Arboreal Group Theory pp 189-230 | Cite as

# The Boundary of Outer Space in Rank Two

Conference paper

## Abstract

In [ where

**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$$ {\left| g \right|_\alpha }\; = \;\mathop {\inf }\limits_{x \in T} \;d\left( {x,gx} \right) $$

*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 }.## Keywords

Conjugacy Class Fundamental Domain Outer Space Primitive Element Division Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York, Inc. 1991