In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group Fn is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on Fn. The Outer space, denoted Xn or CVn, comes equipped with a natural action of the group of outer automorphisms Out(Fn) of Fn. 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(Fn) and to obtain information about algebraic, geometric and dynamical properties of Out(Fn), of its subgroups and individual outer automorphisms of Fn. The space Xn can also be thought of as the set of 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.
The Outer space
Xn
\operatorname{Out}(Fn)
Xn
Xn
In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of
Xn
Pl{C
l{C}
Fn
\overlineXn
Xn
Pl{C
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
\overlineXn
\overline{CV}n
Fn
R
Let n ≥ 2. For the free group Fn fix a "rose" Rn, that is a wedge, of n circles wedged at a vertex v, and fix an isomorphism between Fn and the fundamental group 1(Rn, v) of Rn. From this point on we identify Fn and 1(Rn, v) via this isomorphism.
A marking on Fn consists of a homotopy equivalence f : Rn → 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
A marked metric graph structure on Fn consists of a marking f : Rn → Γ together with a metric graph structure L on Γ.
Two marked metric graph structures f1 : Rn → Γ1 and f2 : Rn → 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 Xn consists of equivalence classes of all the volume-one marked metric graph structures on Fn.
Let f : Rn → 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, ..., ek. Let
\Deltak=\left\{(x1,...,xk)\inRk|
| k | |
| \sum | |
| i=1 |
xi=1,xi>0fori=1,...,k\right\}
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. Note that two open simplices in Xn either are disjoint or coincide.
Let f : Rn → 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, ..., ek. Define Δk′ ⊆ Rk as the set of all x = (x1, ..., xk) ∈ Rk, such that
| k | |
| style{\sum | |
| i=1 |
xi=1}
\Gamma
The map j : Δk → Xn extends to a map h : Δk′ → Xn as follows. For x in Δk put h(x) = j(x). For x ∈ Δk′ - Δk the point h(x) of Xn is obtained by taking the marking f, contracting all edges ei of
\Gamma
It can be shown that for every marking f the map h : Δk′ → Xn is still injective. The image of h is called the closed simplex in Xn corresponding to f and is denoted by S′(f). Every point in Xn belongs to only finitely many closed simplices and a point of Xn represented by a marking f : Rn → Γ where the graph Γ is tri-valent belongs to a unique closed simplex in Xn, namely S′(f).
The weak topology on the Outer space Xn is defined by saying that a subset C of Xn is closed if and only if for every marking f : Rn → Γ the set h-1(C) is closed in Δk′. In particular, the map h : Δk′ → Xn is a topological embedding.
Let x be a point in Xn given by a marking f : Rn → 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 Fn = 1(Rn) and 1(Γ), we also obtain an isometric action of Fn on T with T/Fn = 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 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:
\ellT:Fn\toR, \ellT(g)=mint\ind(t,gt), forg\inFn.
\ellT(g)>0
\ellT(g)
| -1 | |
| \ell | |
| T(ugu |
)=\ellT(g)
\ellT
In the marked metric graph model of Outer space translation length functions can be interpreted as follows. Let T in Xn be represented by a marking f : Rn → Γ with a volume-one metric graph structure L on Γ. Let g ∈ Fn = 1(Rn). 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
\ellT(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. Hence the map
T\mapsto\ellT
One defines the length function topology or axes topology on Xn as follows. For every T in Xn, every finite subset K of Fn and every ε > 0 let
VT(K,\epsilon)=\{T'\inXn:|\ellT(g)-\ellT'(g)|<\epsilonforeveryg\inK\}.
Convergence of sequences in the length function topology can be characterized as follows. For T in Xn and a sequence Ti in Xn we have
\limi\toinftyTi=T
\limi\toinfty
| \ell | |
| Ti |
(g)=\ellT(g).
Another topology on
Xn
When defining the Gromov topology, one should think of points of
Xn
Fn
R
T\inXn
T'\inXn
T
Y\subseteqT,Y'\subseteqT'\inXn
B\subseteqFn
Y'
Y
B
Y'
Y
An important basic result states that the Gromov topology, the weak topology and the length function topology on Xn coincide.[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 α ∈ Aut(Fn) be an automorphism of Fn. Let x be a point of Xn given by a marking f : Rn → Γ with a volume-one metric graph structure L on Γ. Let τ : Rn → Rn be a homotopy equivalence whose induced homomorphism at the fundamental group level is the automorphism α of Fn = 1(Rn). The element xα of Xn is given by the marking f ∘ τ : Rn → 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 Xn be a real tree with a minimal free and discrete co-volume-one isometric action of Fn. Let α ∈ Aut(Fn). As a metric space, Tα is equal to T. The action of Fn is twisted by α. Namely, for any t in T and g in Fn we have:
g\underset{T\alpha}{ ⋅ }t=\alpha(g)\underset{T}{ ⋅ }t.
At the level of translation length functions the tree Tα is given as:
\ellT\alpha(g)=\ellT(\alpha(g)) forg\inFn.
One then checks that for the above action of Aut(Fn) on Outer space Xn the subgroup of inner automorphisms Inn(Fn) is contained in the kernel of this action, that is every inner automorphism acts trivially on Xn. It follows that the action of Aut(Fn) on Xn quotients through to an action of Out(Fn) = Aut(Fn)/Inn(Fn) on Xn. namely, if φ ∈ Out(Fn) is an outer automorphism of Fn and if α in Aut(Fn) is an actual automorphism representing φ then for any x in Xn we have xφ = xα.
The right action of Out(Fn) on Xn can be turned into a left action via a standard conversion procedure. Namely, for φ ∈ Out(Fn) and x in Xn set
φx = xφ−1.
This left action of Out(Fn) on Xn is also sometimes considered in the literature although most sources work with the right action.
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 quotient topology on Mn is the same as that given by the Gromov–Hausdorff distance between metric graphs representing points of Mn. 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.
The unprojectivized Outer space
cvn
cvn
Xn
\operatorname{Out}(Fn)
R>0
cvn
Topologically,
cvn
Xn x (0,infty)
cvn
The projectivized Outer space is the quotient space
CVn:=cvn/R>0
R>0
cvn
CVn
T\incvn
[T]=\{cT\midc>0\}\subseteqcvn
\operatorname{Out}(Fn)
cvn
\operatorname{Out}(Fn)
CVn
\phi\in\operatorname{Out}(Fn)
T\incvn
[T]\phi:=[T\phi]
A key observation is that the map
Xn\toCVn,T\mapsto[T]
\operatorname{Out}(Fn)
Xn
CVn
The Lipschitz distance,[7] named for Rudolf Lipschitz, for Outer space corresponds to the Thurston metric in Teichmüller space. For two points
x,y
dR
x
y
ΛR(x,y):=\sup
| \gamma\inFn\setminus\{1\ |
dR(x,y):=logΛR(x,y)
dR(x,y)=dR(y,x)
d(x,y):=dR(x,y)+dR(y,x)
ΛR(x,y)
x
ΛR(x,y)=max\gamma(x)}
| \elly(\gamma) | |
| \ellx(\gamma) |
Where
\operatorname{cand}(x)
x
The stretching factor also equals the minimal Lipschitz constant of a homotopy equivalence carrying over the marking, i.e.
ΛR(x,y)=minhLip(h)
H(x,y)
h:x\toy
fx
x
h\circfx
fy
y
The induced topology is the same as the weak topology and the isometry group is
\operatorname{Out}(Fn)
\overline{cv}n
cvn
Fn
R
\overline{cv}n
\overline{CV}n
CVn
Xn
An
An
\operatorname{Aut}(Fn)
Out(Fn)