Singular value gap estimates for free products of semigroups

We establish lower estimates for singular value gaps of free products of $1$-divergent semigroups $\Gamma_1,\Gamma_2\subset \mathsf{GL}_d(\mathbb{K})$ which are in ping-pong position. As an application, we prove that if $\Gamma_1$ and $\Gamma_2$ are quasi-isometrically embedded subgroups in ping pong position, then the group they generate $\langle \Gamma_1,\Gamma_2\rangle$ is also quasi-isometrically embedded. In addition, we establish that the class of linear finitely generated groups, admitting a faithful linear representation over $\mathbb{R}$ which is a quasi-isometric embedding, is closed under free products.