Torsion-free nilpotent groups with balanced presentations and Hirsch length >3 have β1(G;ℚ)=2, with one example at h=4, none at h=5, and a construction at h=6 where β2=β1 over every field.
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Nilpotent groups with balanced presentations
Torsion-free nilpotent groups with balanced presentations and Hirsch length >3 have β1(G;ℚ)=2, with one example at h=4, none at h=5, and a construction at h=6 where β2=β1 over every field.