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 $\mathcal{F}\mathcal{S}(\mathrm{\Gamma },\mathcal{A})$ and their actions by relative outer automorphism groups $\text{Out}(\mathrm{\Gamma };\mathcal{A})$. We obtain quantitative relations between the stable translation length ${\tau }_{\phi }$ and the relative train track dynamics of~$\phi \in \text{Out}(\mathrm{\Gamma };\text{A})$. First, if $\phi$ has an orbit with diameter bounded below by a certain constant $\mathrm{\Omega }(\mathrm{\Gamma };\mathcal{A})\ge 1$ then $\phi$ has a filling attracting lamination. Also, there is a positive lower bound ${\tau }_{\phi }\ge A(\mathrm{\Gamma };\mathcal{A})>0$ 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 $\text{Out}(F_n)$.