Let Mg,[n]β, for 2gβ2+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 {\cal M}^\l of Mg,[n]β, defined over a sub-p-adic field 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({\cal M}^\l_{\ol{k}}) be the
geometric algebraic fundamental group of {\cal M}^\l and let
{Out}^*(\pi_1({\cal M}^\l_{\ol{k}})) be the group of its exterior
automorphisms which preserve the conjugacy classes of elements corresponding to
simple loops around the Deligne-Mumford boundary of {\cal M}^\l (this is the
"β-condition" motivating the "almost" above). Let us denote by
{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})) the subgroup consisting of
elements which commute with the natural action of the absolute Galois group
Gkβ of k. Let us assume, moreover, that the generic point of the D-M stack
{\cal M}^\l has a trivial automorphisms group. Then, there is a natural
isomorphism: {Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal
M}^\l_{\ol{k}})). This partially extends to moduli spaces of curves the
anabelian properties proved by Mochizuki for hyperbolic curves over
sub-p-adic fields.Comment: This paper has been withdrawn because of a flaw in the paper
"Profinite Teichm\"uller theory" of the first author, on which this paper
built o