Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 275–281

A FUNDAMENTAL DOMAIN FOR THE MODULAR GROUP OF RIEMANN SURFACES OF TYPE (0,n)

Min Chen

Suzhou University, Department of Mathematics Suzhou 215006, China; mchen@mailbox.suda.edu.cn

Abstract. In this paper we introduce the concept of extreme loop structures, and basing on it we present an explicit fundamental domain for the action of modular group in the deformation space of Riemann surfaces of genus 0 with n (n ≥ 4) boundary components.

Introduction

There have been several well-known ways of parametrizing the deformation space of a surface: for example, Fricke cooordinates, Teichmüller coordinates and Fenchel–Nielsen coordinates. Though these parametrizations provide an overview of all possible Riemann surfaces, some basic questions remain unsolved. We cannot decide whether two Riemann surfaces, when expressed by the parameters, are isometric or not. This is mainly a problem of marking a surface in different ways, then read off the parameters to get a hold of the action of the modular group on the deformation space.

The present work is an attempt towards an understanding of this problem. We present an explicit fundamental domain for the action of the modular group on the deformation space of Riemann surfaces of genus zero with n = m + 1 (m ≥ 3) boundary components. Similar exploration for the topological types (0, 4) and (1, 1) were considered in [M1] and [M2].

In Section 2, we introduce the concept of extreme loop structures. An extreme loop structure is mapped to an extreme loop structure under a conformal mapping, and with certain restricting conditions, it is unique. Even though an extreme loop structure is defined only by finitely many length inequalities, it has a certain global control ability (Theorem 1). Moreover, one can write an algorithm to search all extremal loop structures on a Riemann surface of type (0, n).

As a natural corollary, the inequalities that define our extreme loop structure yield the required fundamental domain.

Part of this paper was written while the author was visiting Bielefeld university in 1996. The author thanks Professor H. Helling who made the visit possible. The author also would like to thank the referee whose suggestions improve the paper; especially, he (or she) pointed out an error in the original proof of Theorem 1.

1991 Mathematics Subject Classification: Primary 30F, 32G, 34C.

Supported in part by the Jianshou provincial natural science foundation.

1. Preliminaries

Throughout this paper G will denote a Fuchsian group acting on the upper half plane H and S = H/G a Riemann surface of genus zero with m + 1 (m ≥ 3) boundary components A_{0}, A_{1}, …, A_{m}.

Assume that α and β are two closed curves on S. Their geometric intersection number #(α, β) is the infimum of the cardinality of α′ ∩ β′ for all α′ ∼ α, β′ ∼ β where ∼ denotes homotopy. This infimum is realized by α′ ∩ β′ if α′ ∼ α, β′ ∼ β and both α′ and β′ are distinct closed geodesics.

Let l be a closed curve on S. If g ∈ G has the property that any curve in H with z and g(z) as its terminal points is projected to a closed curve on S which is freely homotopic to l, then we say that g covers l. Suppose that g covers l and l′ ∼ l, then g also covers l′. Moreover, the set {g′ ∘ g ∘ g^{′−1} : g′ ∈ G} contains all elements in G which cover l. Assume that

g(z) = \frac{az + b}{cz + d}, ad − bc = 1.

Define | g |=| a+d |. Note that |l| = 2 log(\sqrt{|g|^{2} − 4} +|g|)−log 4 is the hyperbolic length of the shortest geodesic in the homotopy class of l.

Provide l with a certain orientation, and let x and y be two distinct points on l. We denote by l(x, y) the path on l which starts at x, follows the orientation, and ends at y. Suppose that l_{1} and l_{2} are two oriented curves such that l_{2} starts at the terminal point of l_{1} . Then l_{2} + l_{1} will represent the curve which starts at the starting point of l_{1} , goes along l_{1} and l_{2} , and ends at the terminal point of l_{2} .

