Notes of Armando Martino’s talk

The Lipschitz metric on Outer space and spectral rigidity

1. The Lipschitz metric

I sometimes consider unprojectivized Culler Vogtmann space, i.e. the space of marked metric graphs of genus ${n}$ up to equivalence (isometry and homotopy). Alternatively, it is the space of free, isometric, simplicial, minimal action of ${F_n}$ on metric trees, up to equivariant isometry.

1.1. Definition

Each point ${A}$ of that space has a translation length function ${\|.\|_A :F_n \rightarrow{\mathbb R}}$, ${\|g\|_A}$ is the minimum displacement of points of the tree by ${g}$.

Theorem 1 (Culler-Morgan, Alperin-Bass) This leads to an embedding of Outer space into ${P({\mathbb R}^{F_n})}$.

Outer space is contractible, finite dimensional.

Definition 2 The Lipschitz metric on unprojectivized Outer space is

$\displaystyle \begin{array}{rcl} d_R (T,T'):=\log D(T,T')=\log \sup_{g\in F_n}\frac{\|g\|_{T'}}{\|g\|_T}. \end{array}$

It descends to a nonsymmetric metric on Outer space inducing the usual topology. It is ${Out(F_n)}$-invariant. The symmetrized version ${d(T,T')=d_R (T,T')+d_R (T',T)}$ is a honest distance. It is proper (Francaviglia).

1.2. Basic properties

If ${f:T\rightarrow T'}$ is an equivariant Lipschitz map, then, for all ${g\in F_n}$,

$\displaystyle \begin{array}{rcl} \|g\|_{T'}\leq Lip(f)\|g\|_T , \end{array}$

so

$\displaystyle \begin{array}{rcl} D(T,T')\leq Lip(f). \end{array}$

Proposition 3 Let ${L=\inf Lip(f)}$ over all equivariant Lipschitz maps ${T\rightarrow T'}$. Then the ${\inf}$ is achieved, ${L=D(T,T')}$ and ${D(T,T')}$ is achieved on some ${g\in F_n}$.

Theorem 4 (Bestvina) Suppose that the trabslation length of ${g\in Out(F_n)}$ is achieved, i.e. there exists a tree ${T}$ on which ${D(T,T\phi)}$ is minimum. Then ${\phi}$ has a train track representative.

2. Computing the Lipschitz distance

Lemma 5 (Sausage Lemma) Given trees ${A}$, ${B}$, then

$\displaystyle \begin{array}{rcl} D(A,B)=\frac{\|g\|_{B}}{\|g\|_A} \end{array}$

where ${g}$ is embedded in ${A}$ as one of

• a cycle ${o}$;
• a figure eight ${\infty}$;
• a dumbbell ${o-o}$.

Theorem 6 (Smillie, Vogtmann) Given a finite set ${S\subset F_n}$, there exist trees ${T}$, ${T'}$ such that ${\|g\|_{T'}=\|g\|_{T}}$ for all ${g\in S}$. So the Culler-Morgan, Alperin-Bass embedding really requires all of ${F_n}$.

2.1. Isometry groups

Theorem 7 (Francaviglia, Martino) For all ${n\geq 3}$, the isometry group of Outer space (for ${d}$ or ${d_R}$) is exactly ${Out(F_n)}$. Morally true, but not quite true, for ${n=2}$.

Proof strategy:

1. Show that isometries preserve the simplicial structure.

2. Use a result of Bridson and Vogtmann stating that all simplicial automorphisms of Outer space are in ${Out(F_n)}$.

3. An isometry preserving simplices is the identity.

The third step is not straightforward. Indeed, the metric is ugly enough that there might exist non affine isometries stabilizing a simplex and each of its faces.

Example: Case of the rose simplex. For ${n=2}$, it has dimension ${1}$ and is isometric to the full Euclidean line. For ${n\geq n-1}$, it is homeomorphic to ${{\mathbb R}^{n-1}}$ with isometry group a semi-direct product ${{\mathbb R}^{n-1}\times}$finite group. Indeed, multiplying the length of each petal by a different number is an isometry. Furthermore, geodesics are not unique.

Example: Case of multi-theta simplices (i.e. graphs with ${2}$ vertices and all edges joining them). The group of isometries which maps rose faces to rose faces is finite.

3. Spectral rigidity

Work with Carette, Ilya Kapovich and Francaviglia.

Let ${T}$, ${T'}$ be trees. We know that

$\displaystyle \begin{array}{rcl} T=T' \Leftrightarrow \|.\|_T =\|.\|_{T'}. \end{array}$

Let ${P_n}$ be the set of primitive elements of ${F_n}$. Sausage Lemma implies that

$\displaystyle \begin{array}{rcl} T=T' \Leftrightarrow \|g\|_T =\|g\|_{T'} ~\forall g\in P_n . \end{array}$

Theorem 8 Given a tree ${T}$, there exists a finite set ${S_T \subset F_n}$ such that

$\displaystyle \begin{array}{rcl} T=T' \Leftrightarrow \|g\|_T =\|g\|_{T'} ~\forall g\in S_T . \end{array}$

3.1. Comment

Cormac Walsh: the symmetric metric equals the Hilbert metric on the simplex.