The Boundary of Outer Space in Rank Two

Abstract

In [4] a space X n was introduced on which the group Out(F n ) of outer automorphisms of a free group of rank n acts virtually freely. Since then, this space has come to be known as “outer space.” Outer space can be defined as a space of free actions of F n on simplicial ℝ-trees; we require that all actions be minimal, and we identify two actions if they differ only by scaling the metric on the ℝ-tree. To describe the topology on outer space, we associate to each action α: F n × T → T a length function | · | α: F n → ℝ defined by $$ {\left| g \right|_\alpha }\; = \;\mathop {\inf }\limits_{x \in T} \;d\left( {x,gx} \right) $$ where d is the distance in the tree T. We have | g | α = | h −1 gh| α and | · |α ≡ 0 if and only if some point of T is fixed by all of F n . Thus an action with no fixed point determines a point in ℝ c — {0}, where C is the set of conjugacy classes in F n . Since actions differing by a scalar multiple define the same point of outer space, we have a map from X n to the infinite dimensional projective space P c = ℝ c — {0}/ℝ*. It can be shown that this map is injective (see [3] or [1]). We topologize X n as a subspace of P c .