Complete hyperbolic Stein manifolds with prescribed automorphism groups

It is well known that the automorphism group of a hyperbolic manifold is a Lie group. Conversely, it is interesting to see whether or not any Lie group can be prescribed as the automorphism group of a certain complex manifold. When the Lie group G is compact and connected, this problem has been completely solved by Bedford?Dadok and independently by Saerens?Zame in 1987. They have constructed strictly pseudoconvex bounded domains O such that Aut(O)?=?G. For Bedford?Dadok?s O, 0?=?dimCO?-?dimRG?=?1; for generic Saerens?Zame?s O, dimCO?»?dimRG. J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds O such that Aut(O)??=?G. In his construction, it is typical that dimCO?»?dimRG. In this article, we tackle this problem from a different aspect. We prove that for any connected Lie group G (compact or noncompact), there exist complete hyperbolic Stein manifolds O such that Aut(O)?=?G with dimCO?=?dimRG. Working on a natural complexification of the real-analytic manifold G, our construction of O is geometrically concrete and elementary in nature