Irreducible laminations for IWIP Automorphisms of free products and Centralisers

Dionysios Syrigos

Published 2014-10-17Version 1

Let $G = G_{1} \ast ...\ast G_{q} \ast F_{r}$ be a group which splits as free product, where $F_r$ is a finitely generated free group. For every such decomposition, we can associate some (relative) outer space $\mathcal{O}$. In this paper we develop the theory of irreducible laminations for free products of groups. In particular, we examine the action of $Out(G, \mathcal{O}) \leq Out(G)$ (of automorphisms which preserve the conjugacy classes of $G_i$'s) on the set of laminations. We generalise the theory of irreducible laminations corresponding to finitely generated free groups. The strategy is the same as in the classical case, but some statements are slightly different because of the non-trivial kernel of the action. As a corollary, we prove that the centraliser of an IWIP modulo its intersection with the kernel of the action is virtually cyclic, which is a generalisation of a well-known theorem in the free case.