Singular value gap estimates for free products of semigroups

We establish lower estimates for singular value gaps of free products of $1$-divergent semigroups ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\subset {\mathsf{G}\mathsf{L}}_d(\mathbb{K})$ which are in ping-pong position. As an application, we prove that if ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are quasi-isometrically embedded subgroups in ping pong position, then the group they generate $⟨{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2⟩$ 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.