The Grothendieck--Teichm\"{u}ller group of~PSL(2,q)

Abstract

Abstract In this paper, we show that the Grothendieck–Teichmüller group of PSL ⁡ ( 2 , q ) {\operatorname{PSL}(2,q)} , or more precisely the group 𝒢 ⁢ 𝒯 1 ⁢ ( PSL ⁡ ( 2 , q ) ) {\mathcal{G\kern-0.569055ptT}_{\kern-1.707165pt1}(\operatorname{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 ) {\operatorname{PSL}(2,q)} has derived length ≤ 3 {\leq 3} . This paper can serve as an introduction to the general results on the Grothendieck–Teichmüller group of finite groups obtained by the author.