On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let X•=(X,DX)$X^{\bullet }=(X, D_{X})$ be a pointed stable curve over an algebraically closed field of characteristic p>0$p>0$ and ΠX•$\Pi _{X^{\bullet }}$ the admissible fundamental group of X•$X^{\bullet }$. We prove that there exists a group‐theoretical algorithm, whose input datum is the admissible fundamental group ΠX•$\Pi _{X^{\bullet }}$, and whose output data are the topological and the combinatorial structures associated with X•$X^{\bullet }$. This result can be regarded as a mono‐anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.