The Grothendieck-Teichm\"uller group of $PSL(2, q)$

TL;DR

This paper characterizes the structure of the Grothendieck-Teichmüller group associated with the projective special linear group over finite fields, revealing its composition and implications for dessins d'enfant and moduli fields.

Contribution

The paper explicitly determines the structure of the Grothendieck-Teichmüller group for $PSL(2, q)$, showing it is a product of elementary abelian 2-groups and dihedral groups, and discusses its triviality when q is even.

Findings

$GT_1(PSL(2, q))$ is a product of an elementary abelian 2-group and dihedral groups of order 8.

When q is even, $GT_1(PSL(2, q))$ is trivial.

The moduli field of dessins d'enfant with monodromy group $PSL(2, q)$ has derived length less than 4.

Abstract

We show that the Grothendieck-Teichm\"uller group of $PSL(2, q)$, or more precisely the group $GT_1(PSL(2, q))$ as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, when $q$ is even, we show that it is trivial. We explain how it follows that the moduli field of any "dessin d'enfant" whose monodromy group is $PSL(2, q)$ has derived length less than 4. This paper can serve as an introduction to the general results on the Grothendieck-Teichm\"uller group of finite groups obtained by the author.