Overview
- Generalizes Mirzakhani’s curve counting theorem to include non-simple curves
- Develops powerful counting techniques for the study of surfaces
- Features an engaging, pedagogical approach and illuminating illustrations
Part of the book series: Progress in Mathematics (PM, volume 345)
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Table of contents (12 chapters)
Keywords
- Mirzakhani curve counting
- Counting closed geodesics
- Mirzakhani curve counting for non-simple curves
- Geodesic currents
- Geodesic currents of cusped surfaces
- Mapping class group
- Teichmüller space
- Train tracks hyperbolic surfaces
- Immersed train track
- Thurston measure
- Square-tiled surfaces
- Geometric structures on surfaces
- Dynamics induced by group actions
- Statistics of simple curves
- Random pants decomposition
About this book
This monograph presents an approachable proof of Mirzakhani’s 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, Teichmüller 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 Mirzakhani’s curve counting theorem. Further applications include counting square-tiled surfaces and mapping class group orbits, and investigating random geometric structures.
Mirzakhani’s 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.
Reviews
“The book is written with enthusiasm and delightful touches of informality. It includes beautiful illustrations by Hugo Parlier. The authors delight in Mirzakhani's ideas and the surprising applications, and they take pleasure in making the mathematics accessible. They keep the book self-contained … .” (Boris Hasselblatt, Mathematical Reviews, July, 2023)
Authors and Affiliations
About the authors
Viveka Erlandsson is Lecturer of Mathematics at University of Bristol, UK. Her research interests include hyperbolic geometry, low-dimensional topology, and Teichmüller theory.
Juan Souto is Directeur de Recherche at the CNRS, Université de Rennes 1, France. His research interests include hyperbolic geometry, low-dimensional topology, and mapping class groups.
Bibliographic Information
Book Title: Mirzakhani’s Curve Counting and Geodesic Currents
Authors: Viveka Erlandsson, Juan Souto
Series Title: Progress in Mathematics
DOI: https://doi.org/10.1007/978-3-031-08705-9
Publisher: Birkhäuser Cham
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 Switzerland AG 2022
Hardcover ISBN: 978-3-031-08704-2Published: 21 September 2022
Softcover ISBN: 978-3-031-08707-3Published: 22 September 2023
eBook ISBN: 978-3-031-08705-9Published: 20 September 2022
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 1
Number of Pages: XII, 226
Number of Illustrations: 33 b/w illustrations
Topics: Topology, Dynamical Systems and Ergodic Theory, Vibration, Dynamical Systems, Control