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.