Belyi injectivity for outer representations on certain quotients of étale fundamental groups of hyperbolic curves of genus zero

Abstract

In the present paper, we study certain quotients of the étale fundamental group of a hyperbolic curve over a field. We prove that the action of the outer automorphism group of a certain quotient of the étale fundamental group of a hyperbolic curve over an algebraically closed field on its conjugacy classes of open subgroups is faithful. Also, we prove that, if $k$ is either a number field or a $p$-adic local field, then the outer Galois representation associated to a certain quotient of the geometric fundamental group of ${\mathbb{P}}_k^1\backslash \{0,1,\infty \}$ is injective.