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,[1] 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 $X_{n}$ was introduced in a 1986 paper,[1] of Marc Culler and Karen Vogtmann, inspired by analogy with the Teichmüller space of a hyperbolic surface. They showed that the natural action of $\operatorname {Out} (F_{n})$ on $X_{n}$ is properly discontinuous, and that $X_{n}$ is contractible.

In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of $X_{n}$ into the infinite dimensional projective space $\mathbb {P} ^{\mathcal {C}}=\mathbb {R} ^{\mathcal {C}}-\{0\}/\mathbb {R} _{>0}$ , where ${\mathcal {C}}$ is the set of nontrivial conjugacy classes of elements of $F_{n}$ . They also proved that the closure ${\overline {X}}_{n}$ of $X_{n}$ in $\mathbb {P} ^{\mathcal {C}}$ is compact.

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 ${\overline {X}}_{n}$ with the space ${\overline {CV}}_{n}$ of projective classes of "very small" minimal isometric actions of $F_{n}$ on $\mathbb {R}$ -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 $π$ ₁(R_n, v) of R_n. From this point on we identify F_n and $π$ ₁(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_# : $π$ ₁(R_n) → $π$ ₁(Γ), that is by an isomorphism F_n → $π$ ₁(Γ).

A metric graph is a finite connected graph $\gamma$ together with the assignment to every topological edge e of Γ of a positive real number L(e) > 0 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.

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

\Delta _{k}=\left\{(x_{1},\dots ,x_{k})\in \mathbb {R} ^{k}\,{\Big |}\,\sum _{i=1}^{k}x_{i}=1,x_{i}>0{\text{ for }}i=1,\dots ,k\right\}

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 $\sum _{i=1}^{k}x_{i}=1$ , such that each x_i ≥ 0 and such that the set of all edges e_i in $\Gamma$ 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 $\Gamma$ 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 $π$ ₁(Γ) acts on T by covering transformations which are also isometries of (T,d), with the quotient space T/ $π$ ₁(Γ) = Γ. Since the induced homomorphism f_# is an isomorphism between F_n = $π$ ₁(R_n) and $π$ ₁(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:

\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}.

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 $\ell _{T}(g)>0$ . For this reason $\ell _{T}(g)$ is called the translation length of g. For any g, u in F_n we have $\ell _{T}(ugu^{-1})=\ell _{T}(g)$ , that is the function $\ell _{T}$ 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 = $π$ ₁(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 $\ell _{T}(g)$ 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 $T\mapsto \ell _{T}$ 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

V_{T}(K,\epsilon )=\{T'\in X_{n}:|\ell _{T}(g)-\ell _{T'}(g)|<\epsilon {\text{ for every }}g\in K\}.

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 $\lim _{i\to \infty }T_{i}=T$ if and only if for every g in F_n we have $\lim _{i\to \infty }\ell _{T_{i}}(g)=\ell _{T}(g)$ .

Gromov topology

Another topology on $X_{n}$ 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 $X_{n}$ as actions of $F_{n}$ on $\mathbb {R}$ -trees. Informally, given a tree $T\in X_{n}$ , another tree $T'\in X_{n}$ is "close" to $T$ in the Gromov topology, if for some large finite subtrees of $Y\subseteq T,Y'\subseteq T'\in X_{n}$ and a large finite subset $B\subseteq F_{n}$ there exists an "almost isometry" between $Y'$ and $Y$ with respect to which the (partial) actions of $B$ on $Y'$ and $Y$ almost agree. For the formal definition of the Gromov topology see.[5]

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.[6]

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 = $π$ ₁(R_n). The element xα of X_n is given by the marking f o τ : R_n → Γ with the metric structure L on Γ. That is, to get x α 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:

g{\underset {T\alpha }{\cdot }}t=\alpha (g){\underset {T}{\cdot }}t.

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

\ell _{T\alpha }(g)=\ell _{T}(\alpha (g))\quad {\text{ for }}g\in F_{n}.

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

φ x = x φ⁻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[1] 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 $cv_{n}$ 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 $cv_{n}$ 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 $X_{n}$ has, including the coincidence of the three topologies (Gromov, axes, weak), and an $\operatorname {Out} (F_{n})$ -action. In addition, there is a natural action of $\mathbb {R} _{>0}$ on $cv_{n}$ by scalar multiplication.

Topologically, $cv_{n}$ is homeomorphic to $X_{n}\times (0,\infty )$ . In particular, $cv_{n}$ is also contractible.

Projectivized Outer space

The projectivized Outer space is the quotient space $CV_{n}:=cv_{n}/\mathbb {R} _{>0}$ under the action of $\mathbb {R} _{>0}$ on $cv_{n}$ by scalar multiplication. The space $CV_{n}$ is equipped with the quotient topology. For a tree $T\in cv_{n}$ its projective equivalence class is denoted $[T]=\{cT|c>0\}\subseteq cv_{n}$ . The action of $\operatorname {Out} (F_{n})$ on $cv_{n}$ naturally quotients through to the action of $\operatorname {Out} (F_{n})$ on $CV_{n}$ . Namely, for $\phi \in \operatorname {Out} (F_{n})$ and $T\in cv_{n}$ put ${\displaystyle [T]\phi$ .

