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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 freeproduct 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 generatingset and prove quasi isometric embedding of a large class of subgroups. We haveused ideas from works of Mosher-Handel and Alibegovi\'c to provenon-distortion. We have used these subgroups to prove that $\Gamma_n$ is thickfor higher complexities. Thickness was developed by Behrstock-Dru\c{t}u-Mosherand thickness implies that the groups are non-relatively hyperbolic.

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

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