Asymmetry of Outer Space of a free product

TL;DR

This paper investigates the asymmetry of the Lipschitz metric on relative outer spaces of free products, introduces an invariant correction function to achieve quasisymmetry, and extends results on automorphism expansion factors.

Contribution

It generalizes the construction of an asymmetric Finsler norm on outer space, introduces an invariant correction function for quasisymmetry, and extends bounds on automorphism expansion ratios.

Findings

The Lipschitz metric becomes quasisymmetric after correction by a specific invariant function.

Restricted to the thick part, the metric is quasi-symmetric.

Established a uniform bound on expansion ratios for IWIP automorphisms of free products.

Abstract

For every free product decomposition $G = G_{1} \ast ...\ast G_{q} \ast F_{r}$ of a group of finite Kurosh rank $G$, where $F_r$ is a finitely generated free group, we can associate some (relative) outer space $\mathcal{O}$. We study the asymmetry of the Lipschitz metric $d_R$ on the (relative) Outer space $\mathcal{O}$. More specifically, we generalise the construction of Algom-Kfir and Bestvina, introducing an (asymmetric) Finsler norm $\|\cdot\|^{L}$ that induces $d_R$. Let's denote by $Out(G, \mathcal{O})$ the outer automorphisms of $G$ that preserve the set of conjugacy classes of $G_i$'s. Then there is an $Out(G, \mathcal{O})$-invariant function $\Psi : \mathcal{O} \rightarrow \mathbb{R}$ such that when $\| \cdot \|^{L}$ is corrected by $d \Psi$, the resulting norm is quasisymmetric. As an application, we prove that if we restrict $d_R$ to the $\epsilon$-thick part of the relative Outer space for some $\epsilon >0$, is quasi-symmetric . Finally, we generalise for IWIP automorphisms of a free product a theorem of Handel and Mosher, which states that there is a uniform bound which depends only on the group, on the ratio of the relative expansion factors of any IWIP $\phi \in Out(F_n)$ and its inverse.