Realizing Countable Groups

As Automorphism Groups

of Riemann Surfaces

Jörg Winkelmann

Received: July 19, 2001

Communicated by Ulf Rehmann

Abstract. Every countable group can be realized as the full automorphism group of a Riemann surface as well as the full group of isometries of a Riemannian manifold.

2000 Mathematics Subject Classification: Primary 30F99, Secondary 20F29,20B27,32M05 Keywords and Phrases: Countable Groups, Full automorphism group, Riemann surface

Acknowledgement. The author whishes to express his gratitude to the University of Tokyo. This manuscript was written and is based on work done during the stay of the author at the University of Tokyo as visiting Associate Professor. Saerens and Zame [4], and independently Bedford and Dadok [2] proved that every compact real Lie group K can be realized as the group of holomorphic automorphisms of a complex manifold as well as the group of isometries of a Riemannian manifold. Here we deduce a similar result for countable discrete groups. Thus the purpose of this paper is to prove the following theorem:

Theorem. Let G be a (finite or infinite) countable group. Then there exists a (connected) Riemann surface M such that G is isomorphic to the group Aut_{O}(M) of all holomorphic automorphisms of M. Moreover, there exists a Riemannian metric h on M such that Aut_{O}(M) equals the group of all isometries of (M, h).

Our strategy is as follows: Using Galois theory of coverings, we first construct a Riemann surface M_{1} on which G acts. Then we remove a discrete subset S ⊂ M_{1} to kill excess automorphisms. However, we have to show that passing from M_{1} to M_{1} \ S we do not risk enlarging the automorphism group, i.e., we will show that every automorphism from M_{1} \ S extends to M_{1}. For this purpose we employ the Freudenthal’s theory of topological ends. Finally, hyperbolicity of the Riemann surface is exploited to ensure that there is a hermitian metric of constant negative curvature such that the group of all holomorphic automorphisms coincides with the group of all isometries. Let us remark that by uniformization theory it is well-known that the following is the list of all Riemann surface with positive-dimensional automorphism group and that their automorphism groups are well-known:

• ℙ_{1}(ℂ),

• ℂ,

• ℂ^{∗},

• H^{+} = {z ∈ ℂ : ℑ(z) > 0},

• E_{τ} = ℂ/ ⟨1, τ⟩_{ℤ} with τ ∈ H^{+},

• A(r, 1) = {z ∈ ℂ : r < |z| < 1} with 0 ≤ r < +∞.

(See [1] for this and other basic facts on Riemann surfaces.) Therefore our result yields a complete characterization which groups may occur as automorphism group of a Riemann surface. The above list furthermore has the following consequence which we will use later on:

Fact. Let M be a Riemann surface with non-commutative fundamental group π_{1}(M). Then Aut_{O}(M) is discrete and acting properly discontinuously on M. In particular, every orbit is closed.

1 Hyperbolic Riemannian Surfaces

A Riemannian surface M is “hyperbolic” (in the sense of Kobayashi) if and only if its universal covering is isomorphic to the unit disk. In this case the Poincaré metric on the unit disk induces a unique hermitian metric of constant Gaussian curvature −1 on M. Every holomorphic automorphism of M is an isometry and conversely every isometry is either holomorphic or antiholomorphic. Thus the group of holomorphic automorphisms of M is a subgroup of index 1 or 2 in the group of isometries and the group of isometries coincides with the group of all holomorphic or antiholomorphic diffeomorphisms of M.

2 Galois theory of coverings

Proposition 1. Let G be a countable group. Let M_{0} be a Riemann surface whose fundamental group is not finitely generated. Then there exists an unramified covering M_{1} → M_{0} such that there is an effective G-action on M_{1}, Aut_{O}(M_{1}) is discrete and acting properly discontinuously on M_{1}.

