# Topics in the Grothendieck conjecture for hyperbolic polycurves of dimension 2

## TL;DR

This paper investigates the Grothendieck conjecture for hyperbolic polycurves of dimension 2 over sub-p-adic fields, revealing limitations of naive analogues and proving the Hom version under certain assumptions.

## Contribution

It demonstrates the validity of the Hom version of the Grothendieck conjecture for hyperbolic polycurves of dimension 2 assuming the section conjecture for hyperbolic curves.

## Key findings

- Naive analogues of the conjecture fail in 2D cases.
- The Isom version of the pro-p Grothendieck conjecture generally does not hold.
- The Hom version is proven under the assumption of the section conjecture.

## 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.