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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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