Proof. Let F_{∞} be a free group with countably infinitely many generators α_{1}, α_{2}, . . .. By standard results on Riemann surfaces (see e.g. [1]) we have π_{1}(M_{0}) ≃ F_{∞}. Since G is countable, there is a surjective group homomorphism ζ : F_{∞} → G. Furthermore, we may require that α_{1}, α_{2} ∈ ker ζ. Then we obtain a short exact sequence of groups

{e} → N → F_{∞} →^{ζ} G → {e}

where N is non-commutative, because it contains a free group with two generators (viz. α_{1} and α_{2}). By Galois theory of coverings, this implies that there exists an unramified covering M_{1} → M_{0} with π_{1}(M_{1}) ≃ N and an effective G-action on M_{1}. Finally, discreteness of Aut_{O}(M_{1}) as well as the action of Aut_{O}(M_{1}) being properly discontinuous is implied by the “Fact” established above.

3 Topological ends

Let us recall the basic facts from the theory of ends as developed by Freudenthal [3]. Let X be a locally compact topological space. Then the set of “ends” e(X) is defined by

e(X) = lim_{K} π_{0}(X \ K).

Thus, if K_{n} is an exhaustion of X by an increasing sequence of compact subsets, then every end ϵ ∈ e(X) can be represented by a sequence U_{n} of connected components of X \ K_{n} with U_{n} ⊃ U_{n+1}. For every connected component W_{n,i} of X \ K_{n} we now define E_{n,i} as the set of ends ϵ which can be represented with U_{n} = W_{n,i}. Now X̄ = X ∪ e(X) becomes a compact topological space as follows: As a basis of the topology we take the family of all open subsets of X together with V_{n,i} = W_{n,i} ∪ E_{n,i} for all n, i. Then every proper continuous map between locally compact topological spaces X and Y extends to a continuous maps between X̄ and Ȳ. In particular, every homeomorphism of X extends to a homeomorphism of X̄.

Definition. An end ϵ of a Riemann surface X is called a “puncture” if there is an open neighbourhood W of ϵ in X̄ such that

• W \ {ϵ} ⊂ X and

• there is a homeomorphism ξ : W → D = {z ∈ ℂ : |z| < 1} such that ξ|_{W \{ϵ}} is holomorphic.

We now prove that the ends of a certain special class of Riemann surfaces cannot be punctures.

Proposition 2. Let a_{n} be a diverging sequence in D = {z ∈ ℂ : |z| < 1} and r_{n} ∈ ℝ^{>0} such that all the closed balls B_{n} = {z ∈ ℂ : |z −a_{n}| ≤ r_{n}} are disjoint subsets of D. Then A = ∪_{n}B_{n} is a closed subset of D.

Let X_{0} = D \ A and let X_{1} → X_{0} be an unramified covering. Then no end of X_{1} is a puncture.

Proof. First we show that A is indeed closed. Let vol(B_{n}) = πr_{n}^{2} denote the euclidean volume. Since the balls B_{n} are disjoint, we have ∑ vol(B_{n}) ≤ π and therefore lim r_{n} = 0. It follows that for every s < 1 there is a natural number N such that |a_{n}| − r_{n} > s for all n ≥ N. Therefore A = ∪_{n}B_{n} is actually a locally finite union, and closedness of the B_{n} implies that A is closed. Now let us assume that there exists an end which is a puncture. The natural embeddings X_{0} ↪ D ↪ ℂ composed with the projection π : X_{1} → X_{0} yield a bounded holomorphic function f on X_{1}. Let ϵ ∈ e(X_{1}) be a puncture with a connected open neighbourhood W as in the above definition of the notion “puncture”. Then the Riemann extension theorem implies that f extends through ϵ. In other words, lim_{x→ϵ} f(x) = a exists. Evidently a is contained in the closure of X_{0} in ℂ. However, the boundary ∂X_{0} is given by ∂X_{0} = ∂D ∪ (∪_{n}∂B_{n}), and the openness of holomorphic maps implies that a cannot lie on either ∂D or on one of the sets ∂B_{n}. Therefore a ∈ X_{0}. Now choose contractible open neighbourhoods U of a in X_{0} and V of ϵ in W such that V \ {ϵ} ⊂ π^{−1}(U). Since π : X_{1} → X_{0} is an unramified covering and U is simply-connected, we obtain π^{−1}(U) ≃ G × U where G is equipped with the discrete topology. Being connected, V^{∗} = V \ {ϵ} must be contained in one connected component of π^{−1}(U). This implies the following: If a_{n} is a sequence in V^{∗} such that lim π(a_{n}) = a, then there is a point p ∈ X_{1} with lim a_{n} = p. But this contradicts the fact that by construction there is sequence a_{n} in V^{∗} with lim a_{n} = ϵ ∉ X_{1} and lim π(a_{n}) = a. Thus this case can be ruled out as well, i.e., there cannot exist an end which is a puncture.

