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

Abstract

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

83 pages + references. Small changes in support of the Overview (arXiv:2607.19249), of Part I (arXiv:1407.3508), and of Part II (arXiv:2212.09907)