Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 41 min ago
PreprintarXiv licencearXiv

Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points

  • Özlem Ejder
  • Zofia Gołaska
  • Yasemin Kara
  • Leonie Nienhaus
  • Özge Ülkem
Open at sourcePDFDOICite

Abstract

We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function f(x)=2/((x−1)2) defined over a number field k, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of f and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values a ∈ k for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. We also determine the normalizer of the geometric IMG of f and show that the arithmetic IMG equals the normalizer if ζ8 is not contained in k. Finally, we show that when k=ℚ, the constant field contains ℚ(μ2∞), providing the first full study of a PCF quadratic map with non-abelian constant field.

Related

  1. Strongly Choquet-Deny groups and invariant strictly convex renormings
    Sheldon Dantas, Michal Doucha
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv
  2. Galois groups of Apéry-like series modulo primes
    Xavier Caruso, Florian Fürnsinn, Daniel Vargas Montoya +1
    Bulletin of the Australian Mathematical SocietyPeer-reviewedLicence not statedAlgebra & number theoryCombinatorics & discrete mathematicsHAL
  3. Infinite free pro-p products as absolute Galois groups
    Tamar Bar-On
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Galois coinvariants and fundamental groups of arithmetic schemes
    Benjamin Steklov
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Construction of Galois modules describing ideal class groups and applications to Iwasawa theory
    Takenori Kataoka
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{ejder2026arboreal,
  title = {{Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points}},
  author = {Özlem Ejder and Zofia Gołaska and Yasemin Kara and Leonie Nienhaus and Özge Ülkem},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.22466},
  eprint = {2605.22466},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.22466},
}
RIS
TY  - UNPB
TI  - Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points
AU  - Özlem Ejder
AU  - Zofia Gołaska
AU  - Yasemin Kara
AU  - Leonie Nienhaus
AU  - Özge Ülkem
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2605.22466
UR  - https://arxiv.org/abs/2605.22466
AB  - We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function f(x)=2/((x−1)^2) defined over a number field k, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of f and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values a ∈ k for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. We also determine the normalizer of the geometric IMG of f and show that the arithmetic IMG equals the normalizer if ζ_8 is not contained in k. Finally, we show that when k=ℚ, the constant field contains ℚ(μ_2^∞), providing the first full study of a PCF quadratic map with non-abelian constant field.
ER  -