Asymmetry of Outer Space of a free product

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 $Ψ: \mathcal{O} \rightarrow \mathbb{R}$ such that when $\| \cdot \|^{L}$ is corrected by $d Ψ$, the resulting norm is quasisymmetric. As an application, we prove that if we restrict $d_R$ to the $ε$-thick part of the relative Outer space for some $ε>0$, 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 $ϕ\in Out(F_n)$ and its inverse.

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