On the action of relatively irreducible automorphisms on their train tracks

Abstract: Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $\phi$ with exponential growth (relatively to ${\mathcal G}$). Such set is known to coincide with the minimally displaced set $\operatorname{Min}(\phi)$ of $\phi$. Our main result is that $\operatorname{Min}(\phi)$ is co-compact, under the action of the cyclic subgroup generated by $\phi$. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(\phi)$ is in uniform distance from $\operatorname{Min}(\phi^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $\phi$, is discrete. Finally, we get some fine insight about the local topology of relative outer space. As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that $\operatorname{Min}(\phi)$ is quasi-isometric to a line.