Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 187–214

GENERALIZED HECKE GROUPS AND HECKE POLYGONS

Shuechin Huang

No. 17, Lane 42, Sec. 2, Chung-Shan N. Rd., Taipei, Taiwan, R.O.C.; shuang@ceec.edu.tw

Abstract. In this paper, we study certain Fuchsian groups ℋ (p_{1}, …, p_{n}), called generalized Hecke groups. These groups are isomorphic to ^{∏∗} _{j=1}^{n}Z_{p_{j}} . Let Γ be a subgroup of finite index

in ℋ (p_{1}, …, p_{n}). By Kurosh’s theorem, Γ is isomorphic to F_{r} ∗ ^{∏∗} _{i=1}^{k}Z_{m_{i}}, where F_{r} is a free group of rank r, and each m_{i} divides some p_{j}. Moreover, H^{2}/Γ is Riemann surface. The numbers m_{1}, …, m_{k} are branching numbers of the branch points on H^{2}/Γ. The signature of Γ is (g; m_{1}, …, m_{k}; t), where g and t are the genus and the number of cusps of H^{2}/Γ, respectively.

A purpose of this paper is to consider two problems. First, determine the necessary and sufficient conditions for the existence of a subgroup of finite index of a given type in ℋ (p_{1}, …, p_{n}). We also extend this work to extended generalized Hecke groups ℋ ^{∗}(p_{1}, …, p_{n}) which are isomorphic to D_{p_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{p_{n}} (amalgamated over Z_{2} ’s generated by reflections), where each D_{p_{j}} is a dihedral group of order 2p_{j}.

The second problem is the realizability problem for the existence of a subgroup with a given signature in ℋ (p_{1}, …, p_{n}). This is a special case of the Hurwitz problem about the realizability of branched covers. Special cases of this work were also studied by Millington, Singerman, Hoare, Edmonds, Ewing and Kulkarni. Our approach is based on constructing special Poincaré polygons which are the same as fundamental domains for ℋ (p_{1}, …, p_{n}), ℋ ^{∗}(p_{1}, …, p_{n}) and their subgroups.

1. Introduction

Suppose that integers p_{1}, p_{2}, …, p_{n} are given, where each p_{j} ≥ 2. The purpose of this paper is to study the geometry and topology of a Fuchsian group ℋ (p_{1}, …, p_{n}), called a generalized Hecke group, and its certain extension ℋ ^{∗}(p_{1}, …, p_{n}), called an extended generalized Hecke group. As an abstract group, ℋ (p_{1}, …, p_{n}) is isomorphic to ^{∏∗} _{i=1}^{n}Z_{p_{i}}, and ℋ ^{∗}(p_{1}, …, p_{n}) is isomorphic to D_{p_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{p_{n}} (amalgamated over Z_{2} ’s generated by reflections), where throughout the paper ^{∏∗} denotes a free product of groups, each Z_{p_{j}} is a finite cyclic group of order p_{j}, and each D_{p_{j}} is a dihedral group of order 2p_{j}; cf. Section 2.

Let Γ be a subgroup of finite index in ℋ (p_{1}, …, p_{n}). Then H^{2}/Γ is a Riemann surface. Let g and t be the genus and the number of cusps of H^{2}/Γ respectively, and let m_{1}, …, m_{k} be the branching numbers of the branch points on H^{2}/Γ. The signature of Γ is (g; m_{1}, …, m_{k}; t).

It follows from Kurosh’s theorem that a subgroup of a generalized Hecke group ∏∗ _{i=1}^{n}Z_{p_{i}} is isomorphic to F ∗ (^{∏∗} _{i=1}^{k}Z_{m_{i}}), where F is a free group, and each

1991 Mathematics Subject Classification: Primary 30F.

m_{j} divides some p_{i}, for j = 1, …, k. A group ^{∏∗} _{i=1}^{n}Z_{p_{i}} may not always contain a subgroup of a given type. For instance, Z_{2} ∗ Z_{2} ∗ Z_{2} ∗ Z_{2} does not embed in Z_{3} ∗ Z_{6} as a subgroup of index 2. Indeed, it is easy to see that there is a unique normal subgroup of index 2 in Z_{3} ∗ Z_{6} , and it is isomorphic to Z_{3} ∗ Z_{3} ∗ Z_{3} .