Fix a point õ on S and choose a set of generators {α_{i} : i = 0, 1, …, m} of the fundamental group π(S, õ) with the following properties: for each i, α_{i} is freely homotopic to A_{i}, α_{j} ∘ α_{j−1} ∘ ⋯ ∘ α_{0} (0 < j < m) are simple, and α_{m} ∘⋯∘α_{1} ∘α_{0} ∼ õ. By a well-known lifting procedure, we get the corresponding generators g_{0} , g_{1}, …, g_{m} of G, with g_{j} (0 ≤ j ≤ m) covering α_{j}, g_{0} ∘ g_{1} ∘ ⋯ ∘ g_{j} (0 < j < m) covering α_{j} ∘α_{j−1} ∘⋯∘α_{0} , and g_{0} ∘g_{1} ∘⋯∘g_{m} = id. The generators g_{j} (j = 0, 1, …) are called standard generators. See Figure 1, where m = 3.

Let ℳ be the set of all Möbius transformations fixing H. A deformation of G is a monomorphism ψ: G ↦ ℳ for which there is an orientation preserving homeomorphism φ: H ↦ H, with φ ∘ g ∘ φ^{−1} = ψ(g), for all g ∈ G. We may write ψ = (ψ(g_{j})). Two deformations ψ_{1} and ψ_{2} are equivalent if there exists an element A ∈ ℳ , with A ∘ ψ_{1}(g) ∘ A^{−1} = ψ_{2}(g) for all g ∈ G. The set of equivalence classes is the deformation space F. A deformation ψ is called modular if ψ(G) = G and it acts on F in the following way: if ψ_{1} is any deformation, then define ψ(ψ_{1}) = ψ_{1} ∘ ψ. The action defined above induces a group operation on the set M of all modular deformations, which is called the modular group. The quotient F/M is the moduli space, or Riemann space of conformally distinct Riemann surfaces of genus zero, with m + 1 boundary components.

2. Extreme loop structures and the fundamental domain

Definition. Suppose that for 1 ≤ i ≤ 3, C_{i} is either a boundary component of S or a simple closed curve on S. If there exist pants P (C_{1}, C_{2}, C_{3}) on S such that C_{1} , C_{2} and C_{3} are exactly its border components, then we write (C_{1}, C_{2}, C_{3}) = 1.

Let l_{j} (j = 1, 2, …, m) be simple closed curves on S with the following properties:

(1) (A_{0}, A_{j}, l_{j}) = 1, j = 1, 2, …, m;

(2) #(l_{i}, l_{j}) = 2 for i ≠ j.

Imaging A_{j} (j = 1, 2, …, m) as points on a circle centered at the point A_{0} with l_{j} as intervals lying inside it, we can give a clockwise order for {A_{j} : j = 1, 2, …, m}, provided we choose one boundary component, say A_{1} , as the initial element. Without loss of generality, we may assume that the indexes of {A_{j}} are consistent with the order, i.e.,

A_{1} ≺ A_{2} ≺ ⋯ ≺ A_{m},

and call {A_{0}, A_{j}, l_{j}}_{j=1}^{m} a loop structure with the center A_{0} and the initial A_{1} .

Let {A_{0}, A_{j}, l_{j}}_{j=1}^{m} and {A_{0}, A_{j}, l′_{j}}_{j=1}^{m} be two loop structures with the center A_{0} and initial A_{1} . We define {A_{0}, A_{j}, l_{j}}_{j=1}^{m} = {A_{0}, A_{j}, l′_{j}}_{j=1}^{m} if l_{j} or l_{j}^{−1} ∼ l′_{j}, j = 1, 2, …, m.

Definition. A loop structure {A_{0}, A_{j}, l_{j}}_{j=1}^{m} is extreme if

(∗) |l_{j}| ≤ {|l| : (A_{0}, A_{j}, l) = 1, #(l, l_{i}) = 2, i ≠ j},  for j = 1, 2, …, m.

