# Commentarii Mathematici Helvetici

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**Volume 82, Issue 2, 2007, pp. 371–383**

**DOI: 10.4171/CMH/95**

Published online: 2007-06-30

Complete hyperbolic Stein manifolds with prescribed automorphism groups

Su-Jen Kan^{[1]}(1) Academia Sinica, Taipei, Taiwan

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.

*Keywords: *Hyperbolic manifolds, Stein manifolds, automorphism groups

Kan Su-Jen: Complete hyperbolic Stein manifolds with prescribed automorphism groups. *Comment. Math. Helv.* 82 (2007), 371-383. doi: 10.4171/CMH/95