The mapping class group
and the curve complex, Term I, 2008

Course Description

This course will be an elementary introduction to curves on surfaces. Following the work of Masur and Minsky we will investigate the complex of curves. The structure of the complex of curves is hierarchical and tightly interlocked with the structure of the mapping class group, the Teichmuller space and Kleinian groups.

We will start with the very basic notions of surface topology and develop the necessary tools in geometry and algebra as the term progresses.

Topics may include: curves on surfaces, isotopy classes, the bigon criterion, the complex of curves C(S), curve surgery, connectedness of C(S), distance in C(S), projectively measured laminations, Dehn twists, mapping classes (Nielsen-Thurston classification), generation of the mapping class group by twists (Lickorish's theorem), combinatorial rigidity of the curve complex (Ivanov's theorem), coarse geometry of the curve complex (Masur-Minsky distance estimate).


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.


Name Building/Office E-mail Phone Office Hours
Saul Schleimer 35/B2.14 s.schleimer at warwick dot ac dot uk 024 7652 3560 WThF 10-11am and by appointment.

Class meetings

Activity Led by Time Building/Room
Lecture Schleimer Tues 12-1pm 35/B0.15a
Lecture Schleimer Friday 11-12noon 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", available on my research page. However, students are advised to read some of the primary material:

  • Curves on 2-manifolds and isotopies, by Epstein.
  • 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.
  • Boundary structure of the modular group, by Harvey.
  • 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.
  • Geometry of the complex of curves I and II, by Masur and Minsky.

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.
  • 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.

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.


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.