Asymmetry of Outer Space of a free product

For every free product decomposition $G = G_{1} \ast …\ast G_{q} \ast F_{r}$ofagroupoffiniteKuroshrank$G$,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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}^L$ 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}$$suchthatwhen$| \cdot |^{L}$ is corrected by $d\mathrm{\Psi }$, 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 $\varphi \in Out(F_n)$ and its inverse.