Train Tracks, Diffeomorphisms of Surfaces and Automorphisms of Free Groups
July 7 - 18, 2014
Instructor: Mladen Bestvina
In the 1970's Thurston introduced the notion of train tracks as a combinatorial tool for studying simple closed curves on surfaces. This notion played an important role in his study of selfdiffeomorphisms of surfaces. In the 80's, Bestvina and Handel introduced a parallel notion of train tracks for free group automorphisms giving a uniform way to study both the mapping class group and the group of outer automorphisms of free groups. This has proven to be a powerful tool and this course will provide an introduction to the topic for advanced undergraduates given by one of its originators.
The course will cover the following topics:
- Homeomorphisms of the torus. Classification via linear algebra. Invariant foliations for Anosov homes.
- Mapping class groups,basic examples: Dehn twists, rotations, pseudo-Anosov homeos via branched covers. Statement of Thurston's theorem.
- Spines of surfaces, Markov partitions, Perron-Frobenius and symbolic dynamics (e.g. density of orbits).
- Metric graphs and homotoping maps between them to optimal maps. Train track structures on graphs.
- Transforming a self-map of a graph allowing changing the graph up to homotopy. Classification into hyperbolic, parabolic and elliptic self-maps. Examples of each kind.
- Proof of the classification.
- Back to surfaces and the proof of Thurston's classification theorem (in the once punctured case).
- What we really did: Outer space and Teichmuller space.
Application Deadline: February 20, 2014
Further details: here!
