Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves II: Tripods and Combinatorial Cuspidalization

This monograph studies the profinite fundamental groups of configuration spaces associated to hyperbolic curves over algebraically closed fields of characteristic zero using combinatorial anabelian geometry techniques. It reduces issues to hyperbolic curve anabelian geometry and provides group-theoretic characterizations of cuspidal inertia subgroups. It also explores tripod synchronization, which shows different behavior from discrete fundamental groups and can be applied to obtain strong results on profinite Dehn multi-twists. Finally, it develops a theory of localizability for the condition that an outer automorphism of the profinite fundamental group of a stable log curve lifts to an outer automorphism of a corresponding log configuration space.

Format: Paperback / softback
Length: 150 pages
Publication date: 22 May 2022
Publisher: Springer Verlag, Singapore

The present monograph delves deeper into the study of profinite fundamental groups associated to hyperbolic curves over algebraically closed fields of characteristic zero through the techniques of combinatorial anabelian geometry. Building upon a combinatorial anabelian result, it reduces issues related to the anabelian geometry of configuration spaces to those concerning hyperbolic curves and provides purely group-theoretic characterizations of cuspidal inertia subgroups in one-dimensional subquotients of these fundamental groups.

We then explore the fascinating phenomenon of tripod synchronization, which occurs when 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 determined by tripods. This theory reveals that these outer automorphisms exhibit distinct behavior compared to discrete fundamental groups and holds the potential to yield various strong results concerning profinite Dehn multi-twists.

In the final chapter, we develop a theory of localizability on the dual graph of a stable log curve, where an outer automorphism of the profinite fundamental group of the stable log curve lifts 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 group-theoretic framework for understanding the behavior of profinite fundamental groups in the context of hyperbolic curves and their associated configuration spaces.

Through the exploration of these topics, the present monograph contributes significantly to the understanding of profinite groups and their applications in mathematics.

Weight: 284g
Dimension: 156 x 233 x 13 (mm)
ISBN-13: 9789811910951
Edition number: 1st ed. 2022