Millington [11] investigated the existence of subgroups with given signatures in the modular group which is isomorphic to Z_{2} ∗ Z_{3} . We state Millington’s theorem as follows.

Theorem 1.1. Let d, k_{1} , k_{2} , g, t be nonnegative integers, and t, d ≥ 1. If the Riemann–Hurwitz relation

d = 3k_{1} + 4k_{2} + 12g + 6t − 12

holds, the modular group contains a subgroup of index d and with a signature

(g; \underbrace{2, …, 2}_{k_{1}}, \underbrace{3, …, 3}_{k_{2}}; t).

This result was partially extended. A group Γ can be embedded as a subgroup of index d in Z_{p_{1}} ∗ Z_{p_{2}} , where p_{1}, p_{2} are distinct primes if and only if the Euler characteristic condition is satisfied, i.e. χ(Γ) = dχ(Z_{p_{1}} ∗ Z_{p_{2}}) [6, Theorem 5.1], where χ is the Euler characteristic of a group in the sense of Wall; cf. [15]. Notice that this result is partial since we do not know whether the group can be realized as a Fuchsian group with a prescribed signature, subject to Euler characteristic (that is the same as Riemann–Hurwitz) condition. However when p_{1}, …, p_{n} are not distinct primes, the Riemann–Hurwitz condition is not sufficient to embed a group as a subgroup of finite index in ^{∏∗} _{i=1}^{n}Z_{p_{i}}.

In [6], Kulkarni derived a further necessary condition, a diophantine condition, and showed that this condition together with the Riemann–Hurwitz condition is also sufficient to embed a group F_{r} ∗ ^{∏∗}_{m}Z_{m} in ^{∏∗} _{i=1}^{n}Z_{p_{i}} as a subgroup of finite index, where henceforth F_{r} denotes a free group of rank r. We describe this theorem as follows.

Theorem 1.2. Let k, r be nonnegative integers. Let Γ = ^{∏∗} _{i=1}^{n}Z_{p_{i}}, and Φ = F_{r} ∗ ^{∏∗} _{i=1}^{k}Z_{m_{i}}, where each m_{i} divides some p_{j}. Then Φ can be realized as a subgroup of Γ of index d if and only if the following conditions are satisfied:

(i) (The Riemann–Hurwitz condition)

∑_{i=1}^{k} \frac{1}{m_{i}} − (k + r) + 1 = d(∑_{i=1}^{n} \frac{1}{p_{i}} − n + 1).

(ii) (The diophantine condition) Let m_{0} = 1, and let m_{1}, …, m_{s} be the maximal set of distinct m_{i}, where each m_{j}, 1 ≤ j ≤ s, occurs b_{j} times. Set

ε_{ij} = \{ \begin{matrix}0 & if m_{j} |̸ p_{i}, \\ 1 & if m_{j} | p_{i}, \end{matrix} δ_{ij} = \frac{p_{i}}{m_{j}} ε_{ij}.

Then the system

\begin{aligned}∑_{i=1}^{n} ε_{ij}x_{ij} & = b_{j},  j = 1, …, s, \\ ∑_{j=0}^{s} δ_{ij}x_{ij} & = d,  i = 1, …, n \end{aligned}

has a solution for x_{ij} in nonnegative integers.

Moreover Kulkarni [7] extended Millington’s theorem to Z_{p_{1}} ∗ Z_{p_{2}} .

Theorem 1.3. Let k, g, t, r be nonnegative integers, where t ≥ 1, r = 2g + t − 1. Let Γ = Z_{p_{1}} ∗ Z_{p_{2}} , and Φ = F_{r} ∗^{∏∗} _{i=1}^{k}Z_{m_{i}}, where each m_{i} divides p_{1} or p_{2} . Then Φ can be realized as a subgroup of Γ of index d and with a signature (g; m_{1}, …, m_{k}; t) if and only if the following conditions are satisfied:

(i) (The Riemann–Hurwitz condition)

∑_{i=1}^{k} \frac{1}{m_{i}} − (k + r) + 1 = d(\frac{1}{p_{1}} + \frac{1}{p_{2}} − 1).

