turkmath.org

Arboreal representations of absolute Galois groups of number fields are given by profinite groups of automorphisms of regular rooted trees, with the geometry of the tree determined by a polynomial which defines such a representation. Thus arboreal representations give rise to dynamical systems on a Cantor set, and allow to apply the methods of topological dynamics to study problems in number theory. In this talk we consider the conjecture of Boston and Jones, which states that the images of Frobenius elements under arboreal representations have a certain cycle structure. To study this conjecture, we borrow from the Lie group theory the concepts of maximal tori and Weyl groups, and introduce maximal tori and Weyl groups in the profinite setting. We then use this new technique to give a partial answer to the conjecture by Boston and Jones in the case when an arboreal representations is defined by a post-critically finite quadratic polynomial over a number field. Based on a joint work with Maria Isabel Cortez.