Anabelian properties of infinite algebraic extensions of finite fields

The Grothendieck conjecture for hyperbolic curves over finite fields was solved affirmatively by Tamagawa and Mochizuki. On the other hand, (a ``weak version'' of) the Grothendieck conjecture for some hyperbolic curves over algebraic closures of finite fields is also known by Tamagawa and Sarashina. So, it is natural to consider anabelian geometry over (infinite) algebraic extensions of finite fields. In the present paper, we give certain generalizations of the above results of Tamagawa and Sarashina to hyperbolic curves over these fields. Moreover, we give a necessary and sufficient condition for algebraic extensions of finite fields to be (torally) Kummer-faithful in terms of their absolute Galois groups.