The Metric Completion of Outer Space

TL;DR

This paper develops a metric completion theory for Outer Space, characterizes boundary points, and establishes a homeomorphism between the simplicial completion and the free splitting complex, providing new insights into the isometry group of Outer Space.

Contribution

It introduces a metric completion framework for Outer Space, characterizes boundary points, and proves a homeomorphism with the free splitting complex, offering a new proof of the isometry group theorem.

Findings

Boundary points are characterized within the metric completion.

The simplicial completion is homeomorphic to the free splitting complex.

A new proof is provided for the isometry group 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 $n \geq 3$ and equal to $\text{PSL}(2,\mathbb{Z})$ for $n=2$.