# TCC Winter 2015

### Course description

Outer space is a contractible space with a proper action of the group of outer automorphisms of a free group. It should be thought of as analogous to a symmetric space with the action of an arithmetic group or the Teichmuller space of a surface with the action of the mapping class group. This course is an introduction to Outer space and its applications to the study of automorphisms of free groups.

### Lecture Notes

Lectures 1 and 2 updated 2/2/2017

Lecture 1 | Oct. 15, 2014 | Introduction, history Relation of Out(F_n) with GL(n,Z), Mod(S) Example of an automorphism not realizable on a surface Models for F_n: finite graph, punctured surface, doubled handlebody M_n Out(F_n) as homotopy equivalences of a graph, diffeos of M_n |

Lecture 2 | October 22 | Whitehead's algorithm, using 3-manifold model Stallings' folds and generators for Out(F_n) |

Lecture 3 | October 29 |
Three definitions of Outer space and the Out(F_n)-action: |

Lecture 4 | November 5 |
Sphere system proof that Outer space is contractible |

Lecture 5 | November 12 | Local structure of spine: poset lemma, Cohen-Macaulay property Simplicial automorphisms of the spine |

Lecture 6 | November 19 | Cube complex structure of the spine Homology computations Filtrations of the spine |

Lecture 7 | November 26 | Filtrations continued, Lie algebra of symplectic derivations of the free Lie algebra |

Lecture 8 | December 5 | Proof of Kontsevich's theorem Using the abelianization to find cocycles: Morita classes |