In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of the Grothendick's Conjecture in Anabelian Curves is explained.

In the proof he shows that for two hyperbolic curves $X$ and $Y$, if there exist an 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 $\pi_1(X)$, can the hyperbolic curve $X$ be reconstructed? Specifically can the differential sheaf $\omega_X$ be reconstructed using only the algebraic fundamental group $\pi_1(X)$?

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

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 $\pi_1(X)$?

In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of the Grothendick's Conjecture in Anabelian Curves is explained. 

In the proof he shows that for two hyperbolic curves $X$ and $Y$, if there exist an 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 $\pi_1(X)$, can the hyperbolic curve $X$ be reconstructed? Specifically can the differential sheaf $\omega_X$ be reconstructed using only the algebraic fundamental group $\pi_1(X)$?