## MAT1312 Topics in geometry: Hyperbolic surfaces

This is the course webpage for MAT1312 Topics in geometry: Hyperbolic surfaces, taught by Maxime Fortier Bourque at the University of Toronto in Fall 2017.
I will list here some references to be used in the course as well as a summary of what was covered in each lecture.

### Schedule

Tuesday 10-11, Thursday 9-11.

### Summary

• 09-12: Overview of the course.
• 09-14: The disk and upper half-plane model, their geodesics and isometries.
• 09-19: Three descriptions of PSL(2,R), the boundary at infinity, properties of geodesics.
• 09-21: Pairs of geodesics, area of triangles, classification of isometries.
• 09-26: Classification of isometries (part 2), the hyperboloid model.
• 09-28: Trigonometry of triangles and right-angled hexagons.
• 10-03: Hyperbolic surfaces: cusps and annuli.
• 10-06: Gluing ideal triangles, pairs of pants.
• 10-10: Pants decompositions.
• 10-13: Complete hyperbolic surfaces, Fuchsian groups.
• 10-17: Hyperbolic orbifolds.
• 10-20: The 84(g-1) theorem.
• 10-24: The 4g+2 theorem.
• 10-27: The mapping class group, Teichmuller space.
• 10-31: Kravetz's wrong proof of Nielsen realization.
• 11-03: Length functions, Fenchel-Nielsen coordinates.
• 11-14: Kerckhoff's proof of Nielsen realization.
• 11-16: Bers' constant, Mumford's compactness, the mapping class group acts properly discontinuously.
• 11-21: Definition and examples of geodesic laminations.
• 11-23: Area and complementary components of laminations, measured geodesic laminations, train tracks, laminations are carried by train tracks, weigthed simple closed geodesics are dense in ML, Dehn-Thurston coordinates for ML.
• 11-28: Tangents to nearby disjoint geodesics are nearby, shearing along nearby geodesics has nearby effect.
• 11-30: Earthquakes are well-defined and continuous, the derivative of length is the total cosine.
• 12-05: Convexity of length functions.