A simply connected solvable Lie group with a left-invariant complex structure is shown to be non-Stein, providing a counterexample to Hasegawa's 2010 conjecture.
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Simply connected nilpotent Lie groups of dimension 2n with left-invariant complex structures are biholomorphic to C^n.
Simply connected nilpotent Lie groups with left-invariant nilpotent complex structures are biholomorphic to C^n by polynomial maps, so their lattices act freely and cocompactly on C^n by holomorphic polynomial automorphisms.
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A simply connected solvable Lie group with a left-invariant complex structure is shown to be non-Stein, providing a counterexample to Hasegawa's 2010 conjecture.
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Simply connected nilpotent Lie groups of dimension 2n with left-invariant complex structures are biholomorphic to C^n.
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Simply connected nilpotent Lie groups with left-invariant nilpotent complex structures are biholomorphic to C^n by polynomial maps, so their lattices act freely and cocompactly on C^n by holomorphic polynomial automorphisms.