Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem

Abstract

This is the second of a three part study of relative free splitting complexes $\mathcal{F}\mathcal{S}(\Gamma ;\mathcal{A})$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths ${\tau }_{\varphi }\ge 0$ of the simplicial isometries of $\mathcal{F}\mathcal{S}(\Gamma ;\mathcal{A})$ induced by relative outer automorphisms $\varphi \in \text{Out}(\Gamma ;\mathcal{A})$, 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 $\mathcal{F}\mathcal{S}(\Gamma ;\mathcal{A})$, 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 $\mathcal{O}(\Gamma ;\mathcal{A})$ to the relative free splitting complex $\mathcal{F}\mathcal{S}(\Gamma ;\mathcal{A})$ is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~$\mathcal{O}(\Gamma ;\mathcal{A})$. Second, if $\varphi \in \text{Out}(\Gamma ;\mathcal{A})$ has a filling attracting lamination with expansion factor $\lambda >1$ then the stable translation length of $\varphi$ acting on $\mathcal{F}\mathcal{S}(\Gamma ;\mathcal{A})$ has an upper bound of the form~$B\log (\lambda )$.

90 pages + references. v3: Several small changes in support of the Overview (arXiv:2607.19249), of Part I (arXiv:2212.09907), and of Part III (arXiv:2503.07532). v2: Title changed (old title: "Stable Translation Lengths on Free Splitting Complexes I: The Two Over All Theorem and its Applications"). Rewrote Step 3 of the Two Over All Theorem, and the Sewing Needle Lemma