Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909533041852416 |
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| author | Handel, Michael Mosher, Lee |
| author_facet | Handel, Michael Mosher, Lee |
| contents | This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(Γ;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $τ_ϕ\ge 0$ of the simplicial isometries of $\mathcal{FS}(Γ;\mathscr A)$ induced by relative outer automorphisms $ϕ\in \text{Out}(Γ;\mathscr A)$, stating and proving quantitative generalizations of earlier theorems for $\text{Out}(F_n)$. The main technical result proved here in Part~II is the \emph{Two Over All Theorem}, which expresses a uniform exponential flaring property along arbitrary Stallings fold paths in $\mathcal{FS}(Γ;\mathscr A)$, a new result even for $\text{Out}(F_n)$. We give two applications of this theorem. First, the natural map from the relative outer space ${\mathscr O}(Γ;\mathscr A)$ to the relative free splitting complex $\mathcal{FS}(Γ;\mathscr A)$ is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~${\mathscr O}(Γ;\mathscr A)$. Second, if $ϕ\in \text{Out}(Γ;\mathscr A)$ has a filling attracting lamination with expansion factor $λ>1$ then the stable translation length of $ϕ$ acting on $\mathcal{FS}(Γ;\mathscr A)$ has an upper bound of the form~$B \log(λ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09907 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem Handel, Michael Mosher, Lee Group Theory 20F65 (primary) 57M07 (secondary) This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(Γ;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $τ_ϕ\ge 0$ of the simplicial isometries of $\mathcal{FS}(Γ;\mathscr A)$ induced by relative outer automorphisms $ϕ\in \text{Out}(Γ;\mathscr A)$, stating and proving quantitative generalizations of earlier theorems for $\text{Out}(F_n)$. The main technical result proved here in Part~II is the \emph{Two Over All Theorem}, which expresses a uniform exponential flaring property along arbitrary Stallings fold paths in $\mathcal{FS}(Γ;\mathscr A)$, a new result even for $\text{Out}(F_n)$. We give two applications of this theorem. First, the natural map from the relative outer space ${\mathscr O}(Γ;\mathscr A)$ to the relative free splitting complex $\mathcal{FS}(Γ;\mathscr A)$ is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~${\mathscr O}(Γ;\mathscr A)$. Second, if $ϕ\in \text{Out}(Γ;\mathscr A)$ has a filling attracting lamination with expansion factor $λ>1$ then the stable translation length of $ϕ$ acting on $\mathcal{FS}(Γ;\mathscr A)$ has an upper bound of the form~$B \log(λ)$. |
| title | Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem |
| topic | Group Theory 20F65 (primary) 57M07 (secondary) |
| url | https://arxiv.org/abs/2212.09907 |