Outer space (mathematics)

In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group F_n is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on F_n. The Outer space, denoted X_n or CV_n, comes equipped with a natural action of the group of outer automorphisms Out(F_n) of F_n. The Outer space was introduced in a 1986 paper of Marc Culler and Karen Vogtmann, and it serves as a free group analog of the Teichmüller space of a hyperbolic surface. Outer space is used to study homology and cohomology groups of Out(F_n) and to obtain information about algebraic, geometric and dynamical properties of Out(F_n), of its subgroups and individual outer automorphisms of F_n. The space X_n can also be thought of as the set of F_n-equivariant isometry types of minimal free discrete isometric actions of F_n on F_n on R-trees T such that the quotient metric graph T/F_n has volume 1.

History

The Outer space [math]\displaystyle{ X_n }[/math] was introduced in a 1986 paper of Marc Culler and Karen Vogtmann, inspired by analogy with the Teichmüller space of a hyperbolic surface. They showed that the natural action of [math]\displaystyle{ \operatorname{Out}(F_n) }[/math] on [math]\displaystyle{ X_n }[/math] is properly discontinuous, and that [math]\displaystyle{ X_n }[/math] is contractible.

In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of [math]\displaystyle{ X_n }[/math] into the infinite-dimensional projective space [math]\displaystyle{ \mathbb{P}^{\,\mathcal{C}} = \mathbb{R}^{\mathcal{C}} \!-\! \{0\}/\mathbb{R}_{\gt 0} }[/math], where [math]\displaystyle{ \mathcal{C} }[/math] is the set of nontrivial conjugacy classes of elements of [math]\displaystyle{ F_n }[/math]. They also proved that the closure [math]\displaystyle{ \overline X_n }[/math] of [math]\displaystyle{ X_n }[/math] in [math]\displaystyle{ \mathbb{P}^{\,\mathcal{C}} }[/math] is compact.

Later a combination of the results of Cohen and Lustig and of Bestvina and Feighn identified (see Section 1.3 of ) the space [math]\displaystyle{ \overline X_n }[/math] with the space [math]\displaystyle{ \overline{CV}_n }[/math] of projective classes of "very small" minimal isometric actions of [math]\displaystyle{ F_n }[/math] on [math]\displaystyle{ \mathbb R }[/math]-trees.

Formal definition

Marked metric graphs

Let n ≥ 2. For the free group F_n fix a "rose" R_n, that is a wedge, of n circles wedged at a vertex v, and fix an isomorphism between F_n and the fundamental group π₁(R_n, v) of R_n. From this point on we identify F_n and π₁(R_n, v) via this isomorphism.

A marking on F_n consists of a homotopy equivalence f : R_n → Γ where Γ is a finite connected graph without degree-one and degree-two vertices. Up to a (free) homotopy, f is uniquely determined by the isomorphism f_# : π₁(R_n) → π₁(Γ), that is by an isomorphism F_n → π₁(Γ).

A metric graph is a finite connected graph [math]\displaystyle{ \gamma }[/math] 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.

A marked metric graph structure on F_n consists of a marking f : R_n → Γ together with a metric graph structure L on Γ.

Two marked metric graph structures f₁ : R_n → Γ₁ and f₂ : R_n → Γ₂ are equivalent if there exists an isometry θ : Γ₁ → Γ₂ such that, up to free homotopy, we have θ o f₁ = f₂.

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

Weak topology on the Outer space

Open simplices

Let f : R_n → Γ where Γ is a marking and let k be the number of topological edges in Γ. We order the edges of Γ as e₁, ..., e_k. Let

[math]\displaystyle{ \Delta_k = \left\{ (x_1, \dots, x_k)\in \mathbb{R}^k \,\Big|\, \sum_{i=1}^k x_i =1, x_i \gt 0 \text{ for } i=1,\dots, k\right\} }[/math]

be the standard (k − 1)-dimensional open simplex in R^k.

Given f, there is a natural map j : Δ_k → X_n, where for x = (x₁, ..., x_k) ∈ Δ_k, the point j(x) of X_n is given by the marking f together with the metric graph structure L on Γ such that L(e_i) = x_i for i = 1, ..., k.

One can show that j is in fact an injective map, that is, distinct points of Δ_k correspond to non-equivalent marked metric graph structures on F_n.

