Galois groups and Cantor actions

TL;DR

This paper investigates the actions of Galois groups on Cantor sets through profinite iterated monodromy groups, providing a complete classification of their asymptotic discriminants in quadratic cases and exploring how wild discriminants occur.

Contribution

It offers the first comprehensive classification of asymptotic discriminants for quadratic polynomial-generated Galois actions and analyzes mechanisms leading to wild discriminants.

Findings

Complete classification of stable vs. wild asymptotic discriminants for quadratic cases

Identification of conditions leading to wild asymptotic discriminants

Insights into the structure of profinite iterated monodromy groups

Abstract

In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymptotic discriminant can arise.