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Does exist open domain in $\mathbb C$ who has only identity for holomorphic automorphism?

Related question: does exist open domain in $\mathbb C$ so that every holomorphic automorphism has fixed point?

These questions were inspired by (much easier!) question by mathmos6.

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This is the intended link: math.stackexchange.com/questions/16640/… –  Hans Lundmark Jan 7 '11 at 16:08
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2 Answers 2

up vote 6 down vote accepted

Sure. Take $\mathbb{CP}^1$ and delete $5$ points $(0,1,\infty, s, t)$ where $s$ and $t$ are generic. Any automorphism of this domain will extend to an automorphism of $\mathbb{CP}^1$, by Riemann's extension theorem. That automorphism must permute the five points $(0, 1, \infty, s, t)$. For generic $s$ and $t$, there is no nontrivial map in $\mathbb{P}GL_2(\mathbb{C})$ which permutes these points.

If I deleted three points, there would be an $S_3$ of such maps; if I deleted four points, the corresponding group is $\mathbb{Z}/2 \times \mathbb{Z}/2$. So, if you are happy with getting a finite symmetry group, then $3$ points are enough.

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Steven Krantz has studied the automorphism group of domains, and has authored or coauthored some papers and surveys on the subject. In particular, Chapter 12 of his recent book "Geometric Function Theory" deals precisely with this topic and provides additional references. (It is a very nice book, by the way!)

Here are some general comments, they all come from this Chapter:

  1. The automorphism group of a domain is a Lie Group (this is proved in Kobayashi's "Hyperbolic manifolds and holomorphic mappings"), so it makes sense to talk about its dimension.

  2. The automorphism group of the disk is 3-dimensional, and 3 is the largest dimension possible.

  3. The only bounded domain whose automorphism group is 1-dimensional is the annulus.

  4. A domain with "very many holes" is 0-dimensional.

  5. The automorphism group of a domain with at least 2 but only finitely many holes is finite.

This last case, which includes your question, was studied by M. Heins in two papers,

"A note on a theorem of Radó concerning the $(1,m)$ conformal maps of a multiply connected region into itself", Bulletin of the American Mathematical Society 47 (1941), 128-130.

"On the number of 1-1 directly conformal maps which a multiply-connected plane region of finite connectivity $p$ ($>2$) admits onto itself", Bulletin of the American Mathematical Society 52 (1946), 454-457.

Heins found sharp bounds $N(k)$ for the size of the automorphism group of a domain $\Omega_k$ with precisely $k\ge 2$ holes:

  • $N(k)=2k$ if $k\ne 4,6,8,12,20$;
  • $N(4)=12$;
  • $N(6)=N(8)=24$;
  • $N(12)=N(20)=60$.

In an exercise, Krantz describes a domain with trivial automorphism group: Start with the "box" $\{\zeta\in{\mathbb C}\mid|{\rm Re}\zeta|<2,|{\rm Im}\zeta|<2\}$, remove the four closed disks of radius $0.1$ and centered at $\pm1\pm i$, and "perturb one of the holes" by 0.1.

Finally, let me quote a paragraph at the end of the chapter:

A very interesting open problem is to determine which finite groups arise as the automorphism groups of planar domains (there are some results for finitely connected regions). It is known that if $G$ is a compact Lie group, then there is some smoothly bounded domain in some ${\mathbb C}^n$ with automorphism group equal to $G$. But it is difficult to say how large $n$ must be in terms of elementary properties of the group $G$.

Krantz closes by mentioning two references for the above:

Bedford-Dadok, "Bounded domains with prescribed group of automorphisms", Comment. Math. Helv. 62 (1987), 561-572.

Saerens-Zame, "The isometry groups of manifolds and the automorphism groups of domains", Trans. Amer. Math. Soc. 301 (1987), 413-429.

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