Semiconjugacy and Self-Similar Subgroups of pfIMGs

TL;DR

This paper explores the structure of profinite iterated monodromy groups (pfIMGs), classifies their self-similar subgroups via semiconjugacies, and examines implications for polynomial dynamics and group properties.

Contribution

It introduces a classification of self-similar subgroups of pfIMGs through semiconjugacies and constructs self-similar closures, linking group properties to polynomial dynamics.

Findings

Proper open self-similar subgroups correspond to rigid semiconjugacies.

Only twisted Chebyshev maps arise for polynomials.

pfIMGs with certain subgroup properties are characterized or constructed from semiconjugacies.

Abstract

The profinite iterated monodromy group (pfIMG) is a self-similar group associated to dynamical systems. We show that its proper open self-similar subgroups correspond to highly rigid semiconjugacies, which we partly classify in general. For polynomials, we show that only the twisted Chebyshev maps can arise. Next, we define and construct self-similar closures of subgroups of pfIMGs, and show that this preserves many group-theoretic properties of the original subgroup. As a consequence, we conclude that pfIMGs with open subgroups satisfying certain properties (e.g. prosolvable or pronilpotent) either satisfy that property themselves, or arise from one of these exceptional semiconjugacies. This is applied to answer some questions posed in [BGJT25] about open Frattini subgroups of pfIMGs: unicritical polynomials of composite degree do not have an open Frattini subgroup, and a polynomial with an open Frattini subgroup is often pro-\(p\).