(ii) (The diophantine condition) Let m_{0} = 1, and let m_{1}, …, m_{s} be the maximal set of distinct m_{i}, where each m_{j}, 1 ≤ j ≤ s, occurs b_{j} times. Set

ε_{ij} = \{ \begin{matrix}0 & if m_{j} |̸ p_{i}, \\ 1 & if m_{j} | p_{i}, \end{matrix} δ_{ij} = \frac{p_{i}}{m_{j}} ε_{ij}.

Then the system

\begin{aligned}ε_{1j}x_{1j} + ε_{2j}x_{2j} & = b_{j},  j = 1, …, s, \\ ∑_{j=0}^{s} δ_{ij}x_{ij} & = d,  i = 1, 2, \end{aligned}

has a solution for x_{ij} in nonnegative integers.

A motivation of this paper was to study realizability of signatures by subgroups of finite index in ℋ (p_{1}, …, p_{n}) considered as a Fuchsian group.

A noncocompact Fuchsian group Γ is a free product of cyclic groups. A system of generators for Γ is said to be independent if the group is a free product of cyclic subgroups generated by elements in the generating system. This notion is due to Rademacher; cf. [12]. A fundamental domain P for Γ is called an admissible fundamental domain for Γ if the side pairings of P is an independent system of generators for Γ; cf. [7]. A fundamental domain is in general not admissible. Indeed, the usual fundamental domain for the modular group and the well-known fundamental domain constructed by Fricke for congruence subgroups are not admissible. In Section 2, we introduce a special kind of Poincaré polygon, called a Hecke polygon, which is an admissible fundamental domain for the group generated by the side pairings of it.

There is a correspondence between Hecke polygons and subgroups of finite index in ℋ (p_{1}, …, p_{n}). Each subgroup of finite index in ℋ (p_{1}, …, p_{n}) admits an admissible fundamental domain which is a Hecke polygon. From this result, we give new proofs of Theorems 1.2 and 1.3 by constructing a Hecke polygon. Meanwhile the diophantine condition (which is the same as the integrality condition of Theorem 1.4) is interpreted geometrically as the relationship between the index of a subgroup and the number of Ω_{j} -polygons of a Hecke polygon; cf. Section 3. In our set-up Theorem 1.2 is restated as follows.

Theorem 1.4. Let k_{0} = 0, k_{1}, …, k_{n}, r be nonnegative integers, where k_{i} ≤ k_{i+1}, for i = 0, …, n − 1. Let Γ = F_{r} ∗ ^{∏∗} _{j=1}^{n}^{∏∗} _{i=k_{j−1}+1}^{k_{j}}Z_{p_{j}/m_{i}}, where m_{i} | p_{j}, i = k_{j−1} + 1, …, k_{j}, j = 1, …, n. Then Γ can be embedded in ℋ (p_{1}, …, p_{n}) as a subgroup of index d if and only if the following conditions hold:

(i) (The Riemann–Hurwitz condition)

∑_{j=1}^{n} ∑_{i=k_{j−1}+1}^{k_{j}} \frac{m_{i}}{p_{j}} − (k_{n} + r) + 1 = d(∑_{j=1}^{n} \frac{1}{p_{j}} − n + 1).

(ii) (The integrality condition) The numbers s_{1}, …, s_{n} satisfying

s_{j}p_{j} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i} = d, j = 1, …, n,

are nonnegative integers.

In particular, if p_{1}, …, p_{n} are distinct primes, the integrality condition reduces to d ≥ k_{j}, j = 1, …, n, where k_{j} is the number of copies of Z_{p_{j}} ’s in Γ (see Corollary 5.2).

In Section 4, we study a special kind of a NEC (non-euclidean crystallographic) group ℋ ^{∗}(p_{1}, …, p_{n}) in which ℋ (p_{1}, …, p_{n}) is a subgroup of index 2.

The algebraic structure of a NEC group with noncompact quotient space was determined by Macbeath and Hoare [9]. It follows that each subgroup of finite index in D_{p_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{p_{n}} is isomorphic to F_{r} ∗ ^{∏∗}_{m}Z_{m} ∗ ^{∏∗}_{i}(D_{x_{i1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{x_{ik_{i}}}) ∗ ^{∏∗}_{j}E_{j}, where each m divides some p_{j}, each x_{ij} divides some p_{l}, and each E_{j} has a presentation

⟨y_{j}, a_{j1}, …, a_{js_{j}} | a_{j1}y_{j}a_{js_{j}}y_{j}^{−1} = a_{jl}^{2} = a_{jl+1}^{2} = (a_{jl}a_{jl+1})^{u_{jl}} = 1, l = 2, …, s_{j}−1⟩.

