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

A single (large) file with all lectures is available on Open Math Notes

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: *marked graphs, *actions on trees *sphere systems in doubled handlebodies
Lecture 4	November 5	Sphere system proof that Outer space is contractible Definition of spine, proof of cocompactness, finiteness of stabilizers, calculation of dimension Homological consequences: VFL, VCD
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