Teichmuller theory in Outer Space

The talks will be about $Out(F_n)$, the outer automorphism group of a
free group. In the study of this group the comparisons with mapping
class groups are surprisingly fruitful. I will start by introducing
Culler-Vogtmann’s Outer space, which plays the role of Teichmuller
space. It comes with a natural metric, called the Lipschitz metric.
This metric is not symmetric, but should be thought of as an analog of
the Teichmuller metric. I will outline a proof of the classification
of automorphisms of $F_n$, analogous to the Nielsen-Thurston
classification of mapping classes. Stallings’ folding paths play the
role of Teichmuller geodesics and irreducible automorphisms have axes
in Outer space. The current research focuses on developing a kind of
Masur-Minsky theory for $Out(F_n)$. The last part of the lectures will
outline some ideas in the theorem (joint with Feighn) that the complex
of free factors (analogous to the Bruhat-Tits building and the curve
complex) is Gromov hyperbolic.