Profinite iterated monodromy groups arising from quadratic polynomials

Abstract

We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. (When the base field is \C it is the closure of the finitely generated iterated monodromy group for the usual topology which is also often studied.) Among other things we prove that the conjugacy class and hence the isomorphism class of G depends only on the combinatorial type of the post-critical orbit of the polynomial. We represent a chosen instance of G by explicit recursively defined generators. The uniqueness up to conjugacy depends on a certain semirigidity property, which ensures that arbitrary conjugates of these generators under the automorphism group of the tree always generate a subgroup that is conjugate to G. We determine the Hausdorff dimension, the maximal abelian factor group, and the normalizer of G using further explicit generators. The description of the normalizer is then used to describe the arithmetic étale iterated monodromy group of the quadratic polynomial. The methods used are purely group theoretical and do not involve fundamental groups over \C at all.