Modular Embeddings and Rigidity for Fuchsian Groups

Abstract

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