Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 69–74

A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2

Yolanda Fuertes and Gabino González-Diez

Universidad Autónoma de Madrid, Departamento de Matemáticas E-28049 Madrid, Spain; yolanda.fuertes@uam.es, gabino.gonzalez@uam.es

Abstract. The fact that compact Riemann surfaces of genus 2 are always hyperelliptic is usually presented as a corollary of the Riemann–Roch theorem. Here we give a proof which involves only the theory of Fuchsian groups that uniformize them.

The class of hyperelliptic Riemann surfaces in each genus g > 1, because of its particular simplicity, is often taken as a model to illustrate known results or to test new conjectures. These are the surfaces S that appear as double covers of the Riemann sphere. They are characterized by the property that they admit an automorphism J of order 2 whose quotient S/⟨J⟩ has genus zero, the double cover being provided by the quotient map S → S/⟨J⟩. A basic fact, which is part of the widely developed theory of automorphism groups of Riemann surfaces, is that surfaces of genus 2 are always hyperelliptic.

Due to the uniformization theorem, an arbitrary Riemann surface of genus g > 1, S can be viewed as a quotient space S = H/K, where H is the upper half-plane and K is the uniformizing group, a discrete subgroup of the group of conformal automorphisms of H isomorphic to the fundamental group of the surface. Such groups possess a very well known structure and are sometimes termed (Fuchsian) surface groups. From this point of view the hyperellipticity of surfaces of genus 2 is equivalent to the statement that a surface group K of genus 2 is automatically contained, with index 2, in a larger group Γ so that H/Γ ≡ CP^{1} , J being then induced by any element of Γ \ K. In most textbooks this result is presented as a consequence of the Riemann–Roch theorem, a result whose presentation is involved. The purpose of this note is to prove this fact in the framework of the elementary theory of Fuchsian groups.

2000 Mathematics Subject Classification: Primary 30F10, 30F35.

The first author was supported by Grant EX 99 03097348 of the MEC and the second author partially by Grant PB96-0036 of the CICYT.MEC

1. Notation and statement of the result

We now summarize some elementary facts from the theory of Riemann surfaces and Fuchsian groups. We refer to Beardon [1], Farkas–Kra [2] and Jones– Singerman [3] for general background.

1.1. We recall that the group of conformal automorphisms of H agrees with the group of orientation preserving isometries of H equipped with the hyperbolic metric d. This is the well-known group of real Möbius transformations PSL(2, R).

An element C ∈ PSL(2, R) is called hyperbolic if it fixes two points on the boundary ∂H =R ∪ {∞} and none in H (see e.g. [2]). In fact, for any z ∈ H, these two points can be obtained as lim_{n→∞}C^{n}z (attractive fixed point) and lim_{n→−∞}C^{n}z (repulsive fixed point), and the non Euclidean or hyperbolic line connecting them is called the axis of C. The transformation C ∈ PSL(2, R) acts on its axis, axis(C), as a (hyperbolic) translation. Its translation length T_{C} is equal to d(z, Cz) for any point z on its axis. Moreover, C is determined by its axis and its translation length. As a matter of fact, C can be written as the product of two half-turns R_{z} and R_{w} at any pair of points z and w on the axis of C with distance apart equal to \frac{1}{2}T_{C}. More precisely, C = R_{z} ∘R_{w} or C = R_{w} ∘R_{z} according to whether the ray from w to z ends at the attractive or at the repulsive fixed point of C. These facts can be easily proved by conjugation with a suitable isometry so that C fixes 0 and ∞ and hence becomes a transformation of the form

C = ( \begin{matrix}λ & 0 \\ 0 & 1/λ \end{matrix} ) , λ > 0,

whose axis is simply the upper half imaginary axis (see [1, p. 174]).

1.2. The fundamental group π_{1}(S, P) of a compact Riemann surface S of genus 2 at a point P ∈ S is generated by the homotopy classes of simple loops α_{1}, β_{1} ; α_{2}, β_{2} enjoying the following properties:

(i) any pair of them intersect only at P;

(ii) this intersection is transversal (only) for the pairs α_{1}, β_{1} and α_{2}, β_{2} ; (iii) they satisfy the single defining relation [[α_{1}], [β_{1}]][[α_{2}], [β_{2}]] = 1, where

the bracket [ , ] stands for the commutator and [γ] denotes the homotopy class of a loop γ.

Such generators are called canonical.

1.3. The uniformization theorem quoted above shows that S can be viewed as a quotient space S = H/K, where K is a discrete subgroup of the group PSL(2, R), i.e. a Fuchsian group, acting freely on H. In these circumstances K consists entirely of hyperbolic elements (see e.g. [1]).

