The Metric Completion of Outer Space

Abstract

We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simplicial tree actions, is homeomorphic to the free splitting complex. We use this to give a new proof of a theorem by Francaviglia and Martino that the isometry group of Outer Space is homeomorphic to $\text{Out}(F_n)$ for $n\ge 3$ and equal to $\text{PSL}(2,\mathbb{Z})$ for $n=2$.

Added references for quasi-metric spaces. Thoroughly revised version. Section 1 is completely new. Fixed some mistakes in other sections