The set j(Δ_k) is called open simplex in X_n corresponding to f and is denoted S(f). By construction, X_n is the union of open simplices corresponding to all markings on F_n. Note that two open simplices in X_n either are disjoint or coincide.

Closed simplices

Let f : R_n → Γ where Γ is a marking and let k be the number of topological edges in Γ. As before, we order the edges of Γ as e₁, ..., e_k. Define Δ_k′ ⊆ R^k as the set of all x = (x₁, ..., x_k) ∈ R^k, such that [math]\displaystyle{ \textstyle{\sum_{i=1}^k x_i=1} }[/math], such that each x_i ≥ 0 and such that the set of all edges e_i in [math]\displaystyle{ \Gamma }[/math] with x_i = 0 is a subforest in Γ.

The map j : Δ_k → X_n extends to a map h : Δ_k′ → X_n as follows. For x in Δ_k put h(x) = j(x). For x ∈ Δ_k′ − Δ_k the point h(x) of X_n is obtained by taking the marking f, contracting all edges e_i of [math]\displaystyle{ \Gamma }[/math] with x_i = 0 to obtain a new marking f₁ : R_n → Γ₁ and then assigning to each surviving edge e_i of Γ₁ length x_i > 0.

It can be shown that for every marking f the map h : Δ_k′ → X_n is still injective. The image of h is called the closed simplex in X_n corresponding to f and is denoted by S′(f). Every point in X_n belongs to only finitely many closed simplices and a point of X_n represented by a marking f : R_n → Γ where the graph Γ is tri-valent belongs to a unique closed simplex in X_n, namely S′(f).

The weak topology on the Outer space X_n is defined by saying that a subset C of X_n is closed if and only if for every marking f : R_n → Γ the set h⁻¹(C) is closed in Δ_k′. In particular, the map h : Δ_k′ → X_n is a topological embedding.

Points of Outer space as actions on trees

Let x be a point in X_n given by a marking f : R_n → Γ with a volume-one metric graph structure L on Γ. Let T be the universal cover of Γ. Thus T is a simply connected graph, that is T is a topological tree. We can also lift the metric structure L to T by giving every edge of T the same length as the length of its image in Γ. This turns T into a metric space (T, d) which is a real tree. The fundamental group π₁(Γ) acts on T by covering transformations which are also isometries of (T, d), with the quotient space T/π₁(Γ) = Γ. Since the induced homomorphism f_# is an isomorphism between F_n = π₁(R_n) and π₁(Γ), we also obtain an isometric action of F_n on T with T/F_n = Γ. This action is free and discrete. Since Γ is a finite connected graph with no degree-one vertices, this action is also minimal, meaning that T has no proper F_n-invariant subtrees.

Moreover, every minimal free and discrete isometric action of F_n on a real tree with the quotient being a metric graph of volume one arises in this fashion from some point x of X_n. This defines a bijective correspondence between X_n and the set of equivalence classes of minimal free and discrete isometric actions of F_n on a real trees with volume-one quotients. Here two such actions of F_n on real trees T₁ and T₂ are equivalent if there exists an F_n-equivariant isometry between T₁ and T₂.

Length functions

Give an action of F_n on a real tree T as above, one can define the translation length function associate with this action:

[math]\displaystyle{ \ell_T: F_n \to \mathbb R, \quad \ell_T(g)=\min_{t\in T} d(t,gt), \quad\text{ for } g\in F_n. }[/math]

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 [math]\displaystyle{ \ell_T(g)\gt 0 }[/math]. For this reason [math]\displaystyle{ \ell_T(g) }[/math] is called the translation length of g. For any g, u in F_n we have [math]\displaystyle{ \ell_T(ugu^{-1}) = \ell_T(g) }[/math], that is the function [math]\displaystyle{ \ell_T }[/math] 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. Let T in X_n be represented by a marking f : R_n → Γ with a volume-one metric graph structure L on Γ. Let g ∈ F_n = π₁(R_n). First push g forward via f_# to get a closed loop in Γ and then tighten this loop to an immersed circuit in Γ. The L-length of this circuit is the translation length [math]\displaystyle{ \ell_T(g) }[/math] 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 F_n define equal translation length functions on F_n then the two trees are F_n-equivariantly isometric. Hence the map [math]\displaystyle{ T\mapsto \ell_T }[/math] from X_n to the set of R-valued functions on F_n is injective.

