Profinite geometric iterated monodromy groups of postcritically finite polynomials in degree 3

TL;DR

This paper investigates the structure of profinite geometric iterated monodromy groups associated with degree 3 postcritically finite polynomials, revealing their finite generation, dependence on ramification portraits, and branch and torsion properties.

Contribution

It establishes that these groups are finitely invariably generated and determined by the polynomial's ramification portrait, advancing understanding of their algebraic and dynamical features.

Findings

Groups are finitely invariably generated.

Groups are determined by the ramification portrait.

Groups are regular branch over the closure of their commutator subgroup.

Abstract

In this article, we study the properties of profinite geometric iterated monodromy groups associated to polynomials. Such groups can be seen as generic representations of absolute Galois groups of number fields into the automorphism group of a regular rooted tree. Our main result is that, for a degree 3 postcritically finite polynomial over a number field, where each finite postcritical point has at least one preimage outside the critical orbits, the associated profinite geometric iterated monodromy group is finitely invariably generated. Moreover, this group is determined by the isomorphism class of the ramification portrait of the polynomial, up to conjugation by an automorphism of the ternary rooted tree. We also study the group-theoretical properties of such groups, namely their branch and torsion properties. In particular, we show that such groups are regular branch over the closure of their commutator subgroup, and that they contain torsion elements of any order realizable in the ternary tree.