According to elementary covering space theory, for any p ∈ H projecting to P ∈ S via the covering map π: H →S, we have an isomorphism ϕ_{p}: π_{1}(S, P) → K defined by sending [γ] ∈ π_{1}(S, P) to the Möbius transformation which maps p to the endpoint of the lift of γ at p; this lift we denote by γ̃_{p}. By applying ϕ_{p} we obtain canonical generators for K, A_{i} =: ϕ_{p}([α_{i}]), B_{i} =: ϕ_{p}([β_{i}]) subject to the same defining relation [A_{1}, B_{1}][A_{2}, B_{2}] = 1.

Let now γ be any non trivial simple loop based at P and put C = ϕ_{p}([γ]). Let us consider the curve l_{p}(γ) in H obtained by putting together the translates of the arc γ̃_{p} by all powers of C, that is, l_{p}(γ) = ⋃_{n∈Z} C^{n}γ̃_{p}. This is clearly a connected curve since the arc C^{n}γ̃_{p} ends where C^{n+1}γ̃_{p} begins, namely at the point C^{n+1}p. Moreover, l_{p}(γ) does not cross itself, for if we had C^{r}x = C^{m}y for two different points x, y ∈ γ̃_{p} then x and y would project to the same point of S. It would then follow that x and y must be the endpoints of γ̃_{p}, say x = p and y = Cp; hence we would have C^{m+1−r}p = p which, C being fixed point free, can only occur if r = m + 1; in other words the arcs C^{r}γ̃_{p} and C^{m}γ̃_{p} intersect only if they are consecutive and then, at the correct point C^{m+1}p. This implies that l_{p}(γ) divides H into two connected components for on the one hand l_{p}(γ) is a topological line (in fact, the restriction of π: H →S to l_{p}(γ) provides a universal cover for the topological circle γ) and on the other hand l_{p}(γ) hits the boundary of H at two points, namely lim_{n→∞} C^{n}p and lim_{n→−∞} C^{n}p, that is, the fixed points of C.

We next observe that, since α_{i}, β_{i} intersect only at the point P, l_{p}(α_{i}) and l_{p}(β_{i}) intersect only at p ∈ H, for if there were another point p′ ∈ l_{p}(α_{i}) ∩ l_{p}(β_{i}), then, necessarily, π(p′) = P and we would have p′ = A_{i}^{d}(p) = B_{i}^{m}(p) for some integers d and m, but this, in turn, would imply A_{i}^{d} = B_{i}^{m}, a contradiction. Now the fact that l_{p}(α_{i}) and l_{p}(β_{i}) intersect transversally at only one point is equivalent to saying that each of these lines separates the endpoints of the other one. Of course this argument can be equally applied to any pair of topological lines connecting these same endpoints and, in particular, to the axes of A_{i} and B_{i}. We conclude that the axes of A_{i} and B_{i} do intersect.

1.4. Let us consider an arbitrary uniformization S = H/K. As in 1.1 we may assume, by suitable conjugation, that

A_{1} = ( \begin{matrix}λ & 0 \\ 0 & 1/λ \end{matrix} ) , λ > 0,

with axis(A_{1}) equals the upper half imaginary axis. We can even perform an extra normalization and require B_{1} to fix 1 so that

B_{1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,

with a + b = c + d and axis(B_{1}) equals the semi-circle joining its two fixed points, namely 1 and −b/c. Since axis(A_{1}) ∩ axis(B_{1}) ≠ ∅, a simple calculation shows that we must have bc > 0, the intersection point being z_{1} = i\sqrt{b/c} .

Theorem 1. Let S = H/K be an arbitrary compact Riemann surface of genus 2, and let

A_{1} = ( \begin{matrix}λ & 0 \\ 0 & 1/λ \end{matrix} ) ,  λ > 0,
B_{1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,  a + b = c + d, bc > 0,
A_{2} and B_{2},

be a set of (normalized) canonical generators for the uniformizing group K. Then, the Möbius transformation

R_{1}(z) = \frac{−b}{cz}

induces on S = H/K an automorphism J, the hyperelliptic involution, such that the quotient S/⟨J⟩ has genus zero.

