Irreducible laminations for IWIP Automorphisms of free products and Centralisers

Abstract: For every free product decomposition $G = G_{1} \ast ...\ast G_{q} \ast F_{r}$, where $F_r$ is a finitely generated free group, of a group $G$ of finite Kurosh rank, we can associate some (relative) outer space $\mathcal{O}$. In this paper, we develop the theory of (stable) laminations for (relative) irreducible with irreducible powers (IWIP) automorphisms. In particular, we examine the action of $Out(G, \mathcal{O}) \leq Out(G)$ (i.e. the automorphisms which preserve the set of conjugacy classes of $G_i$'s) on the set of laminations. We generalise the theory of the attractive laminations associated to automorphisms of finitely generated free groups. The strategy is the same as in the classical case (see \cite{BFM}), but some statements are slightly different because of the existence of the $G_i$'s. More precisely, we prove that the stabiliser of the lamination of a relative IWIP is a $\mathbb{Z}$-extension of a subgroup that is consisted of virtually elliptic automorphisms. Note that in the free case, virtually elliptic automorphisms are exactly the finite order automorphisms of $Out(F_n)$. As a corollary of the previous theorem, we generalise the fact that the centraliser of an IWIP automorphism of a free group, is virtually cyclic. As a direct corollary, if $Out(G)$ is virtually torsion free and every $Aut(G_i)$ is finite, we prove that the centraliser of an IWIP is virtually cyclic. Finally, we give an example which shows that we cannot expect that in general the centraliser of a relative IWIP (and as a consequence the stabiliser of its stable lamination) is virtually cyclic.