  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.