It is easy to see that we can choose the α_{j} in Section 1 in such a way that they satisfy α_{j} ∘ α_{0} ∼ l_{j} or l_{j}^{−1} . We call such a set {α_{j}} of generators compatible with the loop structure {A_{0}, A_{j}, l_{j}}_{j=1}^{m}. Thus, if {α_{j}} is compatible with {A_{0}, A_{j}, l_{j}}_{j=1}^{m}, the inequalities (∗) can be written in the following way:

|α_{1} ∘ α_{0}| ≤ min{|α_{2} ∘ α_{1}^{−1} ∘ α_{2}^{−1} ∘ α_{0}^{−1}|, |α_{3} ∘ α_{2} ∘ α_{1}^{−1} ∘ α_{2}^{−1} ∘ α_{3}^{−1} ∘ α_{0}^{−1}|,
…, |α_{m−1} ∘ ⋯ ∘ α_{2} ∘ α_{1}^{−1} ∘ α_{2}^{−1} ∘ ⋯ ∘ α_{m−1}^{−1} ∘ α_{0}^{−1}|};
⋯
|α_{m} ∘ α_{0}| ≤ min{|α_{1} ∘ α_{m}^{−1} ∘ α_{1}^{−1} ∘ α_{0}^{−1}|, |α_{2} ∘ α_{1} ∘ α_{m}^{−1} ∘ α_{1}^{−1} ∘ α_{2}^{−1} ∘ α_{0}^{−1}|,
…, |α_{m−2} ∘ ⋯ ∘ α_{1} ∘ α_{m}^{−1} ∘ α_{1}^{−1} ∘ ⋯ ∘ α_{m−2}^{−1} ∘ α_{0}^{−1}|}.

Lemma 1. There exists an extreme loop structure on S with center A_{0} and initial A_{1} .

Proof. Since the length spectrum of closed curves on S is discrete, we see that there are l(A_{j}) (j = 1, 2, …, m) on S such that

(1) (A_{0}, A_{j}, l(A_{j})) = 1, #(l(A_{j}), l(A_{i})) = 2 for i ≠ j;

(2) |l(A_{j})| = min{|l| : (A_{0}, A_{j}, l) = 1, #(l, l(A_{i})) = 2, 1 ≤ i < j}. In the way we used in defining our loop structure, we can give an order for {A_{j}}. If we change the indexes of {A_{j}} according to the order, we see that the lemma follows.

Theorem 1. Let {A_{0}, A_{j}, l_{j}}_{j=1}^{m} be an extreme loop structure. Then
|l_{j}| = min{|l| : (A_{0}, A_{j}, l) = 1}, j = 1, 2, …, m.

Moreover, if all inequalities in (∗) hold strictly, then {A_{0}, A_{j}, l_{j}}_{j=1}^{m} is the unique extreme loop structure on S with the center A_{0} and initial A_{1} .

Proof. For any 1 ≤ j ≤ m, let l be a closed geodesic satisfying l ≁ l_{j} and (A_{0}, A_{j}, l) = 1. We will show that |l_{j}| ≤ |l|. The assertion holds trivially if #(l, l_{i}) = 2 for all i ≠ j. Suppose that #(l, l_{i}) > 2 for some i ≠ j. Since all A_{k} (1 ≤ k, k ≠ j) do not lie in the pants P (A_{0}, A_{j}, l), we see immediately that P (A_{0}, A_{j}, l) ∩ P (A_{0}, A_{i}, l_{i}) has p + 1 connected components for some integer p ≥ 1. It is not difficult to see that i can be chosen so that we may assume that the intersection points p_{k1} , p_{k2} , p_{k3} , p_{k4} ∈ ∂D_{k} ∩ l ∩ l_{i} (k = 1, 2) shown by Figure 3 have the following properties:

1. the inner part of the curve l(P_{11}, P_{12}) meets l_{i′} twice for all 1 ≤ i′ ≤ m;

2. among all intersection components D_{2} is the nearest to A_{0} in P (A_{0}, A_{i}, l_{i}).

