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Braids, conformal module, entropy, and Gromov's Oka principle

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Braids, conformal module, entropy, and Gromov's Oka principle/ by Burglind Jöricke.
Author:	Jöricke, Burglind.
Published:	Cham :Springer Nature Switzerland : : 2025.,
Description:	xv, 428 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
Subject:	Conformal invariants. -
Online resource:	https://doi.org/10.1007/978-3-031-67288-0
ISBN:	9783031672880

Braids, conformal module, entropy, and Gromov's Oka principle
Jöricke, Burglind.

Braids, conformal module, entropy, and Gromov's Oka principle[electronic resource] /by Burglind Jöricke. - Cham :Springer Nature Switzerland :2025. - xv, 428 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v. 23601617-9692 ;. - Lecture notes in mathematics ;1943..

1. Introduction -- 2. Riemann Surfaces, Braids, Mapping Classes, and Teichmueller Theory -- 3. The entropy of surface homeomorphisms -- 4. Conformal invariants of homotopy classes of curves. The Main theorem -- 5. Reducible pure braids. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem -- 6. The general case. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem -- 7. The conformal module and holomorphic families of polynomials -- 8. Gromov's Oka Principle and conformal module -- 9. Gromov's Oka Principle for (g, m)-fiber bundles -- 10. Fundamental groups and bounds for the extremal length -- 11. Counting functions -- 12. Riemann surfaces of second kind and finiteness theorems -- A. Several complex variables -- B. A Lemma on Conjugation -- C. Koebe's Theorem -- Index -- References.

This book studies the relation between conformal invariants and dynamical invariants and their applications, taking the reader on an excursion through a wide range of topics. The conformal invariants, called here the conformal modules of conjugacy classes of elements of the fundamental group, were proposed by Gromov in the case of the twice punctured complex plane. They provide obstructions to Gromov's Oka Principle. The invariants of the space of monic polynomials of degree n appeared earlier in relation to Hilbert's 13th Problem, and are called the conformal modules of conjugacy classes of braids. Interestingly, the conformal module of a conjugacy class of braids is inversely proportional to a popular dynamical invariant, the entropy, which was studied in connection with Thurston's celebrated theory of surface homeomorphisms. This result, proved here for the first time, is another instance of the numerous manifestations of the unity of mathematics, and it has applications. After prerequisites on Riemann surfaces, braids, mapping classes and elements of Teichmüller theory, a detailed introduction to the entropy of braids and mapping classes is given, with thorough, sometimes new proofs. Estimates are provided of Gromov's conformal invariants of the twice punctured complex plane and it is shown that the upper and lower bounds differ by universal multiplicative constants. These imply estimates of the entropy of any pure three-braid, and yield quantitative statements on the limitations of Gromov's Oka Principle in the sense of finiteness theorems, using conformal invariants which are related to elements of the fundamental group (not merely to conjugacy classes). Further applications of the concept of conformal module are discussed. Aimed at graduate students and researchers, the book proposes several research problems.

ISBN: 9783031672880

Standard No.: 10.1007/978-3-031-67288-0doiSubjects--Topical Terms:

892215
Conformal invariants.

LC Class. No.: QC174.52.C66

Dewey Class. No.: 516.36

Braids, conformal module, entropy, and Gromov's Oka principle
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505 0 $a 1. Introduction -- 2. Riemann Surfaces, Braids, Mapping Classes, and Teichmueller Theory -- 3. The entropy of surface homeomorphisms -- 4. Conformal invariants of homotopy classes of curves. The Main theorem -- 5. Reducible pure braids. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem -- 6. The general case. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem -- 7. The conformal module and holomorphic families of polynomials -- 8. Gromov's Oka Principle and conformal module -- 9. Gromov's Oka Principle for (g, m)-fiber bundles -- 10. Fundamental groups and bounds for the extremal length -- 11. Counting functions -- 12. Riemann surfaces of second kind and finiteness theorems -- A. Several complex variables -- B. A Lemma on Conjugation -- C. Koebe's Theorem -- Index -- References.
520 $a This book studies the relation between conformal invariants and dynamical invariants and their applications, taking the reader on an excursion through a wide range of topics. The conformal invariants, called here the conformal modules of conjugacy classes of elements of the fundamental group, were proposed by Gromov in the case of the twice punctured complex plane. They provide obstructions to Gromov's Oka Principle. The invariants of the space of monic polynomials of degree n appeared earlier in relation to Hilbert's 13th Problem, and are called the conformal modules of conjugacy classes of braids. Interestingly, the conformal module of a conjugacy class of braids is inversely proportional to a popular dynamical invariant, the entropy, which was studied in connection with Thurston's celebrated theory of surface homeomorphisms. This result, proved here for the first time, is another instance of the numerous manifestations of the unity of mathematics, and it has applications. After prerequisites on Riemann surfaces, braids, mapping classes and elements of Teichmüller theory, a detailed introduction to the entropy of braids and mapping classes is given, with thorough, sometimes new proofs. Estimates are provided of Gromov's conformal invariants of the twice punctured complex plane and it is shown that the upper and lower bounds differ by universal multiplicative constants. These imply estimates of the entropy of any pure three-braid, and yield quantitative statements on the limitations of Gromov's Oka Principle in the sense of finiteness theorems, using conformal invariants which are related to elements of the fundamental group (not merely to conjugacy classes). Further applications of the concept of conformal module are discussed. Aimed at graduate students and researchers, the book proposes several research problems.
650 0 $a Conformal invariants. $3 892215
650 1 4 $a Several Complex Variables and Analytic Spaces. $3 672032
650 2 4 $a Functions of a Complex Variable. $3 672126
650 2 4 $a Dynamical Systems. $3 1366074
650 2 4 $a Algebraic Geometry. $3 670184
650 2 4 $a Manifolds and Cell Complexes. $3 1366170
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