For aspherical normal complex algebraic varieties, a virtually nilpotent fundamental group is forced to be virtually two-step nilpotent.
On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven.arXiv preprint arXiv:2510.09026,
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Topology of higher Albanese maps and aspherical varieties with nilpotent fundamental group
For aspherical normal complex algebraic varieties, a virtually nilpotent fundamental group is forced to be virtually two-step nilpotent.