It may happen that D_{1} = D_{2} .

Figure 3.

Then

|l| + |l_{i}| > |l(p_{11}, p_{13}) + l_{i}(p_{13}, p_{11})| + |l(p_{23}, p_{21}) + l_{i}(p_{21}, p_{23})|.

Since |l(p_{11}, p_{13}) + l_{i}(p_{13}, p_{11})| ≥ |l_{i}| by our assumption, we get |l| > |l(p_{23}, p_{21}) + l_{i}(p_{21}, p_{23})|. Denoting by l′ the geodesic in the homotopy class of l(p_{23}, p_{21}) + l_{i}(p_{21}, p_{23}), we see that

(A_{0}, A_{j}, l′) = 1,  ∑_{i=1}^{m} #(l′, l_{i}) < ∑_{i=1}^{m} #(l, l_{i}),  and  |l′| < |l|.

Repeating the same procedure on l′, we get |l| > |l_{j}| after finitely many steps. Suppose the inequalities in the definition of our extreme loop structure hold strictly. Let {A_{0}, A_{k(i)}, l′}_{i=1}^{m} be an extreme loop structure on S with the center A_{0} and initial A_{1} , where k(i) ∈ {1, 2, …, m} and k(1) = 1. Then l′_{i} ∼ l_{k(i)} or l_{k(i)}^{−1} (i = 1, 2, …, m), and the assertion follows.

Since conformal mappings preserve length, the following corollary is obvious.

Corollary. Let {A_{0}, A_{j}, l_{j}}_{j=1}^{m} be an extreme loop structure on S and f: S ↦ S_{0} a conformal homeomorphism. Then {f(A_{0}), f(A_{j}), f(l_{j})}_{j=1}^{m} is an extreme loop structure on S_{0} . Furthermore, if all inequalities in (∗) hold strictly, then {f(A_{0}), f(A_{j}), f(l_{j})}_{j=1}^{m} is the unique extreme structure on S_{0} with the center f(A_{0}) and initial f(A_{1}).

Remark. Let S′ be a Riemann surface of topological type (0, m + 1), and {A_{0}, A_{j}, l_{j}}_{j=1}^{m} and {B_{0}, B_{j}, l′_{j}}_{j=1}^{m} be extreme loop structures on S and S′, respectively. Suppose that {α_{j}} and {α′_{j}} are sets of generators compatible with {A_{0}, A_{j}, l_{j}}_{j=1}^{m} and {B_{0}, B_{j}, l′_{j}}_{j=1}^{m}, respectively. Then by Theorem 4.1 in [O], we see that S is conformally equivalent to S′, provided the following equalities are true:

\begin{aligned}|α_{j}| & = |α′_{j}|, \\ |l_{j}| & = |l′_{j}|, \\ |α_{j} ∘ α_{1}| & = |α′_{j} ∘ α′_{1}|, \end{aligned}  \begin{aligned}0 & ≤ j ≤ m; \\ 1 & ≤ j ≤ m; \\ 2 & ≤ j ≤ m. \end{aligned}

Since there are only finitely many extreme loop structures on S and S′, it is easy to see that the above discussion leads to an algorithm of comparing two Riemann surfaces of topological type (0, m + 1).

Theorem 2. The fundamental domain for the action of the modular group M on F can be realized by the following inequalities:

