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

TL;DR

This paper studies the geometry of relative free splitting complexes, proving a key exponential flaring property and deriving bounds on translation lengths of automorphisms, extending results known for free groups to more general settings.

Contribution

It introduces the Two Over All Theorem, a new exponential flaring result for relative free splitting complexes, and applies it to bound translation lengths of automorphisms.

Findings

Proves the Two Over All Theorem for relative free splitting complexes.

Shows the natural map from relative outer space to the splitting complex is coarsely Lipschitz.

Establishes an upper bound on stable translation lengths based on lamination expansion factors.

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~$B \log(\lambda)$.