One defines the length function topology or axes topology on X_n as follows. For every T in X_n, every finite subset K of F_n and every ε > 0 let

[math]\displaystyle{ V_T(K,\epsilon)= \{ T'\in X_n: |\ell_T(g)-\ell_{T'}(g)| \lt \epsilon \text{ for every } g\in K\}. }[/math]

In the length function topology for every T in X_n a basis of neighborhoods of T in X_n is given by the family V_T(K, ε) where K is a finite subset of F_n and where ε > 0.

Convergence of sequences in the length function topology can be characterized as follows. For T in X_n and a sequence T_i in X_n we have [math]\displaystyle{ \lim_{i\to\infty} T_i = T }[/math] if and only if for every g in F_n we have [math]\displaystyle{ \lim_{i\to\infty} \ell_{T_i}(g) = \ell_T(g). }[/math]

Gromov topology

Another topology on [math]\displaystyle{ X_n }[/math] is the so-called Gromov topology or the equivariant Gromov–Hausdorff convergence topology, which provides a version of Gromov–Hausdorff convergence adapted to the setting of an isometric group action.

When defining the Gromov topology, one should think of points of [math]\displaystyle{ X_n }[/math] as actions of [math]\displaystyle{ F_n }[/math] on [math]\displaystyle{ \mathbb R }[/math]-trees. Informally, given a tree [math]\displaystyle{ T\in X_n }[/math], another tree [math]\displaystyle{ T'\in X_n }[/math] is "close" to [math]\displaystyle{ T }[/math] in the Gromov topology, if for some large finite subtrees of [math]\displaystyle{ Y\subseteq T, Y'\subseteq T'\in X_n }[/math] and a large finite subset [math]\displaystyle{ B\subseteq F_n }[/math] there exists an "almost isometry" between [math]\displaystyle{ Y' }[/math] and [math]\displaystyle{ Y }[/math] with respect to which the (partial) actions of [math]\displaystyle{ B }[/math] on [math]\displaystyle{ Y' }[/math] and [math]\displaystyle{ Y }[/math] almost agree. For the formal definition of the Gromov topology see.

Coincidence of the weak, the length function and Gromov topologies

An important basic result states that the Gromov topology, the weak topology and the length function topology on X_n coincide.

Action of Out(F_n) on Outer space

The group Out(F_n) admits a natural right action by homeomorphisms on X_n.

First we define the action of the automorphism group Aut(F_n) on X_n. Let α ∈ Aut(F_n) be an automorphism of F_n. Let x be a point of X_n given by a marking f : R_n → Γ with a volume-one metric graph structure L on Γ. Let τ : R_n → R_n be a homotopy equivalence whose induced homomorphism at the fundamental group level is the automorphism α of F_n = π₁(R_n). The element xα of X_n is given by the marking f ∘ τ : R_n → Γ with the metric structure L on Γ. That is, to get xα from x we simply precompose the marking defining x with τ.

In the real tree model this action can be described as follows. Let T in X_n be a real tree with a minimal free and discrete co-volume-one isometric action of F_n. Let α ∈ Aut(F_n). As a metric space, Tα is equal to T. The action of F_n is twisted by α. Namely, for any t in T and g in F_n we have:

[math]\displaystyle{ g\underset{T\alpha}{\cdot} t= \alpha(g) \underset{T}{\cdot} t. }[/math]

At the level of translation length functions the tree Tα is given as:

[math]\displaystyle{ \ell_{T\alpha}(g)=\ell_T(\alpha(g)) \quad \text{ for } g\in F_n. }[/math]

One then checks that for the above action of Aut(F_n) on Outer space X_n the subgroup of inner automorphisms Inn(F_n) is contained in the kernel of this action, that is every inner automorphism acts trivially on X_n. It follows that the action of Aut(F_n) on X_n quotients through to an action of Out(F_n) = Aut(F_n)/Inn(F_n) on X_n. namely, if φ ∈ Out(F_n) is an outer automorphism of F_n and if α in Aut(F_n) is an actual automorphism representing φ then for any x in X_n we have xφ = xα.

