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.