In the paper [Curves and their Fundamental Groups][1] written by Gerd Faltings, is exposed the Mochizuki's proof about the Grothendick's Conjecture in Anabelian Curves. In it proof he show that for for two hyperbolic curves $X$ and $Y$, if there exist a isomorphism between the algebraic fundamental groups $\pi_1(X)$ and $\pi_1(Y)$ then this curves are isomorphic. My question is: With only the fundamental group $\pi_1(X)$, the hyperbolic curve $X$ can be reconstructed?. Specifically the differential sheaf $\omega_X$ can be reconstructed using only the algebraic fundamental group $\pi_1(X)$?

  [1]: http://www.numdam.org/article/SB_1997-1998__40__131_0.pdf