The Semi-absolute Anabelian Geometry of Geometrically Pro-$p$ Arithmetic Fundamental Groups of Associated Low-Dimensional Configuration Spaces

Abstract

Let $p$ be a prime number. In the present paper, we study geometrically pro-$p$ arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a $p$-adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro-$p$ arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space.