Thickness of $\mathsf{Out}(A_1\ast ...\ast A_n)$

Abstract

In this paper we have examined hyperbolicity and relative hyperbolicity of ${\Gamma }_n:=\mathsf{Out}(G_n)$ , where $G_n=A_1\ast ...\ast A_n$, is a finite free product and each $A_i$ is a finite group. We have used the $\mathsf{Out}(G_n)$ action on the Guirardel-Levitt deformation space, to find a virtual generating set and prove quasi isometric embedding of a large class of subgroups. We have used ideas from works of Mosher-Handel and Alibegović to prove non-distortion. We have used these subgroups to prove that ${\Gamma }_n$ is thick for higher complexities. Thickness was developed by Behrstock-Druţu-Mosher and thickness implies that the groups are non-relatively hyperbolic.