We extend Theorem 1.4 in the case of subgroups of finite index in ℋ ^{∗}(p_{1}, …, p_{n}). In this case, the necessary and sufficient conditions are still called the Riemann– Hurwitz and diophantine conditions (see Theorem 4.2). When p_{1}, …, p_{n} are distinct primes, the diophantine condition can be stated in a more concise way (see Theorem 5.1).

Singerman [14] gave a permutation-theoretic approach to the realizability problem for signatures of subgroups of finitely generated Fuchsian groups. A generalization to NEC groups was done by Hoare [4]. Singerman’s theorem is as follows.

Theorem 1.5. Suppose that Γ has a presentation

\begin{aligned}⟨a_{1}, b_{1}, …, a_{g}, b_{g}, x_{1}, …, x_{r}, f_{1}, …, f_{t} | x_{1}^{m_{1}} & = ⋯ = x_{r}^{m_{r}} \\ & = ∏_{i=1}^{g}[a_{i}, b_{i}] ∏_{j=1}^{r} x_{j} ∏_{k=1}^{t} f_{k} = 1⟩ \end{aligned}

with a signature (g; m_{1}, …, m_{r}; t). Then Γ contains a subgroup Φ of index d with a signature (h; n_{11}, n_{12}, …, n_{1ρ_{1}}, …, n_{r1}, n_{r2}, …, n_{rρ_{r}}; s) if and only if there exists a finite permutation group G transitive on d points, and an epimorphism θ: Γ → G satisfying the following conditions:

(i) The permutation θ(x_{j}) has precisely ρ_{j} disjoint cycles of lengths m_{j}/n_{j1} , …, m_{j}/n_{jρ_{j}}.

(ii) If δ(f) denotes the number of cycles in the permutation θ(f), then s = ∑_{i=1}^{t} δ(f_{i}).

In Section 6, we show how to associate a system of permutations to a Hecke polygon such that the signature of the group generated by the side pairings of this polygon can be determined from the action of those permutations. The permutations which we construct (in the special case of generalized Hecke groups) are different from the ones in Singerman’s theorem. In particular, we use permutations to construct the appropriate Hecke polygon, and in fact get an explicit geometric realization of the corresponding surface.

It is of interest to note that the Riemann–Hurwitz and diophantine conditions are not sufficient for the existence of a subgroup with a prescribed signature in ℋ (p_{1}, …, p_{n}) if n ≥ 3. An obvious additional necessary end-condition for the existence of a subgroup Γ of index d in a group is that the number t of cusps of the quotient space H^{2}/Γ is at most d. This condition does not follow from the Riemann–Hurwitz or diophantine condition; cf. the example in Section 7. The realizability problem for the existence of a subgroup of ℋ (p_{1}, …, p_{n}) with a given signature for any possible t ≤ d is still open. Indeed even for torsion free subgroups, this problem appears to be difficult. Curiously, in the cocompact case for the torsion free subgroups, only Riemann–Hurwitz condition is sufficient; cf. [2]. In our case, the result in [2] implies that if n ≥ 3 and t | d, there exists a torsion free subgroup of index d whose corresponding surface has t cusps; cf. Theorem 7.2. Here we use a different approach and consider the realizability of torsion free subgroups with t ≤ d. Special cases are dissussed in Section 7. Some further cases for groups with torsions in the cocompact case are dealt with in [3].

