Arboreal Galois groups of postcritically finite quadratic polynomials: The strictly preperiodic case

TL;DR

This paper extends the understanding of arboreal Galois groups for postcritically finite quadratic polynomials to include cases where the critical point is strictly preperiodic, completing previous classifications.

Contribution

It provides a comprehensive description of arboreal Galois groups for all postcritically finite quadratic polynomials, including the strictly preperiodic critical point case.

Findings

Explicit description of arboreal Galois groups for strictly preperiodic cases

Completes the classification for all postcritically finite quadratic polynomials

Builds on previous work for periodic critical points

Abstract

In a previous paper, we provided an explicit description of the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 +c$ in the special case when the critical point $0$ is periodic under the action of $f(z)$. In the current paper, we complete the picture for all postcritically finite polynomials by addressing the cases when $0$ is strictly preperiodic for the polynomial $f(z)$.