  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
could be prescribed asthe automorphism group of certain complex manifold.
Whenthe Lie group $G$ is compact and connected, this problem has been
completelysolved by Bedford-Dadok and independently by Saerens-Zame on 1987.
Theyhave constructed \spc bounded domains $\Omega$ such that $Aut(\Omega)=G$.
For Bedford-Dadok's $\Omega, 0\le dim_{\Bbb C}\Omega- dim_{\Bbb R}G\le 1$; for
generic Saerens-Zame's$\Omega,dim_{\Bbb C}\Omega \gg dim_{\Bbb R}G$.J.
Winkelmann has answered affirmatively to noncompact connected Liegroups in
recent years. He showed there exist Stein complete hyperbolic manifolds
$\Omega$ such that $Aut(\Omega)=G$.In his construction, it is typical that
$dim_{\Bbb C}\Omega\gg dim_{\Bbb R}G$.In this article, we tackle this problem
from a different aspect. We provethat for any connected Lie group $G$ (compact
or noncompact), there exist completehyperbolic Stein manifolds $\Omega$ such
that $Aut(\Omega)=G$ with $dim_{\BbbC}\Omega=dim_{\Bbb R}G.$ Working on a
natural complexification of the real-analyticmanifold $G$, our construction of
$\Omega$ is geometrically concrete andelementary in nature.
