Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 25–32

A NOTE ON ISOLATED POINTS IN THE BRANCH LOCUS OF THE MODULI SPACE OF COMPACT RIEMANN SURFACES

Emilio Bujalance, Antonio F. Costa and Milagros Izquierdo

Universidad Nacional de Educacion Distancia, Departamento de Matemáticas Fundamentales

E-28040 Madrid, Spain Mälardalen University, Department of Mathematics S-721 23 Västerås, Sweden

Abstract. Let g be an integer ≥ 3 and let ℬ_{g} = {X ∈ ℳ_{g} | Aut X ≠ e}, where ℳ_{g}

denotes the moduli space of a compact Riemann surface.

The geometric structure of ℬ_{g} is of substantial interest because ℬ_{g} corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus g (see [H]).

Surprisingly R.S. Kulkarni [K] has found isolated points in ℬ_{g}. He showed that they appear if and only if 2g +1 is an odd prime distinct from 7. The aim of this paper is to find a geometrical explanation of this phenomenon using the fact that the isolated points are given by surfaces admitting anticonformal involutions (symmetries). The points in the Teichmüller space, corresponding to groups uniformizing surfaces with a symmetry, is a (non disjoint) union of submanifolds. We shall obtain that the isolated intersections of such submanifolds give us the isolated points in the branch loci.

Also we prove that there are no isolated points in the moduli space of Klein surfaces which are not Riemann surfaces.

1. Preliminaries

A non-euclidean crystallographic group, (NEC) group, is a discrete group Γ of the group 𝒢 of isometries of the hyperbolic plane H with compact quotient space H/Γ. If the group Γ is a subgroup of the group 𝒢^{+} of orientation-preserving isometries of H, then it is called a Fuchsian group. Otherwise Γ^{+} = Γ ∩ 𝒢^{+} is a subgroup of index 2 in Γ, called its canonical Fuchsian subgroup.

Let Γ be an NEC group. Then there is a fundamental region P for Γ which is a polygon in H whose perimeter, described counterclockwise, is one of the following:

(1) ε_{1}ε′_{1} ⋯ ε_{r}ε′_{r}δ_{1}γ_{10} ⋯ γ_{1s_{1}}δ′_{1} ⋯ δ_{k}γ_{k0} ⋯ γ_{ks_{k}}δ′_{k}α_{1}β_{1}α′_{1}β′_{1} ⋯ α_{h}β_{h}α′_{h}β′_{h},

(2) ε_{1}ε′_{1} ⋯ ε_{r}ε′_{r}δ_{1}γ_{10} ⋯ γ_{1s_{1}}δ′_{1} ⋯ δ_{k}γ_{k0} ⋯ γ_{ks_{k}}δ′_{k}α_{1}α_{1}^{∗} ⋯ α_{h}α_{h}^{∗}.

1991 Mathematics Subject Classification: Primary 30F10, 30F50; Secondary 20H10.

The first and second authors were partially supported by DGICYT PB 95-0017 and CEE CHRX-CT93-0408Λ, and the third author by the Swedish Natural Science Research Council (NFR).

In these symbols each letter denotes an oriented side of the polygon P. The apostrophe means that the corresponding sides of P are identified by generators of the group which preserve the orientation, and the asterisk means that the corresponding sides are identified by generators of the group which reverse the orientation. As a consequence, if we identify corresponding points on the related edges of the polygon, we obtain from P a surface with boundary. In the case (1), the surface will be a sphere with k disks removed and g handles added. In the case (2), the surface will be a sphere with k disks removed and h cross-caps added. The angle determined by the sides ε_{i}ε′_{i} is 2π/m_{i}, i = 1, …, r, and the angle given by γ_{ij}γ_{ij+1} is π/n_{ij}, i = 1, …, k, j = 0, …, s_{i−1} . In the group Γ there are elliptic elements that identify ε_{i} with ε′_{i} and reflections whose axe contain the sides γ_{ij}. The fundamental region P determines a presentation for Γ. The side pairing and the reflections give rise to a set of generators and the relations are given by the way as the images of P by elements of Γ fit together around the vertices of P.

