maskit.dvi CUBO A Mathematical Journal Vol.12, No¯ 01, (175–179). March 2010 A Short Note On M-Symmetric Hyperelliptic Riemann Surfaces ∗ Rubén A. Hidalgo Departamento de Matemáticas, Universidad Técnica Federico Santa Maŕıa, Valparáıso, Chile email : ruben.hidalgo@usm.cl ABSTRACT We provide an argument, based on Schottky groups, of a result due to B. Maskit which states a necessary and sufficient condition for the double oriented cover of a planar compact Klein surface of algebraic genus at least two to be a hyperelliptic Riemann surface. RESUMEN Damos un argumento, basado en grupos de Schottky, de un resultado debido a B. Maskit el cual establece una condición necesária y suficiente para el cubrimiento duplo orientado de una superficie de Klein compacta planar de genero algebrico al menos dos ser una superficie de Riemann hipereliptica. Key words and phrases: Schottky groups, Hyperelliptic Riemann surfaces. Math. Subj. Class.: 30F10, 30F40. ∗Partially supported by projects Fondecyt 1070271 and UTFSM 12.09.02 176 Rubén A. Hidalgo CUBO 12, 1 (2010) 1 Preliminaries Let us consider a collection of (g + 1) pairwise disjoint round circles on the Riemann sphere, say C1,..., Cg+1, bounding a common domain D of connectivity (g +1). If we denote by τj the reflection on the circle Cj , then the group G = 〈τ1, ..., τg+1〉 is an extended Kleinian group, isomorphic to the free product of (g + 1) copies of Z2. We say that G is a planar extended Schottky group of rank g. The region of discontinuity Ω of a planar extended Schottky group of rank g is connected (the complement of a Cantor set for g ≥ 2) and S = Ω/G is a planar compact Klein surface of algebraic genus g, that is, holomorphically equivalent to the closure of D. Quasiconformal deformation theory asserts that every planar compact Klein surface of algebraic genus g is obtained in this way. Let G = 〈τ1, ..., τg+1〉 a planar extended Schottky group of rank g. Let G + be its index two subgroup of orientation preserving transformations. It turns out that G+ is a (classical) Schottky group of genus g, freely generated by the transformations aj = τg+1τj , for j = 1, ..., g. The closed Riemann surface S+ = Ω/G+ is the double oriented cover of the planar compact Klein surface S = Ω/G. Any of the transformation in G − G+ induces an anticonformal involution τ : S+ → S+ (that is, a real structure on S+) so that S = S+/〈τ〉. It follows that the number of ovals of τ (its connected components of fixed points) is equal to (g + 1), in particular, (S+, τ ) is a M -symmetric Riemann surface. Let us denote by π : S+ → S the two-fold (branched) Klein cover induced by τ and by P : Ω → S+ the Schottky covering of S+ induced by the Schottky group G+. In [4] B. Maskit proved the following result. Theorem 1.1. Let G be a planar extended Schottky group of rank g ≥ 2, defined by tye circles C1,..., Cg+1. Then the Riemann surface Ω/G + is hyperelliptic if and only if there is a circle which is orthogonal to all Cj , j = 1, ..., g + 1. The aim of this note is to provide a different proof of Theorem 1.1 relaying more on the Schottky groups spirit. We need to recall some extra definitions. Let Σ1,... Σg+1 be pairwise disjoint simple loops on the Riemann sphere, all of them bounding a common domain D of connectivity g + 1. Assume that for each j = 1, ..., g + 1, there is a Möbius transformation of order 2, say Ej , so that Ej permutes both topological discs discs bounded by Σj (in particular, both fixed points of Ej belong to Σj ). The group K = 〈E1, ..., Eg+1〉 is a Kleinian group, isomorphic to a free product of g + 1 copies of Z2, called a Whittaker group of rank g [3]. If Ω is the region of discontinuity of K, then Ω is connected (the complement of a Cantor set for g ≥ 2) and S = Ω/K is an orbifold of signature (0, 2g + 2; 2, ..., 2), that is, the Riemann sphere with exactly 2(g + 1) conical points, all of them of conical order 2. Inside K there is exactly one index two torsion free subgroup, say K(2). It turns out that K(2) is a Schottky group of rank g, called a hyperelliptic Schottky group, which is freely generated by the transformations Eg+1E1,..., Eg+1Eg. In this case, S (2) = Ω/K(2) turns out to CUBO 12, 1 (2010) A Short Note On M -Symmetric Hyperelliptic ... 177 be a hyperelliptic Riemann surface, the hyperellitic involution (unique for g ≥ 2 [2]) is induced by any of the transformations in K − K(2). The projection of the fixed points of E1,..., Eg+1 to S (2) provides the 2(g + 1) fixed points of the hyperelliptic involution. 2 The Necessary Part Let us consider a planar extended Schottky group G of rank g ≥ 2, say generated by the reflections τ1,..., τg+1 on a collection of (g+1) pairwise disjoint round circles on the Riemann sphere, say C1,..., Cg+1, bounding a common domain D of connectivity (g + 1). Let G + be the index two orientation preserving Schottky subgroup and let S+ = Ω/G+, where Ω is the region of discontinuity of G (the same as for G+). As before, we denote by τ : S+ → S+ the real structure induced on S+ by the action of G. Let us denote by O1 = P (C1),...., Og+1 = P (Cg+1) the ovals of τ . Let us assume S+ is a hyperelliptic Riemann surface and let j : S+ → S+ be its hyperelliptic involution. As the hyperelliptic involution is unique [2], j and τ should commute, in particular, the collection of ovals of τ is invariant under j. The Schottky group G+ is