Torsion-free elementary amenable groups of Hirsch length ≤3 are solvable of derived length ≤3; solvable groups of cohomological dimension 3 are polycyclic, BS(1,n)⋊ℤ, or properly ascending HNN extensions over ℤ² or π₁(Kb).
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Elementary amenable groups of cohomological dimension 3
Torsion-free elementary amenable groups of Hirsch length ≤3 are solvable of derived length ≤3; solvable groups of cohomological dimension 3 are polycyclic, BS(1,n)⋊ℤ, or properly ascending HNN extensions over ℤ² or π₁(Kb).