On complex exponential groups (Q1000633)

The main result is that a connected complex Lie group has surjective exponential function if and only if its center is connected and its adjoint group has surjective exponential function. This rounds off a number of partial results obtained in several articles by Lai, Wüstner, and Chatterjee. The theorem generalizes a result of the first author on algebraic groups [J. Algebra 186, No.~1, 20--31 (1996; Zbl 0866.22009)]. As a by-product of the proof, it is shown that the groups satisfying the above conditions are semi-algebraic, which means that the radical of their Lie algebra splits as a semidirect sum of an abelian subalgebra centralized by a Levi factor and a nilpotent ideal on which the action of the abelian summand is semisimple. The proof involves a thorough structural analysis and stepwise reduction to more and more special cases.