Then the algebraic structure of an NEC group is determined by its signature

(1.1)  s(Γ) = (h; ±; [m_{1}, …, m_{r}]; {(n_{11}, …, n_{1s_{1}}), …, (n_{k1}, …, n_{ks_{k}})}).

The presentation for Γ associated to the signature (1.1) have generators

x_{1}, …, x_{r},
e_{1}, …, e_{k},
c_{ij}, 1 ≤ i ≤ k, 0 ≤ j ≤ s_{i},
a_{1}, b_{1}, …, a_{h}, b_{h} if H/Γ is orientable or
a_{1}, …, a_{h}, if H/Γ is non-orientable

and relators

x_{i}^{m_{i}}, i = 1, …, r,
c_{ij−1}^{2}, c_{ij}^{2}, (c_{ij−1}c_{ij})^{n_{ij}}, i = 1, …, k, j = 1, …, s_{i},
c_{i0}e_{i}^{−1}c_{is_{i}}e_{i},
x_{1}x_{2} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{h}^{−1}b_{h}^{−1}, if H/Γ is orientable or
x_{1}x_{2} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}^{2} ⋯ a_{h}^{2}, if H/Γ is non-orientable.

These last two relations are sometimes called the long relation. In these presentations, the only elements of finite order are the elliptic elements and the reflections. The elliptic elements are conjugate of powers of the x_{i} or c_{ij−1}c_{ij} and the reflections are conjugate of the c_{ij}. The e_{i} generators are orientation preserving. They are called the connecting generators.

The hyperbolic area of H/Γ, which we shall call μ(Γ), is equal to the hyperbolic area of a fundamental polygon P, then

(1.2)  μ(Γ) = 2π(εh − 2 + k + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} )),

where ε = 2 if there is a + sign and ε = 1 if there is a − sign. If Γ^{∗} is a subgroup of Γ of finite index then the Riemann–Hurwitz formula holds:

(1.3)  |Γ : Γ^{∗}| = \frac{μ(Γ^{∗})}{μ(Γ)}.

Let Γ^{+} be the canonical Fuchsian of an NEC group Γ. Then H/Γ^{+} is a 2sheeted covering of H/Γ, called its complex double. The genus h^{+} = εh + k − 1 of H/Γ^{+} is the algebraic genus of H/Γ. An NEC group Γ without elliptic elements is called a surface group. Its signature is (g; ±; [ ]; {( )^{k}}). A Klein surface whose complex double has genus greater than one can be expressed as H/Γ where Γ is an NEC surface group. An orientable Klein surface without boundary can be thought as a Riemann surface. If G is a finite group, then G is a group of automorphisms of a Klein surface H/Γ if and only if there exists an NEC group Γ′ and a homomorphism from Γ′ on to G having Γ as the kernel (see [BEGG]). Then for each symmetry there is a group Γ′ containing Γ as index two subgroup.

Given an NEC group Γ, we denote by R(Γ) the set of monomorphisms r: Γ → 𝒢 such that r(Γ) is discrete and H/r(Γ) is compact. Two elements r_{1}, r_{2} ∈ R(Γ) are said to be equivalent if there exists g ∈ 𝒢 such that for each γ ∈ Γ, r_{1}(γ) = gr_{2}(γ)g^{−1} . The orbit space T(Γ) is called the Teichmüller space of Γ and it is homeomorphic to a real ball considering in T(Γ) the Teichmüller metric (see [MS]).