defined by the ovals O1,..., Og+1, that is, by the normalizer (in the fundamental group) of them. It follows that the hyperelliptic involution lifts to a conformal automorphism ĵ : Ω → Ω under P : Ω → S+, that is, jP = P ĵ. We have that ĵ2 ∈ G+. As j has fixed point, we may assume that ĵ also has fixed points, in particular, ĵ2 = I. It is known that Ω is of class OAD; that is, it admits no holomorphic function with finite Dirichlet norm (see [1, pg 241]). It follows from this (see [1, pg 200]) that every conformal map from Ω into the Riemann sphere is a Möbius transformation. In this way, ĵ is the restriction of a Möbius transformation of order two. Lemma 2.1. Each oval has exactly two fixed points of j and each fixed point belongs to some oval. Moreover, each oval is invariant under j. Proof. Let us denote by D1 and D2 the two connected components of S + − ∪ g+1 j=1Oj . If one of the fixed points of j is not contained in ∪ g+1 j=1Oj , then we should have that j(D1) = D1. But as D1 is planar (isomorphic to the closure of D) we will have that the restriction of j onto D1 coincides with a Möbius transformation of order 2. It will follows then that j must have at most 4 fixed points on S, a contradiction. In particular, every fixed point of j is contained in some oval. Also, if the oval Ok contains a fixed point of j, then we should have that j(Ok) = Ok. In that case, we have that Ok should have exactly two fixed points of j. As j contains exactly 2(g + 1) fixed points and we have exactly (g + 1) ovals, we have that: (i) each oval has exactly two fixed points of j and (ii) each oval is invariant unde! r j. By the previous lemma, for each k ∈ {1, ..., g + 1}, j(Ok) = Ok. It follows that we may choose liftings ĵ1,..., ĵg+1, of the hyperelliptic involution, each one of order 2 so that ĵk(Ck) = Ck and 178 Rubén A. Hidalgo CUBO 12, 1 (2010) both fixed point of ĵk are contained in Ck. Let us consider the Whittaker group Ĝ = 〈ĵ1, ..., ĵg+1〉, and its hyperelliptic Schottky group Ĝ(2) = 〈ĵg+1ĵ1, ...., ĵg+1ĵg〉. We have, by the construction, that G+ = Ĝ(2). Lemma 2.2. ĵg+1ĵk = τg+1τk, k = 1, ..., g. Proof. Let us first observe that the circles Ck, C ′ k = τg+1(Ck) and ĵg+1(Ck) are lifting of the oval Ok. The circles C1, C ′ 1,..., Cg, C ′ g (respectively, the circles C1, ĵg+1(C1),..., Cg and ĵg+1(Cg)) form a standard fundamental domain for G+. As ĵg+1(Ck) must belong to the disc bounded by the circle Cg+1 which does not contains the circle Ck, we should have that ĵg+1(Ck) should be one of the discs C ′ 1,...., C ′ g. But as mentioned, the only such disc which is a lifting of Ok is exactly C ′ k . We have that ĵg+1(Ck) = C ′ k . Now, we have that the loxodromic transformations ĵg+1ĵk, τg+1τk ∈ G + send Ck onto C ′ k and each maps the exterior of Ck onto the interior of C ′ k . It follows that the transformation η = τkτg+1ĵg+1ĵk ∈ G + keeps invariant the circle Ck and each of its bounded discs. As Ck is contained on the region of discontinuity of G+, it follows that η cannot be loxodromic. As G+ only contains loxodromic transformations besides the identity, we should have η = I. Let us recall that, if C is a circle on the Riemann sphere and p and q are any two different points on it, then there is a unique orthogonal circle to it passing through these two given points. The previous fact together with the fact that a circle is uniquely determined by 3 points on it and the following lemma asserts the existence of a common orthogonal circle as desired to prove the necessary part of the theorem. Lemma 2.3. Let us consider two pairwise disjoint circles, say C1 and C2. Let σj be the reflection of Cj and tj be an elliptic transformation of order 2 preserving Cj whose fixed points belong to Cj . Then σ2σ1 = t2t1 if and only if there is a circle C such that: (i) the fixed points of t1 and t2 belong to C and (ii) C is orthogonal to both C1 and C2. Proof. We may normalize by a suitable Möbius transformation in order to assume that C1 is the unit circle and C2 is the circle centered at the origin and a positive radius r > 1. In this case we CUBO 12, 1 (2010) A Short Note On M -Symmetric Hyperelliptic ... 179 have that σ2σ1(z) = r 2z. The equality t2t1 = σ2σ1 then obligates to have that the fixed points of both t1 and t2 on a line through 0. 3 The Sufficiency Part Let us assume we have (g + 1) circles, say C1,..., Cg+1, each one of them orthogonal to a common circle C0. Let us denote by τk the reflection on the circle Ck, for k = 0, 1, .., g + 1. Let us denote by ηk = τ0τk, for k = 1, ..., g + 1, which are elliptic transformations of order 2. Let G be the planar extended Schottky group generated by the reflections τ1,..., τg+1 and let Ĝ be the Whittaker group generated by the involutions η1,..., ηg+1. It easy to see that G + is the hyperelliptic subgroup of Ĝ. It follows then that the uniformized surface by G+ is hyperelliptic, with hyperelliptic involution induced by η0. Received: September 2008. Revised: January 2009. References [1] Ahlfors, L. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton NJ, 1960. [2] Farkas, H. and Kra, I., Riemann Surfaces, Second edition. Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992. [3] Keen, L., On Hyperelliptic Schottky groups, Ann. Acad. Sci. Fenn. Series A.I. Mathematica, 5, 1980. [4] Maskit, B., Remarks on m-symmetric Riemann surfaces, Contemporary Math., 211 (1997), 433–445.