Partial actions of groups on profinite spaces

TL;DR

This paper investigates partial group actions on profinite spaces, establishing conditions under which orbit spaces and enveloping spaces are profinite, and explores the existence of continuous sections and categorical properties of such actions.

Contribution

It provides new results on the structure of partial actions on profinite spaces, including conditions for the profiniteness of orbit and enveloping spaces, and analyzes categorical relationships.

Findings

Orbit space $X/\sim_G$ is profinite for partial actions with closed domain.

Enveloping space $X_G$ is profinite when $G$ is profinite.

Conditions for the existence of continuous sections of the quotient map.

Abstract

We show that for a partial action $\eta$ with closed domain of a compact group $G$ on a profinite space $X$ the space of orbits $X/\!\sim_G$ is profinite, this leads to the fact that when $G$ is profinite the enveloping space $X_G$ is also profinite. Moreover, we provide conditions for the induced quotient map $\pi_G: X\to X/\!\sim_G$ of $\eta$ to have a continuous section. Relations between continuous sections of $\pi_G$ and continuous sections of the quotient map induced by the enveloping action of $\eta$ are also considered. At the end of this work we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.