Mirzakhani’s Curve Counting and Geodesic Currents

Description

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. 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 Weighted Multicurves Measured Laminations The Space of Geodesic Currents Interesting Subsets of C(Σ) Intersection Form Characterization of Measured Laminations Filling Currents The Mapping Class Group Action Mapping Class Group Action on the Set of Filling Currents Comments (The Liouville Current) 4 Train Tracks Train Tracks as Smoothly Embedded Complexes Almost Geodesic Train Tracks Band Complexes and Train Tracks Curves and Laminations Carried by Train Tracks Weights and Recurrence Switch Equations Thurston Measure First Counting Result 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 Radallas Carrying the Curves in Θσ0 Reducing to a Linear Algebra Problem Proof of Proposition 7.2 Comments 8 The Main Theorem Notation and the Constants The Theorem The Fundamental Domain The Meat The Not Ker-Invariant Case Comments 9 Counting Curves Main Curve Counting Theorem Frequencies and Relative Frequencies Functions on the Space of Currents 10 Counting Square-Tiled Surfaces Square-Tiled Surfaces Recovering Square-Tiled Surface Structures from Multicurves Square-Tiled Surfaces of Given Type Counting Comments 11 Statistics of Simple Curves Credit to Kasra Hard Data A Formula for c(α0) for Simple Curves Punctured Spheres Separating Curves in Closed Surfaces Non-Separating Curves in Closed Surfaces Combinatorial Computation Changing Weights Comments 12 Smörgåsbord Random Pants Decompositions Lattice Point Counting in Teichmüller Space Genericty of pseudo-Anosov Elements A Radon Measures Point-Set Topology Weak-*-Topology Sequential Compactness B Computing Thurston Volumes Punctured Spheres Separating Curves in Closed Surfaces Non-separating Curves in Closed Surfaces References Index