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<p>We prove that arboreal Galois extensions of number fields are never abelian for postcritically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of Andrews-Petsche. Together with a result of Ferraguti-Ostafe-Zannier, this result implies that counterexamples to the conjecture, if they exist, are sparse. We also prove an auxiliary result on places of periodic reduction for rational maps, which may be of independent interest.</p>

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Non-abelian arboreal Galois groups associated to PCF rational maps

  • 期刊论文

  • 发表日期：

    2025年5月30日

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  • Chifan Leung,
  • Clayton Petsche

摘要

We prove that arboreal Galois extensions of number fields are never abelian for postcritically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of Andrews-Petsche. Together with a result of Ferraguti-Ostafe-Zannier, this result implies that counterexamples to the conjecture, if they exist, are sparse. We also prove an auxiliary result on places of periodic reduction for rational maps, which may be of independent interest.