Let A(Γ) denote the automorphism group of Γ, A(Γ)^{+} be the orientation preserving automorphism group if Γ is a Fuchsian group, and I(Γ) the subgroup of inner automorphisms. The modular group M(Γ) = A(Γ)/I(Γ) or M(Γ)^{+} = A(Γ)^{+}/I(Γ) if Γ is a Fuchsian group, acts on T(Γ) as follows. If [r] ∈ T(Γ) and [α] ∈ M(Γ), then [α][r] = [r ∘ α]. The moduli space of Γ is the quotient space

ℳ_{g} = T(Γ)/M(Γ)

and ℳ_{g} = T(Γ)/M(Γ)^{+} if Γ is a Fuchsian group (see [MS]). Let

π: T(Γ) → ℳ_{g}

be the natural projection.

As an application of the extended Nielsen theorem we have that any automorphism of a surface group can be geometrically realized as a homeomorphism of the surface. Hence we can identify the branch locus of the action of M(Γ) on T(Γ) with the set

ℬ_{g} = {X ∈ ℳ_{g} | Aut X ≠ e}.

This is called the branch locus of ℳ_{g}.

For surfaces X with algebraic genus 2, ℬ_{2} = {X ∈ ℳ_{g} | Aut X ≠ Z_{2}}.

Let now Γ ≤ Γ′ be NEC groups and let i: Γ → Γ′ be the inclusion mapping. Then i induces m: T(Γ′) → T(Γ) defined by m[r] = [r ∘ i], where m is an isometric embedding (see [MS]). Then m(T(Γ′)) is a submanifold of T(Γ). So each Γ^{∗} containing Γ as subgroup of index two and each i: Γ → Γ^{∗}, give rise to a submanifold in T(Γ).

2. Isolated points in the branch locus of Riemann surfaces

We consider the set N of classes [r, s, t] of unordered triples {r, s, t} of numbers counted mod q such that none of them is ≡ 0 mod q and r + s + t ≡ 0 mod q under the action of the multiplicative group Z_{q} −{0}. There is a bijection from N onto the set of Riemann surfaces X admitting Z_{q} -action with quotient orbifold a sphere with three cone points of order q. Thus we can associate a symbol [r, s, t] in an one-to-one manner to such Riemann surfaces. The symbol [r, s, t] is called the characteristic symbol of the Riemann surface X.

In [K] it is shown that

Theorem 2.1. The number of isolated points in ℬ_{g} is 1 if g = 2, [(g −2)/3] if q = 2g + 1 is a prime > 7, and 0 otherwise. The isolated point in ℬ_{2} is the hyperelliptic surface w^{2} = z^{5} − 1. For g ≥ 5 the isolated Riemann surfaces are precisely those X′ for which Aut^{+} X′ is isomorphic to Z_{q} and such that in their characteristic symbols [r, s, t] no two of r, s, t are equal, and if q ≡ 1 mod 3 and λ is a root of unity then [r, s, t] ≠ [1, λ, λ^{2}].

Here for a real number x, [x] denotes the greatest integer ≤ x.

The following algorithm, with q = 2g + 1 prime, yields geometric models for Riemann surfaces which correspond to isolated points in ℬ_{g}; see [K]. Its characteristic symbol is chosen to be [1, s, t] without loss of generality.

Geometrical description. Let T be an hyperbolic triangle with angles π/q, π/q, π/q. We label the vertices with the capital letters N, U, V and the opposite sides with n, u, v. We paste 2q copies of T around the vertices labelled N and identifying u_{2i−1} with u_{2i} and v_{2i} with v_{2i+1} , and then U_{2i−1} with U_{2i} and V_{2i} with V_{2i+1} . In this way we obtain an hyperbolic polygon P. The surface X with characteristic symbol [1, s, t] is obtained from P by the identification of the side n_{2i−1} with n_{2i+2s}.