(∗′) \begin{aligned}|(ψ(g_{0})| & ≤ |ψ(g_{1})| ≤ |(ψ(g_{j})|,  2 ≤ j ≤ m; \\ |(ψ(g_{0} ∘ g_{1})| & ≤ min\{|(ψ(g_{0}^{−1} ∘ g_{2}^{−1} ∘ g_{1}^{−1} ∘ g_{2})|, \\ & |ψ(g_{0}^{−1} ∘ g_{3}^{−1} ∘ g_{2}^{−1} ∘ g_{1}^{−1} ∘ g_{2} ∘ g_{3})|, \\ & …, |(ψ(g_{0}^{−1} ∘ g_{m−1}^{−1} ∘ ⋯ ∘ g_{2}^{−1} ∘ g_{1}^{−1} ∘ g_{2} ∘ ⋯ ∘ g_{m−1})|\}; \\ & ⋯ \\ |ψ(g_{0} ∘ g_{m})| & ≤ min\{|(ψ(g_{0}^{−1} ∘ g_{1}^{−1} ∘ g_{m}^{−1} ∘ g_{1})|, \\ & |ψ(g_{0}^{−1} ∘ g_{2}^{−1} ∘ g_{1}^{−1} ∘ g_{m}^{−1} ∘ g_{1} ∘ g_{2})|, \\ & …, |(ψ(g_{0}^{−1} ∘ g_{m−2}^{−1} ∘ ⋯ ∘ ∘g_{1}^{−1} ∘ g_{m}^{−1} ∘ g_{1} ∘ ⋯ ∘ g_{m−2})|\}. \end{aligned}

Proof. Let q = (ψ(g_{i})) ∈ F and denote by S_{q} the Riemann surface attached to q. Choose 0 ≤ i_{0} , i_{1} ≤ m, i_{0} ≠ i_{1} , such that |ψ(g_{i_{0}})| ≤ |ψ(g_{i_{1}})| ≤ |ψ(g_{j})| for all j ≠ i_{0}, i_{1} . According to Lemma 1, there exists an extreme loop structure on S_{q} with the center and initial covered by ψ(g_{i_{0}}) and ψ(g_{i_{1}}), respectively. Thus, applying the lifting procedure, we get a set of standard generators {ψ′(g_{j}) : 0 ≤ j ≤ m} in the covering group such that (∗′) hold, which implies that q is modular equivalent to a point satisfying our restricting condition (∗′). Now let q_{1} = (ψ_{1}(g_{i})) and q_{2} = (ψ_{2}(g_{i})) be two distinct points in F. Suppose

A fundamental domain for the modular group of Riemann surfaces 281

that for these two points, all inequalities in (∗′) hold strictly. Assume ψ_{i}(g) = φ_{i} ∘ g ∘ φ_{i}^{−1} , g ∈ G, where φ_{i} (i = 1, 2) are the associated orientation preserving homeomorphisms. Denote by φ̃_{i} the projections of φ_{i} (i = 1, 2), i.e., φ̃_{i}: S → S_{q_{i}}. It is easy to see that ψ_{i}(g_{j}) cover φ̃_{i}(α_{j}) (0 ≤ j ≤ m) and ψ_{i}(g_{0} ∘ g_{k}) cover φ̃_{i}(α_{k}∘α_{0}) (1 ≤ k ≤ m). Hence the inequalities in (∗′) guarantee that for i = 1, 2, E_{i} = {φ̃_{i}(A_{0}), φ̃_{i}(A_{j}), φ̃_{i}(α_{j} ∘α_{0})}_{j=1}^{m} is an extreme loop structure on S_{q_{i}} with the center φ̃_{i}(A_{0}) and initial φ̃_{i}(A_{1}). Suppose there is a conformal homeomorphism f: S_{q_{1}} → S_{q_{2}} . Then from the corollary of Theorem 1 and (∗′), it is easy to verify that f(φ̃_{1}(α_{j})) ∼ φ̃_{2}(α_{j}), 0 ≤ j ≤ m, and f(φ̃_{1}(α_{k} ∘ α_{0})) ∼ φ̃_{2}(α_{k} ∘ α_{0}), 1 ≤ k ≤ m. It follows that f ∘ φ̃_{1} ∼ φ̃_{2} , which implies that for all g ∈ G, ψ_{1}(g) differ from ψ_{2}(g) only by a conjugation of an element in ℳ . Thus q_{1} = q_{2} , which is a contradiction, and the theorem is proved.

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Received 12 March 1996