Relative Free Splitting Complexes III: Stable Translation Lengths and Filling Paths

This is the last of a three part work about relative free splitting complexes $\mathcal{FS}(\Gamma,\mathscr{A})$ and their actions by relative outer automorphism groups $\text{Out}(\Gamma;\mathscr{A})$. We obtain quantitative relations between the stable translation length $\tau_\phi$ and the relative train track dynamics of~$\phi \in \Out(\Gamma;\A)$. First, if $\phi$ has an orbit with diameter bounded below by a certain constant $\Omega(\Gamma;\mathscr{A}) \ge 1$ then $\phi$ has a filling attracting lamination. Also, there is a positive lower bound $\tau_\phi \ge A(\Gamma;\mathscr{A}) > 0$ amongst all $\phi$ which have a filling attracting lamination. Both proofs rely on a study of \emph{filling paths} in a free splitting. These results are all new even for $\text{Out}(F_n)$.