On the cuspidalization problem for hyperbolic curves over finite fields

Abstract

Abstract. In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobeniuspreserving isomorphism between the geometrically pro-l fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-l fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.