The polygon P admits as symmetry group a dihedral group D_{q} containing the reflections on the diagonals of P and rotations of order q. The symmetries of P are compatible with the identifications of the sides that produce X. One of the rotations, R, of D_{q} transforms the side n_{x} in the side n_{x+2} . The rotation R is compatible with the sides identification because n_{2i−1+2} = n_{2(i+1)−1} is identified with n_{2i+2s+2} = n_{2(i+1)+2s}. One of the reflections, F, of D_{q} transforms the side n_{x} in the side n_{q−x}. To prove that F is compatible with the side identifications it is enough to remark that, if 0 < j ≤ q is such that j ≡ \frac{1}{2}(−2i − 2s + 1) mod q then n_{q−(2i−1)} = n_{2j+2s} is identified with n_{q−2i−2s} = n_{2j−1} . Then the surface X admits q symmetries (anticonformal involutions). In particular there are two symmetries s_{1} and s_{2} that are conjugate in Aut(X) and that generate D_{q} as the automorphism group of X.

The above proves the following

Lemma 2.2. An isolated point X in ℬ_{g} admits two symmetries which generate D_{q} as the automorphism group of X.

Note. There are two possible signatures of NEC groups, namely s(Λ_{1}) = (0; +; [ ]; {(q, q, q)}), s(Λ_{2}) = (0; +; [q]; {(q)}) corresponding to groups that admit the Fuchsian group Λ with signature (0; +; [q, q, q]; { }) as their canonical Fuchsian subgroup. Only for the first signature we have that the epimorphisms θ: Λ → Z_{q} with kernel a surface group extend to an epimorphism θ_{1}: Λ_{1} → D_{q}. Hence if X represents an isolated point in ℬ_{g}, then Aut X = D_{q} and the orbifold X/ Aut X is a disc with three corner points of order q.

Let X be a Riemann surface representing an isolated point of ℬ_{g}. By Lemma 2.2 X has an automorphism group D_{q} then there is an NEC group Δ and an epimorphism θ: Δ → D_{q} such that X = H/ Ker θ. Let s ∈ D_{q} be a symmetry of X and θ^{−1}{1, s} = Γ_{s} then Ker θ ⊂ Γ_{s}. Let Γ be the abstract Fuchsian group such that Γ is isomorphic to Ker θ and let Γ^{∗} be the abstract group isomorphic to Γ_{s}. Let t: Γ^{∗} → Γ_{s} be an isomorphism and i: Γ → Γ^{∗} be a monomorphism such that t ∘ i(Γ) = Ker θ. Then i induces m: T(Γ^{∗}) → T(Γ). Hence mT(Γ^{∗}) give us a submanifold in T(Γ) containing X.

Let v be a symmetry of D_{q} different from s. Then there is w ∈ D_{q} such that v = wsw^{−1} . Let g ∈ Δ such that θ(g) = w. Conjugation by g induces an automorphism in Ker θ, c_{g}, and

(2.1)  α = (t ∘ i)^{−1} ∘ c_{g} ∘ t ∘ i

is an automorphism of Γ. Let [α] be the element of the modular group represented by α.

Lemma 2.3. [t ∘ i] ∈ m(T(Γ^{∗})) ∩ [α]m(T(Γ^{∗})).

Proof. By the definition of m we have [t ∘ i] = m[t] and by the definition of the action of [α] we have: [α]m[t] = [α][t∘i] = [t∘i∘α] = [t∘i∘(t ∘ i)^{−1} ∘c_{g} ∘t∘i] = [c_{g} ∘ t ∘ i] = [t ∘ i].

Lemma 2.4. There exists ε > 0 such that, if B_{ε}([t ∘ i]) is the ball of radius ε around [t ∘ i] in T(Γ) , then

{[t ∘ i]} = m(T(Γ^{∗})) ∩ [α]m(T(Γ^{∗})) ∩ B_{ε}([t ∘ i]).

Proof. Since X is an isolated point in ℬ_{g} there is an ε > 0 such that Aut^{+}(H/r ∗ (Γ)) = {identity} for all r∗ such that r∗ ∈ B_{ε}([t ∘ i]) − {[t ∘ i]}.

Assume that there exists

[r∗] ∈ m(T(Γ^{∗})) ∩ [α]m(T(Γ^{∗})) ∩ (B_{ε}([t ∘ i]) − {[t ∘ i]}).