The right action of Out(F_n) on X_n can be turned into a left action via a standard conversion procedure. Namely, for φ ∈ Out(F_n) and x in X_n set

φx = xφ⁻¹.

This left action of Out(F_n) on X_n is also sometimes considered in the literature although most sources work with the right action.

Moduli space

The quotient space M_n = X_n/Out(F_n) 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 F_n (that is, with the first Betti number equal to n) equipped with volume-one metric structures. The quotient topology on M_n is the same as that given by the Gromov–Hausdorff distance between metric graphs representing points of M_n. The moduli space M_n is not compact and the "cusps" in M_n arise from decreasing towards zero lengths of edges for homotopically nontrivial subgraphs (e.g. an essential circuit) of a metric graph Γ.

Basic properties and facts about Outer space

• Outer space X_n is contractible and the action of Out(F_n) on X_n is properly discontinuous, as was proved by Culler and Vogtmann in their original 1986 paper where Outer space was introduced.
• The space X_n has topological dimension 3n − 4. The reason is that if Γ is a finite connected graph without degree-one and degree-two vertices with fundamental group isomorphic to F_n, then Γ has at most 3n − 3 edges and it has exactly 3n − 3 edges when Γ is trivalent. Hence the top-dimensional open simplex in X_n has dimension 3n − 4.
• Outer space X_n contains a specific deformation retract K_n of X_n, called the spine of Outer space. The spine K_n has dimension 2n − 3, is Out(F_n)-invariant and has compact quotient under the action of Out(F_n).

Unprojectivized Outer space

The unprojectivized Outer space [math]\displaystyle{ cv_n }[/math] consists of equivalence classes of all marked metric graph structures on F_n where the volume of the metric graph in the marking is allowed to be any positive real number. The space [math]\displaystyle{ cv_n }[/math] can also be thought of as the set of all free minimal discrete isometric actions of F_n on R-trees, considered up to F_n-equivariant isometry. The unprojectivized Outer space inherits the same structures that [math]\displaystyle{ X_n }[/math] has, including the coincidence of the three topologies (Gromov, axes, weak), and an [math]\displaystyle{ \operatorname{Out}(F_n) }[/math]-action. In addition, there is a natural action of [math]\displaystyle{ \mathbb R_{\gt 0} }[/math] on [math]\displaystyle{ cv_n }[/math] by scalar multiplication.

Topologically, [math]\displaystyle{ cv_n }[/math] is homeomorphic to [math]\displaystyle{ X_n\times (0,\infty) }[/math]. In particular, [math]\displaystyle{ cv_n }[/math] is also contractible.

Projectivized Outer space

The projectivized Outer space is the quotient space [math]\displaystyle{ CV_n:=cv_n/\mathbb R_{\gt 0} }[/math] under the action of [math]\displaystyle{ \mathbb R_{\gt 0} }[/math] on [math]\displaystyle{ cv_n }[/math] by scalar multiplication. The space [math]\displaystyle{ CV_n }[/math] is equipped with the quotient topology. For a tree [math]\displaystyle{ T\in cv_n }[/math] its projective equivalence class is denoted [math]\displaystyle{ [T]=\{cT \mid c\gt 0\}\subseteq cv_n }[/math]. The action of [math]\displaystyle{ \operatorname{Out}(F_n) }[/math] on [math]\displaystyle{ cv_n }[/math] naturally quotients through to the action of [math]\displaystyle{ \operatorname{Out}(F_n) }[/math] on [math]\displaystyle{ CV_n }[/math]. Namely, for [math]\displaystyle{ \phi\in \operatorname{Out}(F_n) }[/math] and [math]\displaystyle{ T\in cv_n }[/math] put [math]\displaystyle{ [T]\phi:= [T\phi] }[/math].

A key observation is that the map [math]\displaystyle{ X_n \to CV_n, T\mapsto [T] }[/math] is an [math]\displaystyle{ \operatorname{Out}(F_n) }[/math]-equivariant homeomorphism. For this reason the spaces [math]\displaystyle{ X_n }[/math] and [math]\displaystyle{ CV_n }[/math] are often identified.

