We study smooth complex hypersurfaces in a direct product of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental group. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Gromov's question of which subgroups of a direct product of surface groups are K\"ahler for two classes: subdirect products of three surface groups; and subgroups arising as kernel of a homomorphism from the product of surface groups to $\mathbb{Z}^3$.
Three manifold groups, Kaehler groups and complex surfaces
Kaehlerian three-manifold groups
On normal subgroups in the fundamental groups of complex surfaces
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