Wednesday, September 19, 2012
Yuichiro Hoshi, Shinichi Mochizuki
Let Σ be a nonempty set of prime numbers. In the present paper, we continue our study of the pro- Σ fundamental groups of hyperbolic curves and their associated con guration spaces over algebraically closed elds of characteristic zero. Our first main result asserts, roughly speaking, that if an F-admissible automorphism [i.e., an automorphism that preserves the fi ber subgroups that arise as kernels associated to the various natural projections of the configuration space under consideration to con guration spaces of lower dimension] of a configuration space group arises from an F-admissible automorphism of a configuration space group [arising from a con figuration 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.
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Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists
ReplyDeleteby Yuichiro Hoshi and Shinichi Mochizuki
http://www.kurims.kyoto-u.ac.jp/~motizuki/Combinatorial%20Anabelian%20Topics%20I.pdf
After discussing various abstract pro nite 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 pro nite Dehn twists, i.e., an abstract pro nite combinatorial analogue of classical Dehn twists associated to cycles on topological surfaces. This theory of pro nite 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 pro nite Dehn twists to prove a “geometric 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.
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