ON THE ACTION OF RELATIVELY IRREDUCIBLE AUTOMORPHISMS ON THEIR TRAIN TRACKS

Let 𝐺 be a group and let G be a free factor system of 𝐺, namely a free splitting of 𝐺 as 𝐺 =𝐺1 ∗⋯ ∗𝐺𝑘 ∗𝐹𝑟. In this paper, we study the set of train track points for G-irreducible automorphisms 𝜙 with exponential growth. Such set is known to coincide with the minimally displaced set Min⁡(𝜙) of 𝜙, in the relative deformation space corresponding to the splitting. The theory of such relative spaces, even if it is more general by its own nature, is crucial to understanding reducible automorphisms of free groups, as any automorphism is relatively irreducible with respect to some free factor system G. Our main result is that Min⁡(𝜙) is co-compact, under the action of the cyclic subgroup generate by 𝜙. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of Min⁡(𝜙) is in uniform distance from Min⁡(𝜙−1). We also prove that the action of 𝐺 on the product of the attracting and the repelling trees for 𝜙, is discrete. Finally, we get some fine insight about the local topology of relative outer space. Some applications of co-compactness are discussed. In particular 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. From this, we deduce that centralisers of elements of Out⁡(𝐹3) are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that the set Min⁡(𝜙) is always quasi-isometric to a line.