Group-theoretic characterizations of almost open immersions of curves

Abstract

Let p be a prime number and k either a finite field of characteristic p or a generalized sub-p-adic field. Let X1 and X2 be hyperbolic curves over k. In the present paper, we introduce a kind of morphism between X1 and X2 called an almost open immersion, and give some group-theoretic characterizations for the set of almost open immersions between X1 and X2 via their arithmetic fundamental groups. This result generalizes the Isom-version of Grothendieck's anabelian conjecture for curves over k which has been proven by S. Mochizuki and A. Tamagawa to the case of almost open immersions.