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 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 Fn-equivariant 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.
History The Outer space X n {\displaystyle 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 Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} on X n {\displaystyle X_{n}} is properly discontinuous, and that X n {\displaystyle X_{n}} is contractible. In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of X n {\displaystyle X_{n}} into the infinite-dimensional projective space P C = R C − { 0 } / R > 0 {\displaystyle \mathbb {P} ^{\,{\mathcal {C}}}=\mathbb {R} ^{\mathcal {C}}\!-\!\{0\}/\mathbb {R} _{>0}} , where C {\displaystyle {\mathcal {C}}} is the set of nontrivial conjugacy classes of elements of F n {\displaystyle F_{n}} . They also proved that the closure X ¯ n {\displaystyle {\overline {X}}_{n}} of X n {\displaystyle X_{n}} in P C {\displaystyle \mathbb {P} ^{\,{\mathcal {C}}}} 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 X ¯ n {\displaystyle {\overline {X}}_{n}} with the space C V ¯ n {\displaystyle {\overline {CV}}_{n}} of projective classes of "very small" minimal isometric actions of F n {\displaystyle F_{n}} on R {\displaystyle \mathbb {R} } -trees.
Formal definition
Marked metric graphs 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 π1(Rn, v) of Rn. From this point on we identify Fn and π1(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# : π1(Rn) → π1(Γ), that is by an isomorphism Fn → π1(Γ). A metric graph is a finite connected graph γ {\displaystyle \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 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 ∘ f1 = f2. The Outer space Xn consists of equivalence classes of all the volume-one marked metric graph structures on Fn.
Weak topology on the Outer space
Open simplices 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
… excerpt ends here. Continue reading the full article.
