Lipschitz metric isometries between Outer Spaces of virtually free groups

TL;DR

This work extends Dowdall–Taylor’s Lipschitz-geometry results from free groups to finitely generated virtually free groups by embedding $\mathscr{PT}(F)$ into $\mathscr{PT}(H)$ for finite-index $H\le F$ via a length-scaling map, preserving folding paths and Lipschitz distances. It introduces a robust volume notion to fix shortcomings of the Lipschitz pseudometric and establishes the existence of finite, graph-of-groups–level candidates that realize the Lipschitz distance, generalizing Francaviglia–Martino’s approach to the virtually free setting. The paper then identifies a deformation retraction of the spine of Outer Space for the virtually free group to the fixed-point subcomplex $L_\alpha$, proving an isometry-induced simplicial equivalence $L(F)\to L_\alpha$ and giving a structural link between automorphism groups via a short exact sequence; this yields a precise description of how normalizers act on the spine and clarifies the relationship between $\mathrm{Aut}^0(E)$ and $\mathrm{Out}^0(E)$. Overall, the results provide both computationally tractable distance certificates in $\mathscr{PT}(F)$ and a clearer geometric picture of spine deformation in the virtually free case, with implications for the structure of centralizers and normalizers in $\mathrm{Out}(F)$.

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 Krstić and Vogtmann.

Theorems & Definitions (13)

• Theorem A
• Theorem B: cf. Theorem 9.10 of FrancavigliaMartinoTrainTracks
• Theorem C
• Proposition 1.1
• proof
• Theorem 1.2
• proof
• Lemma 2.1: "Sausages Lemma" cf. Lemma 3.14 of FrancavigliaMartino
• proof
• Theorem 2.2