Outer space (mathematics)

The Outer space was introduced in a 1986 paper[1] of Marc Culler and Karen Vogtmann, and it serves as a free group analog of the Teichmüller space of a hyperbolic surface.

The space Xn can also be thought of as the set of Fn-equivariant isometry types of minimal free discrete isometric actions of Fn on R-trees T such that the quotient metric graph T/Fn has volume 1.

was introduced in a 1986 paper[1] of Marc Culler and Karen Vogtmann, inspired by analogy with the Teichmüller space of a hyperbolic surface.

In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of

Later a combination of the results of Cohen and Lustig[2] and of Bestvina and Feighn[3] identified (see Section 1.3 of [4]) the space

together with the assignment to every topological edge e of Γ of a positive real number L(e) called the length of e. The volume of a metric graph is the sum of the lengths of its topological edges.

The Outer space Xn consists of equivalence classes of all the volume-one marked metric graph structures on Fn.

Given f, there is a natural map j : Δk → Xn, where for x = (x1, ..., xk) ∈ Δk, the point j(x) of Xn is given by the marking f together with the metric graph structure L on Γ such that L(ei) = xi for i = 1, ..., k. One can show that j is in fact an injective map, that is, distinct points of Δk correspond to non-equivalent marked metric graph structures on Fn.

The set j(Δk) is called open simplex in Xn corresponding to f and is denoted S(f).

By construction, Xn is the union of open simplices corresponding to all markings on Fn.

For x ∈ Δk′ − Δk the point h(x) of Xn is obtained by taking the marking f, contracting all edges ei of

The image of h is called the closed simplex in Xn corresponding to f and is denoted by S′(f).

Since Γ is a finite connected graph with no degree-one vertices, this action is also minimal, meaning that T has no proper Fn-invariant subtrees.

Moreover, every minimal free and discrete isometric action of Fn on a real tree with the quotient being a metric graph of volume one arises in this fashion from some point x of Xn.

This defines a bijective correspondence between Xn and the set of equivalence classes of minimal free and discrete isometric actions of Fn on a real trees with volume-one quotients.

Here two such actions of Fn on real trees T1 and T2 are equivalent if there exists an Fn-equivariant isometry between T1 and T2.

Give an action of Fn on a real tree T as above, one can define the translation length function associate with this action: For g ≠ 1 there is a (unique) isometrically embedded copy of R in T, called the axis of g, such that g acts on this axis by a translation of magnitude

is constant on each conjugacy class in G. In the marked metric graph model of Outer space translation length functions can be interpreted as follows.

of g. A basic general fact from the theory of group actions on real trees says that a point of the Outer space is uniquely determined by its translation length function.

Namely if two trees with minimal free isometric actions of Fn define equal translation length functions on Fn then the two trees are Fn-equivariantly isometric.

[6] The group Out(Fn) admits a natural right action by homeomorphisms on Xn.

First we define the action of the automorphism group Aut(Fn) on Xn.

Let T in Xn be a real tree with a minimal free and discrete co-volume-one isometric action of Fn.

The quotient space Mn = Xn/Out(Fn) is the moduli space which consists of isometry types of finite connected graphs Γ without degree-one and degree-two vertices, with fundamental groups isomorphic to Fn (that is, with the first Betti number equal to n) equipped with volume-one metric structures.

The moduli space Mn is not compact and the "cusps" in Mn arise from decreasing towards zero lengths of edges for homotopically nontrivial subgraphs (e.g. an essential circuit) of a metric graph Γ.

consists of equivalence classes of all marked metric graph structures on Fn where the volume of the metric graph in the marking is allowed to be any positive real number.

can also be thought of as the set of all free minimal discrete isometric actions of Fn on R-trees, considered up to Fn-equivariant isometry.

is defined as the (natural) logarithm of the maximally stretched closed path from

is the finite set of conjugacy classes in Fn which correspond to embeddings of a simple loop, a figure of eight, or a barbell into

The stretching factor also equals the minimal Lipschitz constant of a homotopy equivalence carrying over the marking, i.e. Where