Modular Embeddings and Rigidity for Fuchsian Groups

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$ are two semi-arithmetic lattices in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\text{\hspace{0.17em}⁣}:{\mathrm{\Gamma }}_1\to {\mathrm{\Gamma }}_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automorphism of $\mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{R})$.