Finitely Presented Normal Subgroups of a Product of Fuchsian Groups

We show that if Γ is a finitely presented normal subgroup of a product G1 × G2 of Fuchsian groups which projects nontrivially to each factor, then Γ has finite index in G1 × G2. The proof uses the authorʹs previous reduction of the description of normal subdirect products to the abelian case and an unpublished result of N. C. Carr on the rigidity of normal subdirect products of diagonal rank one.