TITLE: Is $\mathcal M _{g,n}$ anabelian?
QUESTION [7 upvotes]: Are the moduli spaces $\mathcal M _{g,n}$ expected to be anabelian? Is there anything known in that direction? 
Thank you very much for your help in any case!

REPLY [10 votes]: Grothendieck expected the moduli spaces $\mathcal{M}_{g,n}$ over $\mathbb{Q}$ to be the basic examples of anabelian varieties (besides hyperbolic curves, which was proved by Mochizuki, even over number fields, or more generally over sub-$p$-adic fields). The first non-trivial case is $(g,n)=(0,4)$, where we have $\mathcal{M}_{0,4}=\mathbb{P}^1_{\mathbb{Q}}\setminus \{0,1,\infty\}$. The only other moduli space of dimension $1$ is $\mathcal{M}_{1,1}$, the moduli space of elliptic curves. The geometry of these spaces has been described explicitly by cutting them into cells, enumerated by "fatgraphs" (equivalent to dessins d'enfants), and they are conjectured to be anabelian (I don't know further results, but I am not in this field. Certainly the question is very ambitious).