In the paper [Curves and their Fundamental Groups][1] written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.

In the proof, he shows that for two hyperbolic curves $X$ and $Y$, if there exists an isomorphism between the algebraic fundamental groups $\pi_1(X)$ and $\pi_1(Y)$ then these curves are isomorphic. 

My question is: With only the fundamental group $\pi_1(X)$, can the hyperbolic curve $X$ be reconstructed? Specifically can the differential sheaf $\omega_X$ be reconstructed using only the algebraic fundamental group $\pi_1(X)$?

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