Lipschitz metric isometries between Outer Spaces of virtually free groups

TL;DR

This paper extends the understanding of Lipschitz metric isometries from free groups to virtually free groups, showing embeddings, candidate distances, and deformation retractions in their Outer Spaces.

Contribution

It generalizes known results about free groups to virtually free groups, including isometric embeddings and the existence of Lipschitz distance candidates.

Findings

Embedding of Outer Space via covers is an isometry

Existence of 'candidates' for Lipschitz distance in virtually free groups

Deformation retraction identified with Krstić and Vogtmann's space

Abstract

Dowdall and Taylor observed that given a finite-index subgroup of a free group, taking covers induces an embedding from the Outer Space of the free group to the Outer Space of the subgroup, that this embedding is an isometry with respect to the (asymmetric) Lipschitz metric, and that the embedding sends folding paths to folding paths. The purpose of this note is to extend this result to virtually free groups. We further extend a result Francaviglia and Martino, proving the existence of "candidates" for the Lipschitz distance between points in the Outer Space of the virtually free group. Additionally we identify a deformation retraction of the spine of the Outer Space for the virtually free group with the space considered by Krsti\'c and Vogtmann.