Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists

Abstract

Let $\Sigma$ be a nonempty set of prime numbers. In the present paper, we continue our study of the pro-$\Sigma$ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields of characteristic zero. Our first main result asserts, roughly speaking, that if an F-admissible automorphism [i.e., an automorphism that preserves the fiber subgroups that arise as kernels associated to the various natural projections of the configuration space under consideration to configuration spaces of lower dimension] of a configuration space group arises from an F-admissible automorphism of a configuration space group [arising from a configuration space] of strictly higher dimension, then it is necessarily FC-admissible, i.e., preserves the cuspidal inertia subgroups of the various subquotients corresponding to surface groups. After discussing various abstract profinite combinatorial technical tools involving semi-graphs of anabelioids of PSC-type that are motivated by the well-known classical theory of topological surfaces, we proceed to develop a theory of profinite Dehn twists, i.e., an abstract profinite combinatorial analogue of classical Dehn twists associated to cycles on topological surfaces. This theory of profinite Dehn twists leads naturally to comparison results between the abstract combinatorial machinery developed in the present paper and more classical scheme-theoretic constructions. In particular, we obtain a purely combinatorial description of the Galois action associated to a [scheme-theoretic!] degenerating family of hyperbolic curves over a complete equicharacteristic discrete valuation ring of characteristic zero. Finally, we apply the theory of profinite Dehn twists to prove a version of the Grothendieck Conjecture for—i.e., put another way, we compute the centralizer of the geometric monodromy associated to—the tautological curve over the moduli stack of pointed smooth curves.