In the paper Curves and their Fundamental Groups 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 $\pi_1(X)$?