论文标题
相对自由分裂复合物II：稳定的翻译长度和所有定理的两个
Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem
论文作者
论文摘要
这是相对自由分裂复合物的三部分研究中的第二份$ \ Mathcal {fs}（γ; \ Mathscr a）$，从第〜i部分已知是Gromov双曲线。在此处和第三部分中，我们专注于稳定的翻译长度$τ_D\ ge 0 $ $ \ Mathcal {fs}（fs}（γ; \ mathscr a）$由相对外部自动形态引起的$ \ text \ in \ text {out}（γ;; \ m nation $ nitiatiate and and anditiated and s anditiated and and s and and s and s and sateers and sateers sate sate and sate sate sate and sate nip and temanccr {fs}（fs; \ mathscr a）$ $ \ text {out}（f_n）$。第〜II部分在此处证明的主要技术结果是\ emph {两个phos the theorem}，它在$ \ mathcal {fs}（γ; \ mathscr a）$中表达沿任意失速折叠路径沿任意失速的折叠路径的均匀指数的flaring属性，即使对于$ \ \ text {out}（f_n）$。我们提供了该定理的两个应用。首先，来自相对外部空间的自然图$ {\ mathscr o}（γ; \ mathscr a）$到相对免费分裂复杂的$ \ MATHCAL {fs}（fs; \ mathscr a）$是log-lipschitz semimetric on〜$ $ {$ {$ {$ {$ o）的lig-lipschitz semimetric，是log-lipschitz semimetric oon lig-lipschitz的c.其次，如果$ ϕ \ in \ text {out}（γ; \ m mthscr a）$具有填充吸引的层压，则用膨胀因子$λ> 1 $ $ 1 $，则稳定的翻译长度$ ϕ $作用于$ \ nathcal {fs}（fs}}（γ; \ nathscr a）$具有上限的上限。
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~$B \log(λ)$.
