Abstract: The Maximum Principle in one complex variable implies that every holomorphic function on any compact complex manifold is constant. One can then ask the question: which non-compact complex manifolds have no non-constant holomorphic functions? In full generality this is difficult to answer. However, if one ask this of complex Lie groups, then one is considering Cousin groups (also called toroidal groups). It turns out that these groups play a central role in the structure theory of some non-compact homogeneous complex manifolds - a subtitle of the talk could be: "How I came to know and love Cousin groups".

In our talk we recall the notions of Lie groups, Lie algebras, and the exponential maps between them. We will show how to construct one particular Cousin group and prove all holomorphic functions on it are constant by using Liouville's theorem, the density of one set in another, and the Identity Principle. From this construction one sees what the structure of all Cousin groups must be and can then classify Abelian complex Lie groups (they are direct products of copies of $\mathbb C$ and $\mathbb C^*$ with Cousin groups). We also define and investigate properties of holomorphic reductions of complex Lie groups and complex homogeneous spaces (with examples, particularly in the nilpotent and solvable cases). In an analogous way one can get reductions relative to other analytic objects on our manifolds, e.g., bounded holomorphic functions and analytic hypersurfaces.