Then [r∗] = m[r_{1}] = [r_{1} ∘ i] = [α]m[r_{2}] = [r_{2} ∘ i ∘ α], where [r_{1}] and [r_{2}] ∈ m(T(Γ^{∗})). Hence there is h ∈ 𝒢^{+} such that r_{1} ∘ i = c_{h} ∘ r_{2} ∘ i ∘ α, where c_{h} is defined by c_{h}(x) = hxh^{−1} . Let r_{3} = c_{h} ∘ r_{2} . Then

(2.2)  r_{1} ∘ i = r_{3} ∘ i ∘ α,  r_{1} ∘ i(Γ) ⊂ r_{1}(Γ^{∗})

and

r_{3} ∘ i(Γ) = r_{3} ∘ i ∘ α(Γ) ⊂ r_{3}(Γ^{∗}).

If r_{1}(Γ^{∗}) ≠ r_{3}(Γ^{∗}) then there are two symmetries a and b of Y = H/r_{1} ∘ i(Γ) and then there is a conformal automorphism of Y , ab, different from the identity in contradiction with the choice of ε. Thus r_{1}(Γ^{∗}) = r_{3}(Γ^{∗}). Hence

(2.3)  β = r_{3}^{−1} ∘ r_{1}

is an automorphism of Γ^{∗}. Then by (2.2) and (2.3)

(2.4)  β ∘ i = r_{3}^{−1} ∘ r_{1} ∘ i = r_{3}^{−1} ∘ r_{3} ∘ i ∘ α = i ∘ α.

Given an NEC group, Λ, we define

E(Λ) = {ge_{i}g^{−1} : gεΛ and e_{i} is a connecting generator}.

We have E(Γ_{s}) ⊂ Ker θ and E(c_{g}(Γ_{s})) ⊂ Ker θ. Since Γ_{s} and c_{g}(Γ_{s}) produce different symmetries on X then

(2.5)  E(Γ_{s}) ∩ E(c_{g}(Γ_{s})) = Φ.

On the other hand

(2.6)  c_{g}(E(Γ_{s})) = E(c_{g}(Γ_{s})).

We shall translate the situation to Γ using t ∘ i from (2.5) we have

(t ∘ i)^{−1}(E(Γ_{s})) ∩ (t ∘ i)^{−1}(E(c_{g}(Γ_{s}))) = Φ,

and from (2.6)

(t ∘ i)^{−1}(c_{g}(E(Γ_{s}))) = (t ∘ i)^{−1}(E(c_{g}(Γ_{s}))),

so from the definition of α, (2.1), we obtain

α ∘ (t ∘ i)^{−1}(E(Γ_{s})) = (t ∘ i)^{−1}(E(c_{g}(Γ_{s}))).

Also E(Γ^{∗}) = t^{−1}(E(Γ_{s})). Then

α ∘ i^{−1}(E(Γ^{∗})) = (t ∘ i)^{−1}(E(c_{g}(Γ_{s}))) = α ∘ i^{−1}(t^{−1}E(Γ_{s})).

So

(2.7)  α ∘ i^{−1}(E(Γ^{∗})) ∩ i^{−1}(E(Γ^{∗})) = Φ.

Since β is an automorphism of Γ^{∗} the following equality holds: β(E(Γ^{∗})) = E(Γ^{∗}). We have E(Γ^{∗}) ⊂ i(Γ), and by (2.4)

E(Γ^{∗}) = β ∘ i(i^{−1}E(Γ^{∗})) = i ∘ α ∘ i^{−1}(E(Γ^{∗})).

Then α ∘ i^{−1}(E(Γ^{∗})) = i^{−1}(E(Γ^{∗})) that is a contradiction with (2.7).

