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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:
  1. Homeomorphisms of the torus. Classification via linear algebra. Invariant foliations for Anosov homes.
  2. Mapping class groups,basic examples: Dehn twists, rotations, pseudo-Anosov homeos via branched covers. Statement of Thurston's theorem.
  3. Spines of surfaces, Markov partitions, Perron-Frobenius and symbolic dynamics (e.g. density of orbits).
  4. Metric graphs and homotoping maps between them to optimal maps. Train track structures on graphs.
  5. 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.
  6. Proof of the classification.
  7. Back to surfaces and the proof of Thurston's classification theorem (in the once punctured case).
  8. What we really did: Outer space and Teichmuller space.

Financial support is available to US citizens and permanent residents and all eligible students who are accepted will have all of their costs associated with attendance covered.

Deadline: March 21, 2014

Course Poster