Settled elements in profinite groups

María Isabel Cortez, Olga Lukina

Published 2021-06-01Version 1

An automorphism of a rooted spherically homogeneous tree is settled if it satisfies certain conditions on the growth of cycles at finite levels of the tree. In this paper, we consider a conjecture by Boston and Jones that the image of an arboreal representation of the absolute Galois group of a number field in the automorphism group of a tree has a dense subset of settled elements. Inspired by analogous notions in theory of compact Lie groups, we introduce the concepts of a maximal torus and a Weyl group for actions of profinite groups on rooted trees, and we show that the Weyl group contains important information about settled elements. We study maximal tori and their Weyl groups in the images of arboreal representations associated to quadratic polynomials over algebraic number fields, and in branch groups.