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Modular embeddings and rigidity for Fuchsian groups. (English) Zbl 1325.20044
Summary: We prove a rigidity theorem for semiarithmetic Fuchsian groups: If $$\Gamma_1$$, $$\Gamma_2$$ are two semiarithmetic lattices in $$\text{PSL}(2,\mathbb R)$$ virtually admitting modular embeddings, and $$f\colon\Gamma_1\to\Gamma_2$$ is a group isomorphism that respects the notion of congruence subgroups, then $$f$$ is induced by an inner automorphism of $$\text{PGL}(2,\mathbb R)$$.
Reviewer: Reviewer (Berlin)

##### MSC:
 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 11F06 Structure of modular groups and generalizations; arithmetic groups 22E40 Discrete subgroups of Lie groups 14G35 Modular and Shimura varieties
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