Abstract
We study the postcritically finite non-polynomial map \(f(x)=\frac{1}{(x-1)^2}\) over a number field k and prove various results about the geometric \(G^{\textrm{geom}}(f)\) and arithmetic \(G^{\textrm{arith}}(f)\) iterated monodromy groups of f. We show that the elements of \(G^{\textrm{geom}}(f)\) are the ones in \(G^{\textrm{arith}}(f)\) that fix certain roots of unity by assuming a conjecture on the size of \(G^{\textrm{geom}}_n(f)\). Furthermore, we describe exactly for which \(a \in k\) the Arboreal Galois group \(G_a(f)\) and \(G^{\textrm{arith}}(f)\) are equal.
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We extend our gratitude to the referees for their invaluable feedback and corrections, particularly in relation to the suggestions concerning the structure and presentation of the introduction.
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Ö. Ejder was supported by part by Tübitak and Marie Skłodowska-Curie actions grant 120C071.
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Ejder, Ö., Kara, Y. & Ozman, E. Iterated monodromy group of a PCF quadratic non-polynomial map. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01549-z
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DOI: https://doi.org/10.1007/s00229-024-01549-z