Arithmetic monodromy groups of 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 1 or 2 2 . Prior to this article, a result of this kind was only known for quadratic maps (Pink) and a few examples in degree 3 3 .