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