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 up to equivalence (isometry and homotopy). Alternatively, it is the space of free, isometric, simplicial, minimal action of on metric trees, up to equivariant isometry.
Each point of that space has a translation length function , is the minimum displacement of points of the tree by .
Theorem 1 (Culler-Morgan, Alperin-Bass) This leads to an embedding of Outer space into .
Outer space is contractible, finite dimensional.
Definition 2 The Lipschitz metric on unprojectivized Outer space is
It descends to a nonsymmetric metric on Outer space inducing the usual topology. It is -invariant. The symmetrized version is a honest distance. It is proper (Francaviglia).
1.2. Basic properties
If is an equivariant Lipschitz map, then, for all ,
Proposition 3 Let over all equivariant Lipschitz maps . Then the is achieved, and is achieved on some .
Theorem 4 (Bestvina) Suppose that the trabslation length of is achieved, i.e. there exists a tree on which is minimum. Then has a train track representative.
2. Computing the Lipschitz distance
Lemma 5 (Sausage Lemma) Given trees , , then
where is embedded in as one of
- a cycle ;
- a figure eight ;
- a dumbbell .
Theorem 6 (Smillie, Vogtmann) Given a finite set , there exist trees , such that for all . So the Culler-Morgan, Alperin-Bass embedding really requires all of .
2.1. Isometry groups
Theorem 7 (Francaviglia, Martino) For all , the isometry group of Outer space (for or ) is exactly . Morally true, but not quite true, for .
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 .
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 , it has dimension and is isometric to the full Euclidean line. For , it is homeomorphic to with isometry group a semi-direct product 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 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 , be trees. We know that
Let be the set of primitive elements of . Sausage Lemma implies that
Theorem 8 Given a tree , there exists a finite set such that
Cormac Walsh: the symmetric metric equals the Hilbert metric on the simplex.