

Findings

Abelian Cantor groups can be used in orbit space constructions for n>1.

The result extends previous work to include abelian groups.

Orbit spaces of certain dimensions can be realized with abelian group actions.

Abstract

By a Cantor group we mean a topological group homeomorphic to the Cantor set. The author earlier proved that every compact metric space of rational cohomological dimension n can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension n. In this paper, we consider actions of abelian Cantor groups and, in particular, we show that in the result mentioned above the Cantor group can be assumed to be abelian for n>1.