## Hyperbolicity of the curve complex, Term II, 2010

### Course Description

This course will be an elementary introduction to the idea of hyperbolic spaces and to curves on surfaces. We will follow Bowditch's treatment of a theorem of Masur and Minsky: the curve complex is Gromov hyperbolic. We will start with very basic notions of surface topology and develop the necessary tools in hyperbolic and Teichmuller geometry as the term progresses.

Topics may include: curves on surfaces, isotopy classes, the bigon criterion, the complex of curves C(S), connectedness of C(S), distance in C(S), the infinite diameter of C(S), Gromov hyperbolicity, quasi-geodesics, Morse stability, isoperimetric inequalities, singular flat metrics, the annulus lemma, Teichmuller geodesics, and the systole map.

### Schedule

The schedule has a list of topics, organized by week. These are subject to change, depending on the whims of the instructor and of the class.

### Instructor

Name Building/Office E-mail Phone Office Hours
Saul Schleimer 35/B2.14 s.schleimer at warwick dot ac dot uk 024 7652 3560 Tu 2:15-3pm and Wed by appointment. (Please email.)

### Class meetings

Activity Led by Time Building/Room
Lecture Schleimer Thur 1:15-2:15pm 35/B0.15a
Lecture Schleimer Friday 10:15-11:15am 35/B0.15a

Please do ask questions in class. I am also more than happy to answer questions via email or in office hours.

### Reference materials

As the course progresses I hope to revise my Notes on the complex of curves. However, students are advised to read some of the primary material:

• Curves on 2-manifolds and isotopies, by Epstein.
• Boundary structure of the modular group, by Harvey.
• Hyperbolic groups, by Gromov.
• The geometry of cycles in the Cayley diagram of a group, by Gilman.
• On the definition of word hyperbolic groups, by Gilman.
• Geometry of the complex of curves I, by Masur and Minsky.
• Intersection numbers and the hyperbolicity of the curve complex, by Bowditch

Other works of interest include:

• Conway's ZIP proof, by Francis and Weeks
• Braids, links, and mapping class groups, by Birman.
• The geometries of 3-manifolds, by Scott.
• A representation of orientable combinatorial 3-manifolds, by Lickorish.
• A finite set of generators for the homeotopy group of a 2-manifold, by Lickorish.
• On the geometry and dynamics of diffeomorphisms of surfaces, by Thurston.
• Automorphisms of Surfaces After Nielsen and Thurston, by Casson and Bleiler.
• Knots, links, braids and 3-manifolds, by Prasolov and Sossinsky.
• Three-dimensional Geometry and Topology, by Thurston and edited by Levy.
• Automorphism of complexes of curves and of Teichmuller spaces, by Ivanov.
• Automorphisms of complexes of curves on punctured spheres and on punctured tori, by Korkmaz.
• Automorphisms of the complex of curves, by Luo.

### Assessed work

One homework exercise per 1.5 weeks. (Exercises can be found in the notes and will also be stated in lecture.) Email solutions to the instructor.

### Mistakes

Please tell me in person, or via email, about any errors on this website or made in class. I am especially keen to hear about mathematical errors, gaffes, or typos appearing in my notes on the curve complex.