Arboreal Cantor actions

Olga Lukina

Published 2018-01-04Version 1

In a recent joint work with Hurder, the author introduced the asymptotic discriminant of an equicontinuous action of a discrete group on a Cantor set. 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 an equicontinuous group action on a Cantor set 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 on the automorphism groups of spherically homogeneous rooted trees. In this paper, given an arboreal representation satisfying certain mild conditions, we associate to it an equicontinuous action of a discrete group, and the asymptotic discriminant. We give examples of arboreal representations with stable asymptotic discriminant and examples with wild asymptotic discriminant.