Arboreal Cantor actions

TL;DR

This paper investigates minimal equicontinuous actions of discrete groups on Cantor sets derived from arboreal representations of absolute Galois groups, focusing on the asymptotic discriminant as an invariant.

Contribution

It introduces a method to associate discrete group actions to arboreal representations and analyzes their asymptotic discriminants, including stable and wild cases.

Findings

Established a link between arboreal Galois representations and Cantor set actions.

Identified conditions for stable and wild asymptotic discriminants.

Provided examples illustrating different discriminant behaviors.

Abstract

In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.