There is a close relation between the Hurwitz problem on realizability of a branched covering of a sphere and the problem of the existence of a subgroup of finite index in ℋ (p_{1}, …, p_{n}) [5]. Given a subgroup Γ of index d in ℋ (p_{1}, …, p_{n}), let π_{Γ}: H^{2} → H^{2}/Γ be the natural projection. Then π_{Γ} and π_{ℋ (p_{1},...,p_{n})} induce a branched covering ϕ: H^{2}/Γ → H^{2}/ℋ (p_{1}, …, p_{n}) of degree d of a punctured sphere H^{2}/ℋ (p_{1}, …, p_{n}). In particular, in any of the cases (i) p_{1} = p_{n} = 2, p_{2} = ⋯ = p_{n−1} = p (ii) n = 4, p_{j} ≥ 4, (j = 1, …, 4) (iii) n = 5, p_{j} ≥ 3, (j = 1, …, 5) (iv) n ≥ 6, p_{j} ≥ 2, (j = 1, …, n), the covering space H^{2}/Γ of genus g with t cusps of a once-punctured sphere H^{2}/ℋ (p_{1}, …, p_{n}), branched at {x_{1}, …, x_{k}} to order {2, …, 2, p …, p} for case (i) and to order {p_{1}, …, p_{1}, …, p_{n}, …, p_{n}} for case (ii), (iii), (iv), can be realized for any t ≤ d; cf. Corollary 7.4 and Corollary 7.8.

I am grateful to Professor Ravi Kulkarni for his assistance on this paper. I would also like to thank Professor Linda Keen and Professor Frederick Gardiner for their encouragement.

2. Hecke polygons

Let p_{1}, p_{2}, …, p_{n}, be integers, where each p_{j} ≥ 2. For each j = 1, …, n − 1, let C_{j} be a circle |z − a_{j}| = δ_{j}, where a_{j} ∈ R, a_{j} < a_{j+1} , and δ_{j}^{2} + δ_{j+1}^{2} ≤ (a_{j} − a_{j+1})^{2} < (δ_{j} + δ_{j+1})^{2} . Then C_{j} intersects only C_{j−1} and C_{j+1} , for j = 2, …, n − 2. Suppose that C_{j−1} and C_{j} intersect at a point b_{j} ∈ H^{2} with an angle π/p_{j}, for j = 2, …, n − 2. Let b_{1} = a_{1} − δ_{1}e^{−πi/p_{1}} and b_{n} = a_{n−1} + δ_{n−1}e^{πi/p_{n}}. Let 𝒟^{∗} be the hyperbolic polygon with vertices at b_{1}, …, b_{n}, and ∞. An extended generalized Hecke group ℋ ^{∗}(p_{1}, …, p_{n}) is a discrete group generated by the reflections in the edges of 𝒟^{∗}. The stabilizers of each vertex b_{j} and each edge of 𝒟^{∗} are D_{p_{j}} and Z_{2} respectively, where Z_{2} ’s are reflections of the dihedral groups D_{p_{j}}, i.e., the elements in the nonidentity coset of the rotation group Z_{p_{j}}. Therefore ℋ ^{∗}(p_{1}, …, p_{n}) is isomorphic to D_{p_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{p_{n}}.

Let ℋ (p_{1}, …, p_{n}) be the subgroup of ℋ ^{∗}(p_{1}, …, p_{n}), called a generalized Hecke group, which consists of all orientation-preserving transformations in ℋ ^{∗}(p_{1}, …, p_{n}). Then ℋ (p_{1}, …, p_{n}) is isomorphic to ^{∏∗} _{j=1}^{n}Z_{p_{j}}.

We will need the following definitions. The elements of the ℋ ^{∗}(p_{1}, p_{2}, …, p_{n})- orbits of b_{j} and ∞ are called the b_{j} -vertices and the cusps, respectively, j = 1, …, n. Suppose that C_{j} and the hyperbolic line through a_{j} and ∞ intersect at a point c_{j}, for j = 1, …, n−1 (see Figure 1). The elements of the ℋ ^{∗}(p_{1}, …, p_{n})- orbits of c_{j} ’s are called the c_{j} -vertices. The elements of the ℋ ^{∗}(p_{1}, …, p_{n})- orbits of the edges joining c_{j} to ∞ are called the c_{j} -edges. The elements of the ℋ ^{∗}(p_{1}, …, p_{n})-orbits of the edges joining b_{j} to ∞ are called b_{j} -edges. The elements of ℋ ^{∗}(p_{1}, …, p_{n})-orbits of the edges joining b_{j} to c_{j} and c_{j} to b_{j+1} respectively, for j = 1, …, n, are called e_{j} -edges and f_{j} -edges respectively. Each of the e_{j} - and f_{j} -edges has finite length, and each of the b_{j} -edges has infinite length. The hyperbolic line joining a_{j} to ∞ consists of two c_{j} -edges, for j = 1, …, n − 1.

