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Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IItripods and combinatorial cuspidalization /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IIby Yuichiro Hoshi, Shinichi Mochizuki.
Reminder of title:	tripods and combinatorial cuspidalization /
Author:	Hoshi, Yuichiro.
other author:	Mochizuki, Shinichi.
Published:	Singapore :Springer Nature Singapore :2022.
Description:	xxiii, 150 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Curves, Algebraic.
Online resource:	https://doi.org/10.1007/978-981-19-1096-8
ISBN:	9789811910968$q(electronic bk.)

Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IItripods and combinatorial cuspidalization /
Hoshi, Yuichiro.

Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IItripods and combinatorial cuspidalization /[electronic resource] :by Yuichiro Hoshi, Shinichi Mochizuki. - Singapore :Springer Nature Singapore :2022. - xxiii, 150 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v. 22991617-9692 ;. - Lecture notes in mathematics ;2035..

1. Combinatorial Anabelian Geometry in the Absence of Group-theoretic Cuspidality -- 2. Partial Combinatorial Cuspidalization for F-admissible Outomorphisms -- 3. Synchronization of Tripods -- 4. Glueability of Combinatorial Cuspidalizations. References.

The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of configuration spaces associated to hyperbolic curves over algebraically closed fields of characteristic zero. The starting point of the theory of the present monograph is a combinatorial anabelian result which allows one to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves, as well as to give purely group-theoretic characterizations of the cuspidal inertia subgroups of one-dimensional subquotients of the profinite fundamental group of a configuration space. We then turn to the study of tripod synchronization, i.e., of the phenomenon that an outer automorphism of the profinite fundamental group of a log configuration space associated to a stable log curve induces the same outer automorphism on certain subquotients of such a fundamental group determined by tripods [i.e., copies of the projective line minus three points]. The theory of tripod synchronization shows that such outer automorphisms exhibit somewhat different behavior from the behavior that occurs in the case of discrete fundamental groups and, moreover, may be applied to obtain various strong results concerning profinite Dehn multi-twists. In the final portion of the monograph, we develop a theory of localizability, on the dual graph of a stable log curve, for the condition that an outer automorphism of the profinite fundamental group of the stable log curve lift to an outer automorphism of the profinite fundamental group of a corresponding log configuration space. This localizability is combined with the theory of tripod synchronization to construct a purely combinatorial analogue of the natural outer surjection from the etale fundamental group of the moduli stack of hyperbolic curves over the field of rational numbers to the absolute Galois group of the field of rational numbers.

ISBN: 9789811910968$q(electronic bk.)

Standard No.: 10.1007/978-981-19-1096-8doiSubjects--Topical Terms:

190871
Curves, Algebraic.

LC Class. No.: QA565 / .H67 2022

Dewey Class. No.: 516.352

Topics surrounding the combinatorial anabelian geometry of hyperbolic curves IItripods and combinatorial cuspidalization /
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520 $a The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of configuration spaces associated to hyperbolic curves over algebraically closed fields of characteristic zero. The starting point of the theory of the present monograph is a combinatorial anabelian result which allows one to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves, as well as to give purely group-theoretic characterizations of the cuspidal inertia subgroups of one-dimensional subquotients of the profinite fundamental group of a configuration space. We then turn to the study of tripod synchronization, i.e., of the phenomenon that an outer automorphism of the profinite fundamental group of a log configuration space associated to a stable log curve induces the same outer automorphism on certain subquotients of such a fundamental group determined by tripods [i.e., copies of the projective line minus three points]. The theory of tripod synchronization shows that such outer automorphisms exhibit somewhat different behavior from the behavior that occurs in the case of discrete fundamental groups and, moreover, may be applied to obtain various strong results concerning profinite Dehn multi-twists. In the final portion of the monograph, we develop a theory of localizability, on the dual graph of a stable log curve, for the condition that an outer automorphism of the profinite fundamental group of the stable log curve lift to an outer automorphism of the profinite fundamental group of a corresponding log configuration space. This localizability is combined with the theory of tripod synchronization to construct a purely combinatorial analogue of the natural outer surjection from the etale fundamental group of the moduli stack of hyperbolic curves over the field of rational numbers to the absolute Galois group of the field of rational numbers.
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