## Notes of Martin Bridson’s first lecture

1. Plan

1. Outer space and its spine, connectedness, folding, change of marking, contractibility, local structure and symmetries, isoperimetric inequalities, curvature.
2. Lipschitz metric, classification of isometries, existence of train tracks, relative train tracks, refinements, selected applications. Growth, isometric inequality for ${F_n \times_{\phi}{\mathbb Z}}$.
3. Torsion in ${Out(F_n)}$, Maps ${Out(F_n)\rightarrow Homeo({\mathbb R}^d)}$, ${Out(F_n)\rightarrow Out(F_m)}$.

4. ${IA_n =ker(Out(F_n)\rightarrow GL(n,{\mathbb Z})}$ Local indicability (Magnus) Lattices acting on free groups.

2. Outer space

2.1. Why ${Out(F_n)}$ ?

Because the fundamental groups of the simplest spaces are ${\{1\}}$, ${{\mathbb Z}}$, ${{\mathbb Z}^n}$, surface groups, and ${F_n}$.

Their outer automorphism groups are ${Gl(n,{\mathbb Z})}$, ${Mod_g}$, the mapping class groups, and ${Out(F_n)}$. Guiding principle : study these 3 sequences of groups in parallel.

One studies these groups via their actions on symmetric space, Teichmüller space, and Outer space respectively.

In each case, one defines the space as the space of marked metrics of constant curvature on ${Y}$ with volume ${1}$.

1. Symmetric space ${=}$ marked flat ${n}$-tori, volume ${=1}$;
2. Teichmüller space ${=}$ marked constant curvature metrics on ${S_g}$, of area ${=1}$;
3. Outer space ${=}$ marked metric graphs (curvature ${=-\infty}$), length ${=1}$.

2.2. Definition of Outer space

Definition 1 A marked metric graph of genus ${n}$ is ${(\Gamma,\mu,g)}$ where ${\Gamma}$ is a finite graph, ${\pi_1 (\Gamma)=F_n}$, connected, no vertices of valence ${1}$ or ${2}$, metric ${g}$ is a length metric, marking ${\mu}$ is a homotopy equivalence of the bouquet of ${n}$ circles to ${\Gamma}$.

Two marked graphs are equivalent if there exists an isometry ${\Gamma_1 \rightarrow\Gamma_2}$ such that the diagram of markings commutes up to homotopy.

Outer space is the space of equivalence classes.

There is a natural right action of ${Out(F_n)}$ on Outer space by precomposition of markings with self-homotopy equivalences of the bouquet.

Remark 1 We could have made all maps base point preserving, this would have led to the Auter Space on which ${Aut(F_n)}$ acts.

2.3. Other viewpoints

1. One can regard marked metric graphs as a recipe for a free minimal cocompact simplicial action of ${F_n}$ on an ${{\mathbb R}}$-tree. Let ${X_n}$ denote the space of such actions.
2. There is a natural equivariant embedding ${X_n \rightarrow {\mathbb R}^{\mathcal{C}}\rightarrow{\mathbb R}^{\mathcal{C}}}$, where ${\mathcal{C}}$ is the set of conjugacy classes in ${F_n}$, which sends a marked metric graph to the collection of lengths ${\lambda(c)}$ of free homotopy classes ${c}$ (length of shortest loop representing ${c}$).

2.4. The spine

The spine ${K_n \subset X_n}$ is an equivariant rectract.

Theorem 2 ${K_n}$ is a simplicial ${E\Gamma}$ for ${\Gamma=Out(F_n)}$. I.e. ${K_n}$ is contractible, ${Out(F_n)}$ acts on it cocompactly, simply with finite stabilizers and every finite subgroup of ${Out(F_n)}$ has a contractible fixed point set.

What does a marked metric graph look like ? A graph with a marked basis of its fundamental group. ${X_n}$ has the structure of a simplicial complex with missing faces. Indeed, one can shrink edges, but not too many at the same time, since this would change the genus. So barycentrically subdivide and push away from missing faces. The resulting result is ${K_n}$. Its dimension is ${2n-3}$, instead of ${3n-3}$.

Remark 2 ${K_n}$ can be regarded as a purely combinatorial object. Vertices of ${K_n}$ are marked graphs with edges all of equal lengths. So ${K_n}$ is the realisation of the poset of marked (not metric) graphs with order relation

$\displaystyle \begin{array}{rcl} \Gamma_1 \leq \Gamma_2 \Leftrigtarrow \exists \textrm{extra edge } e \textrm{ in }\Gamma_2 . \end{array}$

2.5. Moving around in the spine

Change choice of tree. Pick a maximal tree, vary the lengths of its edges. This parametrizes a simplex. Different choices of maximal tree corresponds to adjacent simplices.

Proposition 3 ${K_n}$ is connected.

Follows from

Theorem 4 (Nielsen) ${Out(F_n)}$ is generated by Nielsen moves.

Let ${F_n = Free(a_1 ,\ldots, a_n)}$. Nielsen moves are ${\lambda_{ij}:a_i \mapsto a_j a_i}$, ${\rho_{ij}:a_i \mapsto a_i a_j}$, ${\tau:a_i \mapsto a_i^{-1}}$. In action of ${Out(F_n)}$ on homology, these are mapped to elementary matrices.

There is a nice proof using foldings.

Proof of Proposition. ${K_n}$ contains roses (metric bouquets). Nielsen moves correspond to simplicial paths in ${K_n}$ joining roses to roses.

2.6. Presentation of ${Out(F_n)}$

Gersten (after Nielsen, Magnus and Mc Cool).

1. ${[\rho_{ij},\rho_{kl}]=[\rho_{ij},\lambda_{kl}]=[\lambda_{ij},\lambda_{kl}]=1}$.
2. ${[\rho_{ij},\lambda_{il}]=1}$.
3. ${...}$ (shows that ${Out(F_n)}$ is perfect).
4. same with ${\rho}$‘s replaced by ${\lambda}$‘s.
5. ${(\rho_{ij}\rho_{ji}^{-1}\lambda_{ij})^4=1}$ (Steinberg relation).

To prove it, one can use the action on ${K_n}$.

Need to understand the stars of roses. When ${n=3}$, it is a subset of a cube. The link is a ${2}$-sphere with 4 disks added (glued like hexagonal planar cuts of the cube). Its homotopy type is a wedge of sphere.

The stabilizer of a vertex is a semi-direct product ${W_n =({\mathbb Z}_2^n)\times S_n}$.