Overview
- The theory sheds new light on the Grothendieck-Teichmüller group and arithmetic absolute Galois groups
- The theory developed in the monograph has important applications to the study of such groups
- The monograph only requires a knowledge of graphs, profinite groups, and basic logarithmic algebraic geometry
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2299)
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Table of contents (4 chapters)
Keywords
About this book
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 inducesthe 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 étale 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.
Authors and Affiliations
Bibliographic Information
Book Title: Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves II
Book Subtitle: Tripods and Combinatorial Cuspidalization
Authors: Yuichiro Hoshi, Shinichi Mochizuki
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-981-19-1096-8
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022
Softcover ISBN: 978-981-19-1095-1Published: 22 May 2022
eBook ISBN: 978-981-19-1096-8Published: 20 May 2022
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XXIII, 150
Number of Illustrations: 1 b/w illustrations
Topics: Number Theory, Algebraic Geometry