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 $\mathcal{FS}(Γ;\mathscr A)$, 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~$B \log(λ)$.