## 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.
### Sources

- Svetlana Katok,
**Fuchsian groups**, The University of Chicago Press, 1992.
- Peter Buser,
**Geometry and spectra of compact Riemann surfaces**, Birhauser, 1992.
- Benson Farb and Dan Margalit,
**A primer on mapping class groups**, Princeton University Press, 2012.
- John H. Hubbard,
**Teichmuller theory and applications to geometry, topology, and dynamics**, Volume 1, Matrix Editions, 2006.
- Steven P. Kerckhoff,
**The Nielsen realization problem**, Annals of Mathematics, **117** (1983), 235-265.
- William P. Thurston,
**Earthquakes in 2-dimensional hyperbolic geometry**, in Fundamentals of Hyperbolic Manifolds, Cambridge University Press, 2006.
- Notes from a course by Caroline Series.

### 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.