A key observation is that the map $X_{n}\to CV_{n},T\mapsto [T]$ is an $\operatorname {Out} (F_{n})$ -equivariant homeomorphism. For this reason the spaces $X_{n}$ and $CV_{n}$ are often identified.

Lipschitz distance

The Lipschitz distance,[7] named for Rudolf Lipschitz, for Outer space corresponds to the Thurston metric in Teichmüller space. For two points $x$ , $y$ in X_n the (right) Lipschitz distance $d_{R}$ is defined as the (natural) logarithm of the maximally stretched closed path from $x$ to $y$ :

\Lambda _{R}(x,y):=\sup _{\gamma \in F_{n}\setminus \{1\}}{\frac {\ell _{y}(\gamma )}{\ell _{x}(\gamma )}}

and

d_{R}(x,y):=\log \Lambda _{R}(x,y)

This is an asymmetric metric (also sometimes called a quasimetric), i.e. it only fails symmetry $d_{R}(x,y)=d_{R}(y,x)$ . The symmetric Lipschitz metric normally denotes:

d(x,y):=d_{R}(x,y)+d_{R}(y,x)

The supremum $\Lambda _{R}(x,y)$ is always obtained and can be calculated by a finite set the so called candidates of $x$ .

\Lambda _{R}(x,y)=\max _{\gamma \in cand(x)}{\frac {\ell _{y}(\gamma )}{\ell _{x}(\gamma )}}

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

Where $cand(x)$ 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 $x$ via the marking.

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

\Lambda _{R}(x,y)=\min _{h\in H(x,y)}Lip(h)

Where $H(x,y)$ are the continuous functions $h:x\to y$ such that for the marking $f_{x}$ on $x$ the marking $h\circ f_{x}$ is freely homotopic to the marking $f_{y}$ on $y$ .

The induced topology is the same as the weak topology and the isometry group is $\operatorname {Out} (F_{n})$ for both, the symmetric and asymmetric Lipschitz distance.[8]

Applications and generalizations

The closure ${\overline {cv}}_{n}$ of $cv_{n}$ in the length function topology is known to consists of (F_n-equivariant isometry classes) of all very small minimal isometric actions of F_n on R-trees.[9] Here the closure is taken in the space of all minimal isometric "irreducible" actions of $F_{n}$ on $\mathbb {R}$ -trees, considered up-to equivariant isometry. It is known that the Gromov topology and the axes topology on the space of irreducible actions coincide,[5] so the closure can be understood in either sense. The projectivization of ${\overline {cv}}_{n}$ with respect to multiplication by positive scalars gives the space ${\overline {CV}}_{n}$ which is the length function compactification of $CV_{n}$ and of $X_{n}$ , analogous to Thurston's compactification of the Teichmüller space.
Analogs and generalizations of the Outer space have been developed for free products,[10] for right-angled Artin groups,[11] for the so-called deformation spaces of group actions[6] 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.[12] The Auter space $A_{n}$ shares many properties in common with the Outer space, but $A_{n}$ only comes with an action of $Aut(F_{n})$ .

References

Culler, Marc; Vogtmann, Karen (1986). "Moduli of graphs and automorphisms of free groups" (PDF). Inventiones Mathematicae. 84 (1): 91–119. doi:10.1007/BF01388734.
Cohen, Marshall M.; Lustig, Martin (1995). "Very small group actions on R-trees and Dehn twist automorphisms" (PDF). Topology. 34: 575–617. doi:10.1016/0040-9383(94)00038-m.
Bestvina, Mladen; Feighn, Mark (1994). "Outer limits" (PDF).
Guiradel, Vincent (2000). "Dynamics of $Out(F_{n})$ on the boundary of outer space". Annales Scientifiques de l'École Normale Supérieure. 33 (4): 433–465. doi:10.1016/S0012-9593(00)00117-8.
Frédéric Paulin, The Gromov topology on R-trees. Topology and its Applications 32 (1989), no. 3, 197–221.
Vincent Guirardel, Gilbert Levitt, Deformation spaces of trees. Groups, Geometry, and Dynamics 1 (2007), no. 2, 135–181.
Francaviglia, Stefano; Martino, Armando (2011). "Metric properties of Outer Space". Publicacions Matemàtiques. arXiv:0803.0640v2.
Francaviglia, Stefano; Martino, Armando (2012). "The isometry group of Outer Space". Advances in Mathematics. 231 (3–4): 1940–1973. arXiv:0912.0299. doi:10.1016/j.aim.2012.07.011.
Mladen Bestvina, The topology of Out(F_n). Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 373-384, Higher Ed. Press, Beijing, 2002; ISBN 7-04-008690-5.
Guirardel, Vincent; Levitt, Gilbert (2007). "The outer space of a free product". Proceedings of the London Mathematical Society. 94 (3): 695–714. arXiv:math/0501288. doi:10.1112/plms/pdl026.
Ruth Charney, Nathaniel Stambaugh, Karen Vogtmann, Outer space for untwisted automorphisms of right-angled Artin groups, arXiv:1212.4791, preprint, 2012
Allen Hatcher, and Karen Vogtmann, Cerf theory for graphs. Journal of the London Mathematical Society 58 (1998), no. 3, 633–655.

Outer space (mathematics)

History