Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves

Let ${\mathcal{M}}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected 'etale cover ${\mathcal{M}}^{\text{l}}$ of ${\cal M}_{g,[n]}$,definedoverasub-$p$-adicfield$k$, is “almost” anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let ${\pi }_1({\mathcal{M}}_{\text{ol}k}^{\text{l}})$ be the geometric algebraic fundamental group of ${\mathcal{M}}^{\text{l}}$ and let ${Out}^{\ast }({\pi }_1({\mathcal{M}}_{\text{ol}k}^{\text{l}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${\mathcal{M}}^{\text{l}}$ (this is the “$\ast$-condition” motivating the “almost” above). Let us denote by ${Out}_{G_k}^{\ast }({\pi }_1({\mathcal{M}}_{\text{ol}k}^{\text{l}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${\mathcal{M}}^{\text{l}}$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({\mathcal{M}}^{\text{l}})\cong {Out}_{G_k}^{\ast }({\pi }_1({\mathcal{M}}_{\text{ol}k}^{\text{l}}))\mathrm{.}$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.