Lipschitz distance

The Lipschitz distance, named for Rudolf Lipschitz, for Outer space corresponds to the Thurston metric in Teichmüller space. For two points [math]\displaystyle{ x, y }[/math] in X_n the (right) Lipschitz distance [math]\displaystyle{ d_R }[/math] is defined as the (natural) logarithm of the maximally stretched closed path from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math]:

[math]\displaystyle{ \Lambda_R(x,y):=\sup_{\gamma \in F_n\setminus \{1\}} \frac{\ell_y(\gamma)}{\ell_x(\gamma)} }[/math] and [math]\displaystyle{ d_R(x,y):= \log \Lambda_R(x,y) }[/math]

This is an asymmetric metric (also sometimes called a quasimetric), i.e. it only fails symmetry [math]\displaystyle{ d_R(x,y)=d_R(y,x) }[/math]. The symmetric Lipschitz metric normally denotes:

[math]\displaystyle{ d(x,y):=d_R(x,y)+d_R(y,x) }[/math]

The supremum [math]\displaystyle{ \Lambda_R(x,y) }[/math] is always obtained and can be calculated by a finite set the so called candidates of [math]\displaystyle{ x }[/math].

[math]\displaystyle{ \Lambda_R(x,y) = \max_{\gamma \in \operatorname{cand}(x)} \frac{\ell_y(\gamma)}{\ell_x(\gamma)} }[/math]

A simple loop, a figure of eight, and a barbell

Where [math]\displaystyle{ \operatorname{cand}(x) }[/math] is the finite set of conjugacy classes in F_n which correspond to embeddings of a simple loop, a figure of eight, or a barbell into [math]\displaystyle{ x }[/math] via the marking (see the diagram).

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

[math]\displaystyle{ \Lambda_R(x,y)=\min_{h \in H(x,y)} Lip(h) }[/math]

Where [math]\displaystyle{ H(x,y) }[/math] are the continuous functions [math]\displaystyle{ h: x \to y }[/math] such that for the marking [math]\displaystyle{ f_x }[/math] on [math]\displaystyle{ x }[/math] the marking [math]\displaystyle{ h \circ f_x }[/math] is freely homotopic to the marking [math]\displaystyle{ f_y }[/math] on [math]\displaystyle{ y }[/math].

The induced topology is the same as the weak topology and the isometry group is [math]\displaystyle{ \operatorname{Out}(F_n) }[/math] for both, the symmetric and asymmetric Lipschitz distance.

Applications and generalizations

• The closure [math]\displaystyle{ \overline{cv}_n }[/math] of [math]\displaystyle{ cv_n }[/math] in the length function topology is known to consist of (F_n-equivariant isometry classes of) all very small minimal isometric actions of F_n on R-trees. Here the closure is taken in the space of all minimal isometric "irreducible" actions of [math]\displaystyle{ F_n }[/math] on [math]\displaystyle{ \mathbb R }[/math]-trees, considered up to equivariant isometry. It is known that the Gromov topology and the axes topology on the space of irreducible actions coincide, so the closure can be understood in either sense. The projectivization of [math]\displaystyle{ \overline{cv}_n }[/math] with respect to multiplication by positive scalars gives the space [math]\displaystyle{ \overline{CV}_n }[/math] which is the length function compactification of [math]\displaystyle{ CV_n }[/math] and of [math]\displaystyle{ X_n }[/math], analogous to Thurston's compactification of the Teichmüller space.
• Analogs and generalizations of the Outer space have been developed for free products, for right-angled Artin groups, for the so-called deformation spaces of group actions and in some other contexts.
• A base-pointed version of Outer space, called Auter space, for marked metric graphs with base-points, was constructed by Hatcher and Vogtmann in 1998. The Auter space [math]\displaystyle{ A_n }[/math] shares many properties in common with the Outer space, but [math]\displaystyle{ A_n }[/math] only comes with an action of [math]\displaystyle{ \operatorname{Aut}(F_n) }[/math].

See also

• Geometric group theory
• Mapping class group
• Train track map
• Out(Fn)

Further reading

• Mladen Bestvina, The topology of Out(F_n). Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 373–384, Higher Education Press, Beijing, 2002; ISBN 7-04-008690-5.
• Karen Vogtmann, On the geometry of outer space. Bulletin of the American Mathematical Society 52 (2015), no. 1, 27–46.
• Vogtmann, Karen (2008). "What Is . . . Outer Space?". AMS Notices 55 (7): 784–786. https://www.ams.org/notices/200807/tx080700784p.pdf.