## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 659 - 811

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

Yuichiro Hoshi and Shinichi Mochizuki

#### 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 "*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.

#### Article information

**Dates**

Received: 31 March 2011

Revised: 28 December 2011

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1540417834

**Digital Object Identifier**

doi:10.2969/aspm/06310659

**Mathematical Reviews number (MathSciNet)**

MR3051259

**Zentralblatt MATH identifier**

1321.14018

**Subjects**

Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]

Secondary: 14H10: Families, moduli (algebraic)

**Keywords**

Anabelian geometry combinatorial anabelian geometry profinite Dehn twist semi-graph of anabelioids inertia group hyperbolic curve configurationa space

#### Citation

Hoshi, Yuichiro; Mochizuki, Shinichi. Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists. Galois–Teichmüller Theory and Arithmetic Geometry, 659--811, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310659. https://projecteuclid.org/euclid.aspm/1540417834