Anabelian geometry and m-step reconstruction

The Grothendieck conjecture states that the geometric information of hyperbolic curves can be group-theoretically reconstructed from their arithmetic fundamental groups. Over a field of zero characteristic, this conjecture was proved by H. Nakamura, A. Tamagawa, and S. Mochizuki (~1999), and by A. Tamagawa, J. Stix, and S. Mochizuki over fields of positive characteristic (~2009). However, around this conjecture, many unsolved problems remain. One of them, the m-step solvable Grothendieck conjecture and the main topic of this talk, deals with the group-theoretical reconstruction of geometric information of hyperbolic curves from "near" abelian quotients of the fundamental groups -- more precisely from the maximal geometrically m-step solvable quotient of the arithmetic fundamental groups. In this talk, we will present this conjecture and a part of its proof as obtained by the speaker.