Arithmetic Monodromy Groups of Dynamical Belyi maps

TL;DR

This paper investigates the structure of arithmetic monodromy groups associated with a broad class of dynamical Belyi maps, revealing that their quotient by geometric monodromy groups is always of order 1 or 2, extending previous results beyond quadratic cases.

Contribution

It proves that for a large family of dynamical Belyi maps, the quotient of the arithmetic monodromy group by the geometric monodromy group has order 1 or 2, generalizing earlier specific cases.

Findings

The quotient of arithmetic and geometric monodromy groups is of order 1 or 2.

Extension of known results from quadratic maps to higher degrees.

Provides new insights into the structure of monodromy groups for dynamical Belyi maps.

Abstract

We consider a large family of dynamical Belyi maps of arbitrary degree and study the arithmetic monodromy groups attached to the iterates of such maps. Building on the results of Bouw-Ejder-Karemaker on the geometric monodromy groups of these maps, we show that the quotient of the arithmetic monodromy group by the geometric monodromy group has order either $1$ or $2$. Prior to this article, a result of this kind was only known for quadratic maps (Pink) and a few examples in degree $3$.