2. Proof

(i) First of all we observe that R_{1} is the half-turn fixing the point z_{1} = i\sqrt{b/c} at which the axes of A_{1} and B_{1} meet. Now by 1.1 we can write

(1)  A_{1} = R_{3}R_{1},  B_{1} = R_{1}R_{2},

where R_{3} (respectively R_{2} ) is the half-turn at a certain point z_{3} ∈ axis(A_{1}) (respectively z_{2} ∈ axis(B_{1})). Similarly we put

(2)  A_{2} = R_{6}R_{5},  B_{2} = R_{4}R_{6},

where R_{6} is the half-turn at the point z_{6} ∈ axis(A_{2}) ∩ axis(B_{2}) and R_{5} (respectively R_{4} ) is the half-turn at a suitable point z_{5} ∈ axis(A_{2}) (respectively z_{4} ∈ axis(B_{2})).

(ii) Using identities (1) and (2) we can write [A_{1}, B_{1}] = (R_{3}R_{2}R_{1})^{2} and [A_{2}, B_{2}]^{−1} = (R_{4}R_{5}R_{6})^{2} , thus the defining relation [A_{1}, B_{1}] = [A_{2}, B_{2}]^{−1} becomes (R_{3}R_{2}R_{1})^{2} = (R_{4}R_{5}R_{6})^{2} . Since these elements are both hyperbolic, we deduce that

(3)  (R_{3}R_{2}R_{1}) = (R_{4}R_{5}R_{6})

or equivalently,

(4)  R_{1}R_{2}R_{3}R_{4}R_{5}R_{6} = 1

(iii) Next we claim that the rotations R_{j}, j = 1, …, 6, are all equivalent mod K. By (1) and (2), the statement is clear if we consider the first three or the second three separately. To complete the argument we only need to observe that R_{1}R_{6} ∈ K. Indeed we have R_{1}R_{6} = B_{1}^{−1}A_{1}^{−1}B_{2}A_{2} as it is readily seen using (3).

(iv) In order to prove that R_{1} normalizes K and, hence, that its action (and, by the previous point, that of any other R_{j}) induces an automorphism J on S = H/K we merely need to check that

(5) R_{1}A_{1}R_{1} = A_{1}^{−1}, R_{1}B_{1}R_{1} = B_{1}^{−1},
(6) R_{1}A_{2}R_{1} = (R_{1}R_{6})A_{2}^{−1}(R_{1}R_{6})^{−1}, R_{1}B_{2}R_{1} = (R_{1}R_{6})B_{2}^{−1}(R_{1}R_{6})^{−1}.

(v) Finally we prove that the quotient surface X = S/⟨J⟩ has genus zero. Let us denote by p: S → X the natural quotient map. We have an induced commutative diagram of homomorphisms between homology groups

\xymatrix{H_{1}(S) \ar[rr]^{J_{∗}} \ar[dr]_{p_{∗}} &  & H_{1}(S) \ar[dl]^{p_{∗}} \\  & H_{1}(X)}

If X has genus > 0, then there is on it a loop γ which is homologically non trivial. Consider then a lift of γ^{2} to S. Such a lift, let us call it c, is again a (closed) loop because p has degree 2, and so it defines a homology class which satisfies p_{∗}(c) = γ^{2} . But on the other hand, since the homology group is the abelianization of the fundamental group, it follows from (5) and (6) that J_{∗} = −identity. This contradicts the commutativity of the diagram above.

Remark 1. An alternative way to state our result is to say that the group K is an index 2 subgroup of the Fuchsian group Γ generated by the half-turns R_{j}, j = 1, …, 6, which satisfy the relation (4), thus it is a group of signature (0, 6) (see [3, p. 260]).

The projections π(z_{j}) of the points z_{j}, centers of the rotations R_{j}, are the six points which, according to the Riemann–Hurwitz formula, J fixes. They are called the Weierstrass points of S. Note that they divide the smooth geodesics π(axis(A_{i})), π(axis(B_{i})) into two parts of equal length.

Remark 2. If the genus g is larger than 2 we can still obtain expressions analogous to (1) and (2) for any set of canonical generators A_{1}, …, A_{g}; B_{1}, …, B_{g}. However this way of reasoning will break down when we try to find relations of the kind (5) and (6). In fact it is well known that when g > 2, most surfaces are not hyperelliptic.

Acknowledgment. We are grateful to our colleague Ernesto Girondo for carefully reading the manuscript and to the referee for helping us to improve its presentation. The first author is also grateful to the Faculty of Mathematical Studies, University of Southampton, where she spent the last two years.

74 Yolanda Fuertes and Gabino González-Diez

References

[1] Beardon, A.: The Geometry of Discrete Groups. - Grad. Texts in Math. 91, Springer-Verlag, New York, 1983.

[2] Farkas, H.M., and I. Kra: Riemann Surfaces. - Grad. Texts in Math. 71, Springer-Verlag, New York, 1992.

[3] Jones, G., and D. Singerman: Complex Functions. - Cambridge University Press, 1987.

Received 24 September 2001