# Outer space (mathematics)

In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free groupFn 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 automorphismsOut(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 Fn-equivariant isometry types of minimal free discrete isometric actions of Fn on Fn on R-treesT such that the quotient metric graph T/Fn has volume 1.

## . . . 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

${displaystyle mathbb {R} }$