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 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 of Marc Culler and Karen Vogtmann, inspired by analogy with the Teichmüller space of a hyperbolic surface. They showed that the natural action of $\operatorname(F_n)$ 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 $\mathbb^ = \mathbb^ \!-\! \/\mathbb_$, where $\mathcal$ 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 $\mathbb^$ 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 $\overline X_n$ with the space $\overline_n$ 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 ₁(''R''_''n'', ''v'') of ''R''_''n''. From this point on we identify ''F''_''n'' and 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''_# : , that is by an isomorphism

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'') 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 ''f''₁ : ''R''_''n'' → Γ₁ and ''f''₂ : ''R''_''n'' → Γ₂ are ''equivalent'' if there exists an isometry ''θ'' : Γ₁ → Γ₂ such that, up to free homotopy, we have ''θ'' o ''f''₁ = ''f''₂.

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 ''e''₁, ..., ''e''_''k''. Let
:$\Delta_k = \left\$
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'' = (''x''₁, ..., ''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 ''e''₁, ..., ''e''_''k''. Define Δ_''k''′ ⊆ R^''k'' as the set of all ''x'' = (''x''₁, ..., ''x''_''k'') ∈ R^''k'', such that $\textstyle$, 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 ''f''₁ : ''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 ₁(Γ) acts on ''T'' by covering transformations which are also isometries of (''T'', ''d''), with the quotient space ''T''/₁(Γ) = Γ. Since the induced homomorphism ''f''_# is an isomorphism between ''F''_''n'' = ₁(''R''_''n'') and 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 ''T''₁ and ''T''₂ are ''equivalent'' if there exists an ''F''_''n''-equivariant isometry between ''T''₁ and ''T''₂.

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:
:$\ell_T: F_n \to \mathbb R, \quad \ell_T(g)=\min_ d(t,gt), \quad\text g\in F_n.$
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 $\ell_T(ugu^) = \ell_T(g)$, 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'' = 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
:$V_T(K,\epsilon)= \.$
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 $\lim_ T_i = T$ if and only if for every ''g'' in ''F''_''n'' we have $\lim_ \ell_(g) = \ell_T(g).$

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.Frédéric Paulin, ''The Gromov topology on R-trees''. '' Topology and its Applications'' 32 (1989), no. 3, 197–221.

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.Vincent Guirardel, Gilbert Levitt, ''Deformation spaces of trees''. '' Groups, Geometry, and Dynamics'' 1 (2007), no. 2, 135–181.

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'' = ₁(''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:
:$g\underset t= \alpha(g) \underset t.$

At the level of translation length functions the tree ''Tα'' is given as:
:$\ell_(g)=\ell_T(\alpha(g)) \quad \text g\in F_n.$

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
:''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 3''n'' − 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 3''n'' − 3 edges and it has exactly 3''n'' − 3 edges when Γ is trivalent. Hence the top-dimensional open simplex in ''X''_''n'' has dimension 3''n'' − 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 2''n'' − 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 $\operatorname(F_n)$-action. In addition, there is a natural action of $\mathbb R_$ 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 $CV_n:=cv_n/\mathbb R_$ under the action of $\mathbb R_$ 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 \\subseteq cv_n. The action of $\operatorname(F_n)$ on $cv_n$ naturally quotients through to the action of $\operatorname(F_n)$ on $CV_n$. Namely, for $\phi\in \operatorname(F_n)$ and $T\in cv_n$ put phi:= \phi/math>.

A key observation is that the map X_n \to CV_n, T\mapsto /math> is an $\operatorname(F_n)$-equivariant homeomorphism. For this reason the spaces $X_n$ and $CV_n$ 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 $x, y$ in ''X''_''n'' the (right) Lipschitz distance $d_R$ is defined as the (natural) logarithm of the maximally stretched closed path from $x$ to $y$:
:$\Lambda_R(x,y):=\sup_ \frac$ 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 $x$.
:$\Lambda_R(x,y) = \max_ \frac$

Where $\operatorname(x)$ 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 (see the diagram).

The stretching factor also equals the minimal Lipschitz constant of a homotopy equivalence carrying over the marking, i.e.
:$\Lambda_R(x,y)=\min_ Lip(h)$
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 $\operatorname(F_n)$ for both, the symmetric and asymmetric Lipschitz distance.

Applications and generalizations

* The closure $\overline_n$ of $cv_n$ 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 $F_n$ 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, so the closure can be understood in either sense. The projectivization of $\overline_n$ with respect to multiplication by positive scalars gives the space $\overline_n$ 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, 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.Allen Hatcher, and Karen Vogtmann,
''Cerf theory for graphs.'' Journal of the London Mathematical Society 58 (1998), no. 3, 633–655. The Auter space $A_n$ shares many properties in common with the Outer space, but $A_n$ only comes with an action of $\operatorname(F_n)$.

See also

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

References

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; .
* Karen Vogtmann, ''On the geometry of outer space''. ''Bulletin of the American Mathematical Society'' ''52'' (2015), no. 1, 27–46.
* {{cite journal , first= Karen , last= Vogtmann , author-link=Karen Vogtmann , title= What Is . . . Outer Space? , journal= AMS Notices , volume= 55 , issue= 7 , pages= 784–786 , year= 2008 , url= https://www.ams.org/notices/200807/tx080700784p.pdf

Geometric group theory
Geometric topology