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April 1, 2026Open Access

Arboreal Galois Groups of Postcritically Finite Quadratic Polynomials: The Strictly Preperiodic Case

Key Points

The aim is to characterize arboreal Galois groups of postcritically finite quadratic polynomials when the critical point is strictly preperiodic.
Building on previous work to expand understanding of Galois groups
Analyzing the behavior of critical points in the context of quadratic polynomials
Examining cases where critical points are strictly preperiodic
Identified unique structural properties of arboreal Galois groups for strictly preperiodic cases
Demonstrated that the behavior of critical points significantly influences group characteristics
Completed the classification for all postcritically finite quadratic polynomials

Abstract

In BGJT25b, 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).

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