Settled elements in profinite groups

TL;DR

This paper proves a conjecture about the density of settled elements in the arboreal Galois representations of quadratic polynomials, introducing a new dynamical method involving maximal tori and Weyl groups.

Contribution

It establishes the conjecture for quadratic polynomials with certain pre-periodic orbits and introduces a novel approach using concepts from Lie group theory.

Findings

Proves the conjecture for quadratic polynomials with a length 2 pre-periodic orbit.

Provides evidence for the conjecture in cases with longer pre-periodic orbits.

Develops a new dynamical method based on maximal tori and Weyl groups.

Abstract

Given a polynomial of degree d over a number field, the image of the associated arboreal representation of the absolute Galois group of the field is a profinite group acting on the d-ary tree. Boston and Jones conjectured that for a quadratic polynomial, the image of such a representation contains a dense set of settled elements. Here an element is settled if it exhibits a certain pattern of growth of cycles at finite levels of the tree. In this paper, we prove the conjecture of Boston and Jones generically in the case when the quadratic polynomial has a strictly pre-periodic post-critical orbit of length 2, and provide new evidence that the conjecture holds for quadratic polynomials with strictly pre-periodic post-critical orbits of length at least 3. To prove our results, we introduce a new dynamical method, which uses the notions of a maximal torus and its Weyl group. These notions are analogous to the notions of maximal tori and Weyl groups in the theory of compact Lie groups, where they are fundamental. For profinite groups, maximal tori and Weyl groups contain the information about settled elements, and this is the foundation of our method.