Outer space (mathematics)

The Outer space ${\displaystyle X_{n}}$ 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 ${\displaystyle \operatorname {Out} (F_{n})}$ on ${\displaystyle X_{n}}$ is properly discontinuous, and that ${\displaystyle X_{n}}$ is contractible.

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

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:

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

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

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

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

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.

• 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).

The unprojectivized Outer space ${\displaystyle cv_{n}}$ 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 ${\displaystyle cv_{n}}$ 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 ${\displaystyle X_{n}}$ has, including the coincidence of the three topologies (Gromov, axes, weak), and an ${\displaystyle \operatorname {Out} (F_{n})}$-action. In addition, there is a natural action of ${\displaystyle \mathbb {R} _{>0}}$ on ${\displaystyle cv_{n}}$ by scalar multiplication.

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

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

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

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

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

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

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

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

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

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

Where ${\displaystyle \operatorname {cand} (x)}$ 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 ${\displaystyle x}$ 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.

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

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

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

• The closure ${\displaystyle {\overline {cv}}_{n}}$ of ${\displaystyle cv_{n}}$ 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.[9] Here the closure is taken in the space of all minimal isometric "irreducible" actions of ${\displaystyle F_{n}}$ on ${\displaystyle \mathbb {R} }$-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 ${\displaystyle {\overline {cv}}_{n}}$ with respect to multiplication by positive scalars gives the space ${\displaystyle {\overline {CV}}_{n}}$ which is the length function compactification of ${\displaystyle CV_{n}}$ and of ${\displaystyle X_{n}}$, 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 ${\displaystyle A_{n}}$ shares many properties in common with the Outer space, but ${\displaystyle A_{n}}$ only comes with an action of ${\displaystyle \operatorname {Aut} (F_{n})}$.

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

• 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?" (PDF). AMS Notices. 55 (7): 784–786.