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

Abstract

In this paper we have examined hyperbolicity and relative hyperbolicity of $\Gamma_n := \mathsf{Out}(G_n)$ , where $G_n = A_1*...*A_n$, 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 Alibegovi\'c to prove non-distortion. We have used these subgroups to prove that $\Gamma_n$ is thick for higher complexities. Thickness was developed by Behrstock-Dru\c{t}u-Mosher and thickness implies that the groups are non-relatively hyperbolic.