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The study of the geometry of free group automorphisms began when Thurston proved the Bounded Cancellation Lemma. Today the study focuses on the large scale geometry of Out(Fn) and of Culler-Vogtmann's Outer space, modeled on the success of the Masur-Minsky theory that explains the large scale geometry of mapping class groups and Teichmüller space. The last several years have seen a substantial progress, but the Out(Fn) theory is still lagging behind.
Mladen Bestvina of the University of Utah reviews the main results and challenges for the future in the study of Out(Fn), June 27, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.