Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields

Abstract

In the present paper, we study the cuspidalization problem for fundamental groups of configuration spaces of proper hyperbolic curves over finite fields. The goal of this paper is to show that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves induces an isomorphism between the geometrically pro-$l$ fundamental groups of the associated configuration spaces.