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arXiv · 2308.14867

Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map

Ozlem Ejder·Yasemin Kara·Ekin Ozman
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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 $G^{\text{geom}}(f)$ and arithmetic $G^{\text{arith}}(f)$ iterated monodromy groups of $f$. We show that the elements of $G^{\text{geom}}(f)$ are the ones in $G^{\text{arith}}(f)$ that are fixing the roots of unity by assuming a conjecture on the size of $G^{\text{geom}}_n(f)$. Furthermore, we describe exactly for which $a \in k$ the Arboreal Galois group $G_a(f)$ and $G^{\text{arith}}(f)$ are equal.

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Ozlem Ejder, Yasemin Kara, Ekin Ozman. 2023-08-28. Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map. https://arxiv.org/abs/2308.14867

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