Finitely presented normal subgroups of a product of Fuchsian groups (Q1584602)

The question of whether or not the Baumslag-Roseblade theorem extends to finitely presented subgroups of a direct product of two Fuchsian groups is answered for normal subgroups. It is shown that if \(\Gamma\) is a finitely presented normal subgroup of a product \(G_1\times G_2\) of Fuchsian groups which projects non-trivially onto each factor, then \(\Gamma\) has finite index in \(G_1\times G_2\). In the proof, the author uses a previous paper of himself [Proc. Edinb. Math. Soc., II. Ser. 33, No. 2, 309-319 (1990; Zbl 0704.20022)] concerning the description of normal subdirect products in the Abelian case and also an unpublished result of N. C. Carr on the rigidity of normal subdirect products of diagonal rank one. The main result of the paper is the following: Theorem A. Let \(\Gamma\) be finitely presented normal subgroup of a product \(G_1\times G_2\) of Fuchsian groups; then either (i) \(\Gamma\) has finite index in \(G_1\times G_2\), or (ii) \(\Gamma\) is contained, with finite index, in one of the factors.