Asymmetry of outer space of a free product
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 $\Psi : \mathcal{O} \rightarrow \mathbb{R}$ 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.
Asymmetry of outer space, free product of groups, Lipschitz metric, outer space, train track representatives
1-19
Syrigos, Dionysios
e698e7fe-fb8a-44e8-a9b5-972f772260c1
Syrigos, Dionysios
e698e7fe-fb8a-44e8-a9b5-972f772260c1
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.
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Submitted date: 7 July 2015
Accepted/In Press date: 6 January 2018
e-pub ahead of print date: 17 January 2018
Keywords:
Asymmetry of outer space, free product of groups, Lipschitz metric, outer space, train track representatives
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 381856
URI: http://eprints.soton.ac.uk/id/eprint/381856
ISSN: 0092-7872
PURE UUID: 75c5b12f-92db-47fa-a127-4670e46a06c2
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Date deposited: 14 Oct 2015 13:45
Last modified: 14 Mar 2024 21:22
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Author:
Dionysios Syrigos
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