Outer space
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.