Let $S$ be a non-orientable closed surface with $b_{1}(S)=2$, is $\pi_{1}(S)$ a Kahler group?
2026-03-27 16:03:56.1774627436
Is this group a Kahler group?
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No: https://mathscinet.ams.org/mathscinet-getitem?mr=3033516
It follows from the Biswas-Mj result that a one-relator Kahler group has no torsion in $G^{ab}$. But your group is a one-relator group with torsion in $G^{ab}$.