In the above lemma choosing a symmetry different from s and v we can replace the submanifold [α]m(T(Γ^{∗})) by [α^{j}]m(T(Γ^{∗})) with j ∈ {2, …, q − 1}. Applying Lemma 2.4 to every pair of symmetries of D_{q} we have

Theorem 2.5. Let Γ be a Fuchsian surface group and [r] ∈ T(Γ) such that H/r(Γ) represents an isolated point in the branch loci with automorphism group D_{q}. Let [α^{j}]m(T(Γ^{∗})), j ∈ {1, …, q}, be the submanifolds of T(Γ) given by the groups uniformizing surfaces with a symmetry and such that [r] ∈ [α^{j}]m(T(Γ^{∗})). Then there exists a neigborhood V of [r] such that

{[r]} = V ∩ (∩[α^{j}]m(T(Γ^{∗}))).

Remark. Let S_{1} and S_{2} be two submanifolds of T(Γ) such that S_{1} and S_{2} correspond to groups uniformizing Riemann surfaces with a symmetry. Since dim S_{1} = dim S_{2} = \frac{1}{2} dim T(Γ), if S_{1} and S_{2} cut transversally in a point p then such a point projects by π: T(Γ) → ℳ_{g} in an isolated point of the branch loci.

3. On the non-existence of isolated points for non-orientable surfaces and surfaces with boundary

We shall see in this section that, on the contrary of what happens with Riemann surfaces, there are not isolated points in the branch locus ℬ of Klein surfaces which either are non-orientable or have nonempty boundary.

We first consider non-orientable surfaces Y without boundary. It is well known that if G = Aut Y , then G = Aut^{+} Y_{c}, where Y_{c} denotes the complex double of Y .

Theorem 3.1. There are not isolated points in the branch locus ℬ_{g} of non-orientable Klein surfaces without boundary.

Proof. Assume that Y is a non-orientable surface without boundary of genus g representing an isolated point in ℬ_{g}. Using Theorem 2.1, we obtain that Aut Y = Aut^{+} Y_{c} = Z_{q}, with q = 2g^{+} + 1 a prime, then q = 2g − 1. On the other hand the quotient orbifold Y/ Aut Y must be either a disc with three corner points of order q, or a disc with a corner point of order q and a cone point of order q. Now, as Y is non-orientable and q is an odd integer, then the orbifold Y/Z_{q} is non-orientable and we are in a contradiction.

Let Γ be an NEC surface group with signature (h; ±; [ ]; {( )^{k}}), the notation means that there are k empty period cycles in Γ.

If G′ is a subgroup of Aut(H/Γ), then the G′ -action on H/Γ induces a stratum of the branch locus of the surfaces with group Γ. The stratum has dimension 0 if the quotient orbifold of H/Γ by G′ is the disc with three corner points, or the disc with a corner point of order q and a cone point of order q. But the quotient orbifold of a bordered Klein surface contains a boundary component without corner points or a boundary component with at least two corner points of order 2 (see [BM]). This proves the following

Theorem 3.2. There are no isolated points in the branch locus of Klein surfaces with nonempty boundary.

The authors wish to thank the referee for several helpful comments and suggestions.

References

[BEGG]Bujalance, E., J.J. Etayo, J.M. Gamboa, and G. Gromadzki: A Combinatorial Approach to Groups of Automorphisms of Bordered Klein Surfaces. - Lecture Notes in Math. 1439, Springer-Verlag, 1990.

[BM] Bujalance, E., and E. Martínez: A remark on NEC groups of surfaces with boundary. - Bull. London Math. Soc. 21, 1989, 263–266.

[H] Harvey, J.W.: On branch loci in Teichmüller space. - Trans. Amer. Math. Soc. 153, 1971, 387–399.

[K] Kulkarni, R.S.: Isolated points in the branch locus of the moduli space of compact Riemann surfaces. - Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 1991, 71–81.

[MS] Macbeath, A.M., and D. Singerman: Spaces of subgroups and Teichmüller space. - Proc. London Math. Soc. 31, 1975, 211–256.

Received 30 October 1995