Asymmetry of outer space of a free product

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 𝒪. We study the asymmetry of the Lipschitz metric d_R on the (relative) Outer space 𝒪. More specifically, we generalize the construction of Algom-Kfir and Bestvina, introducing an (asymmetric) Finsler norm ∥⋅∥^L that induces d_R. Let’s denote by Out(G,𝒪) the outer automorphisms of G that preserve the set of conjugacy classes of G_i’s. Then there is an Out(G,𝒪)-invariant function Ψ:𝒪→ℝ such that when ∥⋅∥^L is corrected by dΨ, 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 generalize 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∈Out(F_n) and its inverse.

KEYWORDS:

• Asymmetry of outer space
• free product of groups
• Lipschitz metric
• outer space
• train track representatives

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E06, 20E08, 20F28, 20F65, 20F34

Acknowledgments

I wish to thank my advisor Armando Martino for his help, suggestions and corrections. Also, I would like to thank the anonymous referees for their suggestions, which helped me to improve the exposition of this work and to fill a minor gap of the first version of the present paper.