Mirzakhani's curve counting and geodesic currents / Viveka Erlandsson, Juan Souto.
Erlandsson, Viveka, author.; Souto, Juan, author.
2022
QA565
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Title Mirzakhani's curve counting and geodesic currents / Viveka Erlandsson, Juan Souto.
Author Erlandsson, Viveka, author.
ISBN 9783031087059 (electronic bk.)
3031087054 (electronic bk.)
9783031087042
3031087046
DOI https://doi.org/10.1007/978-3-031-08705-9
Published Cham : Birkhäuser, [2022]
Copyright ©2022
Language English
Description 1 online resource (xii, 226 pages) : illustrations.
Item Number 10.1007/978-3-031-08705-9 doi
Call Number QA565
Dewey Decimal Classification 516.3/52
Summary This monograph presents an approachable proof of Mirzakhanis curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to the area, the presentation builds intuition with elementary examples before progressing to rigorous proofs. This approach illuminates new and established results alike, and produces versatile tools for studying the geometry of hyperbolic surfaces, Teichmuller theory, and mapping class groups. Beginning with the preliminaries of curves and arcs on surfaces, the authors go on to present the theory of geodesic currents in detail. Highlights include a treatment of cusped surfaces and surfaces with boundary, along with a comprehensive discussion of the action of the mapping class group on the space of geodesic currents. A user-friendly account of train tracks follows, providing the foundation for radallas, an immersed variation. From here, the authors apply these tools to great effect, offering simplified proofs of existing results and a new, more general proof of Mirzakhanis curve counting theorem. Further applications include counting square-tiled surfaces and mapping class group orbits, and investigating random geometric structures. Mirzakhanis Curve Counting and Geodesic Currents introduces readers to powerful counting techniques for the study of surfaces. Ideal for graduate students and researchers new to the area, the pedagogical approach, conversational style, and illuminating illustrations bring this exciting field to life. Exercises offer opportunities to engage with the material throughout. Basic familiarity with 2-dimensional topology and hyperbolic geometry, measured laminations, and the mapping class group is assumed.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed October 4, 2022).
Added Author Souto, Juan, author.
Series Progress in mathematics (Boston, Mass.) ; v. 345.
Available in Other Form Print version: 9783031087042
Mirzakhani's curve counting and geodesic currents.
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Intro
Preface
Acknowledgments
Contents
Notation
1 Introduction
Some Things that the Reader Will Not Find in This Book
What the Reader Will Find Here
A Comment on the Way This Book Is Written
2 Read Me
Curves, Multicurves, and Arcs
Intersection Number
Geodesic Representatives
Random Geodesics. Or Length Versus Intersections
Space of Geodesics
Geodesics Projecting into a Fixed Compact Set
Laminations
Hausdorff Limits
Mapping Class Group
Shortening Curves
3 Geodesic Currents
Definition and Examples
Curves as Currents

Masur's Ergodicity Theorem
An Example: The Case of the Once Punctured Torus
Comments
Piecewise Linear Structure
Symplectic Structure
Weil-Petersson Volume Form
Other Interesting Measures on ML(S)
5 Radallas
Terminology
The Game of Cars
Almost Geodesic Radallas
Finding Almost Geodesic Radallas
Generic Curves
Comments: Expected Angle of Self-Intersection
6 Subconvergence of Measures
Idea of the Proof of Theorem 6.1
Local Map
Global Map
New Measures
Comments
7 Approximating the Thurston Measure
Dribbling

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