Topics in the Grothendieck conjecture for hyperbolic polycurves of dimension 2

Abstract

In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension 2 over sub-p-adic fields. In 1-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-p-adic fields and the pro-p version of this conjecture. In 2-dimensional case, a naive analogue of this conjecture does not hold for hyperbolic polycurves over general sub-p-adic fields. Moreover, the Isom version of the pro-p Grothendieck conjecture does not hold in general. We explain these two phenomena and prove the Hom version of the Grothendieck conjecture for hyperbolic polycurves of dimension 2 under the assumption that the Grothendieck section conjecture holds for some hyperbolic curves.