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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 $\pi_1(X)$?

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The way to reconstruct the differential sheaf from the fundamental group is, of course,

$$\pi_1(X)\curvearrowright X\curvearrowright \omega_X$$

But I don't think you can get to the differential sheaf directly, without passing through the underlying curve at some point. At least not with Mochizuki's original result alone.

Perhaps $\omega_X$ can be carved out of $\pi_1(X)$ with the techniques of mono anabelian reconstruction (see Topics in absolute anabelian geometry III), but that's a different story.

Not sure if that helps.

I think the same should apply to any other piece of relevant information from $X$. Ultimately, only $\pi_1(X)$ is anabelian enough.

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    $\begingroup$ What does it mean to 'reconstruct the differential sheaf directly' if you don't know what $X$ is? I don't think this is what the OP is asking. (Although maybe 'anabelian reconstruction' is all the OP is looking for.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 1, 2017 at 13:29

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