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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:
*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