# Complete hyperbolic Stein manifolds with prescribed automorphism groups

### Su-Jen Kan

Academia Sinica, Taipei, Taiwan

## Abstract

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 $Ω$ such that $Aut(Ω) =G$. For Bedford–Dadok’s $Ω$, $0≤dim_{C}Ω−dim_{R}G ≤1$; for generic Saerens–Zame’s $Ω$, $dim_{C}Ω≫dim_{R}G$.

J. Winkelmann has answered affirmatively to noncompact connected Lie groups in recent years. He showed there exist Stein complete hyperbolic manifolds $Ω$ such that $Aut(Ω) =G$. In his construction, it is typical that $dim_{C}Ω≫dim_{R}G$.

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 $Ω$ such that $Aut(Ω) =G$ with $dim_{C}Ω=dim_{R}G$. Working on a natural complexification of the real-analytic manifold $G$, our construction of $Ω$ is geometrically concrete and elementary in nature.

## Cite this article

Su-Jen Kan, Complete hyperbolic Stein manifolds with prescribed automorphism groups. Comment. Math. Helv. 82 (2007), no. 2, pp. 371–383

DOI 10.4171/CMH/95