4 Proof of the theorem

Proof. Let a_{n} be a diverging sequence in D = {z ∈ ℂ : |z| < 1} and r_{n} ∈ ℝ^{>0} such that all the closed balls B_{n} = {z ∈ ℂ : |z−a_{n}| ≤ r_{n}} are disjoint subsets of D. Let A = ∪_{n}B_{n}. Then A is a closed subset of D and the fundamental group of X_{0} = D\A is not finitely generated. Hence, by prop. 1 there is an unramified covering π : X_{1} → X_{0} with an effective G-action on X_{1}. Let Aut_{O}(X_{1}) denote the group of all holomorphic automorphisms of X_{1} and A the group of all diffeomorphisms of X_{1} which are either holomorphic or antiholomorphic. Again by prop. 1 we may assume that Aut_{O}(X_{1}) is discrete and acting properly discontinuously. Since Aut_{O}(X_{1}) is of finite index in A, the group A is likewise discrete and acting properly discontinuously on X_{1}. Now, for every g ∈ A \ {e} the fixed point set X_{1}^{g} = {x ∈ X_{1} : g(x) = x} is a nowhere dense real analytic subset of X_{1}. Hence Σ = ∪_{g∈A\{e}}X_{1}^{g} is a set of measure zero. In particular, Σ ≠ X_{1}. Let p ∈ X_{1} \ Σ, S = G(p) and X = X_{1} \ S. Note that the conditions p ∈ X_{1} \Σ, S = G(p) imply that g(p) ∉ S if g ∈ A\G.

Therefore

G = {g ∈ A : g(S) = S}.

Let h be the unique hermitian metric of constant Gaussian curvature −1 on X and I its isomorphism group. We claim that I = Aut_{O}(X) ≃ G. To show this, it suffices to show that every holomorphic or antiholomorphic automorphism of X extends to a holomorphic or antiholomorphic automorphism of X_{1}. If ϕ is a holomorphic or antiholomorphic automorphism of X, it is in particular a selfhomeomorphism and therefore extends to a homeomorphism ϕ̄ of the compact topological space X̄ = X ∪ e(X) (where e(X) is the set of ends as explained in §3. above). Now e(X) = e(X_{1}) ∪ S. Evidently every end of X given by a point of S is a puncture as defined in §3. On the other hand, due to prop. 2 none of the ends of X_{1} is a puncture. Now ϕ̄|_{e(X)} is a permutation of the elements of e(X) which stabilizes the set of those ends which are punctures. Hence ϕ̄(S) = S. Thus ϕ extends to a continuous self-map of X_{1} = X ∪ S. However, a continuous map which is holomorphic or antiholomorphic everywhere except for some isolated points, is necessarily holomorphic resp. antiholomorphic everywhere (by Riemann extension theorem). Hence every ϕ ∈ I of X extends to a holomorphic or antiholomorphic automorphism of X_{1}. Consequently

Aut_{O}(X) = I = {g ∈ A : g(S) = S} = G.

References

[1] Ahlfors, L.; Sario, E.: Riemann Surfaces. Princeton Mathematical Series, No. 26 Princeton University Press, Princeton, N.J. (1960)

[2] Bedford, E.; Dadok, J.: Bounded domains with prescribed group of automorphisms. Comm. Math. Helv. 62 , 561–572 (1987)

[3] Freudenthal, H.: Über die Enden topologischer Räume und Gruppen. Math. Z. 33, 692–713 (1931)

[4] Saerens, R.; Zame, W.R.: The Isometry Groups of Manifolds and the Automorphism Groups of Domains. Trans. A.M.S. 301, no. 1, 413-429 (1987)

Jörg Winkelmann

Korean Institute for Advanced Studies 207-43 Cheongryangri-dong Dongdaemun-gu Seoul 130-012 Korea jwinkel@member.ams.org 418

Documenta Mathematica 7 (2002)