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 formula_1 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 formula_2 on formula_1 is properly discontinuous, and that formula_1 is contractible.
In the same paper Culler and Vogtmann constructed an embedding, via the "translation length functions" discussed below, of formula_1 into the infinite dimensional projective space formula_6, where formula_7 is the set of nontrivial conjugacy classes of elements of formula_8. They also proved that the closure formula_9 of formula_1 in formula_11 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 formula_9 with the space formula_13 of projective classes of "very small" minimal isometric actions of formula_8 on formula_15-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"# : 1("R""n") → 1(Γ), that is by an isomorphism "F""n" → 1(Γ).
A metric graph is a finite connected graph "formula_16" 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 "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
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 formula_18, such that each "x""i" ≥ 0 and such that the set of all edges "e""i" in "formula_19" 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 "formula_19" 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:
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 formula_22. For this reason formula_23 is called the "translation length" of "g". For any "g", "u" in "F""n" we have formula_24, that is the function formula_25 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 formula_23 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 formula_27 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
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 formula_29 if and only if for every "g" in "F""n" we have formula_30.
Gromov topology.
Another topology on formula_1 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 formula_1 as actions of formula_8 on formula_15-trees.
Informally, given a tree formula_35, another tree formula_36 is "close" to formula_37 in the Gromov topology, if for some large finite subtrees of formula_38 and a large finite subset formula_39 there exists an "almost isometry" between formula_40 and formula_41 with respect to which the (partial) actions of formula_42 on formula_40 and formula_41 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" = 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:
At the level of translation length functions the tree "Tα" is given as:
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
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.
Unprojectivized Outer space.
The "unprojectivized Outer space" formula_47 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 formula_47 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 formula_1 has, including the coincidence of the three topologies (Gromov, axes, weak), and an formula_2-action. In addition, there is a natural action of formula_51 on formula_47 by scalar multiplication.
Topologically, formula_47 is homeomorphic to formula_54. In particular, formula_47 is also contractible.
Projectivized Outer space.
The projectivized Outer space is the quotient space formula_56 under the action of formula_51 on formula_47 by scalar multiplication. The space formula_59 is equipped with the quotient topology. For a tree formula_60 its projective equivalence class is denoted formula_61. The action of formula_2 on formula_63 naturally quotients through to the action of formula_2 on formula_59. Namely, for formula_66 and formula_60 put formula_68.
A key observation is that the map formula_69 is an formula_2-equivariant homeomorphism. For this reason the spaces formula_1 and formula_59 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 formula_73,formula_74 in "X""n" the (right) Lipschitz distance formula_75 is defined as the (natural) logarithm of the maximally stretched closed path from formula_73 to formula_74:
This is an asymmetric metric (also sometimes called a quasimetric), i.e. it only fails symmetry formula_80. The symmetric Lipschitz metric normally denotes:
The supremum formula_82 is always obtained and can be calculated by a finite set the so called candidates of formula_73.
Where formula_85 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 formula_73 via the marking.
The stretching factor also equals the minimal Lipschitz constant of a homotopy equivalence carrying over the marking, i.e.
Where formula_88 are the continuous functions formula_89 such that for the marking formula_90 on formula_73 the marking formula_92 is freely homotopic to the marking formula_93 on formula_74.
The induced topology is the same as the weak topology and the isometry group is formula_2 for both, the symmetric and asymmetric Lipschitz distance.