Notes of Martin Bridson lecture 2

Recall ${K_n}$ denotes the spine of Outer space.

1. Plan revisited

The plan I gave yesterday is for a one year course. After discussing with people, I decided on more realistic contents. In this talk, I intend to discuss various definitions of train tracks. This afternoon, folding. The goal is to prove the following theorems.

1. All simplicial automorphisms of ${K_n}$ belong to ${Out(F_n)}$.
2. Outer space is contractible.
3. The Dehn functions of ${Out(F_n)}$ and ${Aut(F_n)}$ are exponential.
4. The Dehn functions of mapping torus ${F_n \times_\phi {\mathbb Z}}$ is quadratic.
5. If ${d, every topological action of ${Aut(F_n)}$ (and hence ${Out(F_n)}$ and ${Gl(n,{\mathbb Z})}$) on ${{\mathbb R}^d}$ or ${S^{d-1}}$ factors through ${{\mathbb Z}/2}$.
6. If ${n\not=m\leq 2n}$, then every homomorphism ${Out(F_n)\rightarrow Out(F_m)}$ has image at most ${{\mathbb Z}/2}$.
7. If ${\Gamma}$ is a lattice in a higher rank semisimple Lie group, then every homomorphism from ${\Gamma}$ to ${Out(F_n)}$ has finite image.

2. Simplicial automorphisms

2.1. Results

Theorem 1 (Bridson, Vogtmann 2001) If ${n\geq 3}$, all simplicial automorphisms of ${K_n}$ belong to ${Out(F_n)}$.

Note this fails for ${n=2}$. The spine ${K_2}$ is the Farey tree for ${PSL(2,{\mathbb Z})}$.

Here is a consequence, which also followed from earlier work of Dyer and Formanek. In this respect, ${Out(F_n)}$ behaves like a lattice in a semisimple group (Mostow rigidity).

Theorem 2 (Bridson, Vogtmann 2000) If ${n\geq 3}$, ${Out(F_n)}$ is co-Hopfian, i.e. every monomorphism ${Out(F_n)\rightarrow Out(F_n)}$ is an inner automorphism.

2.2. Idea of proof

Need to understand the local structure of ${K_n}$. We have seen that homotopically, the link of vertex ${v}$ is a wedge of ${N(v)}$ spheres of dimension ${2n-4}$.

Lemma 3

1. If ${v}$ corresponds to a trivalent graph (maximal in the partial ordering),

$\displaystyle \begin{array}{rcl} N(v)\leq 2^{n-1}\prod_{i=2}^{n}\log_2 (i). \end{array}$

2. If ${v}$ is a rose (minimal in the partial ordering),

$\displaystyle \begin{array}{rcl} N(v)\geq \prod_{j=1}(2j-1)^2 . \end{array}$

3. If ${v}$ is neither trivalent nor a rose, then the link of ${v}$ is the simplicial join of two simplicial complexes.
4. Links of roses or trivalent graphs are not joins.

Next, one proves that any simplicial automorphism of ${K_n}$ preserves the topological type of labelling graphs.

2.3. Comment

Frédéric Haglund: what is the automorphism group of the link of a rose.

Answer: I don’t know. If I knew, this would make the last steps in the previous proof easier.

3. Outer space is contractible

The spine ${K_n}$ is the union of the stars of the roses.

3.1. The Culler-Vogtmann proof

Use Morse theory. Construct a height function with only one critical point. Fix basis ${\gamma_1 ,\ldots,\gamma_n \in F_n}$. Set

$\displaystyle \begin{array}{rcl} height=\sum_{i=1}^{n}length[\gamma_i]. \end{array}$

At a rose, thought of as a basis ${(x_i)}$ of ${F_n}$, height is the sum of length of ${\gamma_i}$ in the word metric defined by ${(x_i)}$.

Whitehead’s algorithm (or rather Higgins and Lyndon) is a generalisation of Nielsen moves. It amounts to changing maximal tree by swapping one edge (Nielsen moves correspond to the case of one edge trees).

Lemma 4 (Higgins, Lyndon) Let ${\rho_1}$ and ${\rho_2}$ be roses such that ${height(\rho_1). Then there exists a sequence of roses ${\rho_2 =R_1 , R_2 ,\ldots,R_\ell =\rho_1}$ along which height decreases. ${R_{i+1}}$ is obtained from ${R_i}$ by a Whitehead automorphism.

Proposition 5 Let ${(\gamma_i)}$ be a basis of ${F_n}$. The roses of minimal height are of the form

$\displaystyle \{\gamma_i, \gamma_i \gamma_j ,\gamma_i \gamma_j^{-1}\,;\,i\not=j\}.$

Then argue by induction. Fix some ordering on roses such that

$\displaystyle \begin{array}{rcl} \rho<\rho' \Rightarrow height(\rho)\leq height(\rho'). \end{array}$

Show that adding the star of the next rose preserves contractibility.

4. Dehn functions

4.1. Non positive curvature

What we just did sounds like Outer space would have nonpositive curvature. This is not a good picture. If one tries to follow this lines of contraction, one does does not get the weak convexity property required by nonpositive curvature.

One can prove that Outer space is not ${CAT(0)}$ (seem my thesis). This does not rule out a coarser kind of nonpositive curvature. What does really rule it out is nonpolynomial filling function.

4.2. Filling and Dehn functions

Definition 6 By filling area of a loop, we mean the minimal area (i.e. number of ${2}$-cells) of a simplicial disk filling it. The filling function is

$\displaystyle \begin{array}{rcl} \delta(k)=\inf\{FillArea(\ell)\,;\,\ell \textrm{ loop, } length(\ell)\leq k\}. \end{array}$

Theorem 7 (Gromov, Bridson) In a finitely presented group acting properly and cocompactly on a simplicial complex, ${\delta}$ has the same growth type as the Dehn function.

Theorem 8 (Bridson, Vogtmann 1995, Handel, Mosher 2010) If ${n\geq 3}$, the Dehn functions of ${Out(F_n)}$ and ${Aut(F_n)}$ are exponential (both exponential lower bounds and upper bounds).

4.3. Proof

The proof I explain comes from a short recent note by Vogtmann and me, using a crucial idea from Handel and Mosher.

The square diagram. We shall use the spine ${L_n}$ of Auter space. Recall this is the set of marked graphs with minimal cocompact isometric ${F_n}$ action.

If ${m>n}$, ${L_n}$ embeds in ${L_m}$. There is a map ${r:K_m \rightarrow K_n}$ in the opposite direction (restrict action of ${F_m}$ to subgroup ${F_n}$) and pass to minimal invariant subtree (this requires to fix once and for all a subgroup ${F_n \subset F_m}$ and kills . This gives a commuting square diagram.

There is a nice geometric proof that ${Out(F_3)}$ has an exponential Dehn function (lower bound). One can exhibit an explicit family of words ${w_i}$ in generators of ${Aut(F_3)}$ which are hard to fill. These give loops in ${L_3}$ whose images in ${K_3}$ are hard to fill. Push them to ${L_m}$ and then ${K_m}$. This yields loops which are again hard to fill in ${K_m}$.