Figure 1. A fundamental polygon 𝒟^{∗} for ℋ ^{∗}(p_{1}, p_{2}, p_{3}).

Its ℋ ^{∗}(p_{1}, …, p_{n}) translates are called the c_{j} -lines.

The ℋ ^{∗}(p_{1}, …, p_{n}) translates of the polygon with vertices at {b_{1}, c_{1}, ∞}, {c_{1}, b_{2}, c_{2}, ∞}, …, {c_{n−2}, b_{n−1}, c_{n−1}, ∞} and {c_{n−1}, b_{n}, ∞}, respectively, are called the Ω_{j}^{∗} -polygons. The ℋ (p_{1}, …, p_{n}) translates of the polygon with vertices at {b_{1}, a_{1}, ∞}, {a_{1}, b_{2}, a_{2}, ∞}, …, {a_{n−2}, b_{n−1}, a_{n−1}, ∞} and {a_{n−1}, b_{n}, ∞}, respectively, are called the Ω_{j} -polygons. If each p_{j} is greater than 2, then Ω_{1} - and Ω_{n} -polygons are triangles, and the rest of Ω_{j} -polygons are quadrilaterals.

Let Δ_{j}^{∗} and Δ̃_{j}^{∗} be triangles with vertices at {b_{j}, c_{j}, ∞} and {c_{j}, b_{j+1}, ∞}, where j = 1, …, n − 1. For j = 1, …, n − 1, let Δ_{j} = Δ_{j}^{∗} ∪ σ_{j}(Δ_{j}^{∗}) and Δ̃_{j} =

Δ̃_{j}^{∗} ∪ σ_{j} ˜(Δ_{j}^{∗}), where σ_{j} is a reflection in the circle C_{j}.

A usual construction of a fundamental domain for ℋ (p_{1}, …, p_{n}) would be 𝒟^{∗} ∪ σ(𝒟^{∗}), where σ is a reflection in an edge of 𝒟^{∗}. But we find it more convenient to take 𝒟 = ⋃_{j=1}^{n−1}(Δ_{j} ∪ Δ̃_{j}) as a fundamental domain (see Figure 2).

Figure 2. A fundamental polygon for ℋ (p_{1}, p_{2}, p_{3}).

A Hecke polygon is defined to be a convex polygon P whose boundary is a finite union of c_{j} -lines and b_{j} -edges satisfying the following conditions:

S_{1} . Each c_{j} -line in ∂P is paired to another c_{j} -line in ∂P such that one of them is a side of an Ω_{j} -polygon in P, and the other is a side of an Ω_{j+1} -polygon in P.

S_{2} . The b_{j} -edges in ∂P come in pairs. The edges of each pair meet at a b_{j} -vertex with an interior angle 2kπ/p_{j}, where k | p_{j}, and are identified.

S_{3} . a_{1}, …, a_{n−1} , and ∞ are among the vertices of P.

The main point about Hecke polygons is the following theorem.

Theorem 2.1. Let P be a Hecke polygon, and let Γ_{P} be the subgroup of ℋ (p_{1}, …, p_{n}) generated by the side pairing transformations of P. Then P is an admissible fundamental domain for Γ_{P} . Conversely, every subgroup of finite index in ℋ (p_{1}, …, p_{n}) admits an admissible fundamental domain which is a Hecke polygon.

Proof. The argument is similar to the one in Theorem 3.3 [7]. Suppose that P is a Hecke polygon and that Γ_{P} is the subgroup of ℋ (p_{1}, …, p_{n}) generated by the side pairing transformations of P. It follows from the Poincaré polygon theorem [10, Section IV.H] that the set S of the side pairing transformations is an independent set of generators of Γ_{P} , that is, Γ_{P} = ^{∏∗}_{f∈S}⟨f⟩. So the fundamental polygon P is an admissible fundamental domain for Γ_{P} .

Conversely, suppose that Γ is a subgroup of finite index in ℋ (p_{1}, …, p_{n}). Let 𝒯 ^{∗} be the tessellation of H^{2} whose tiles are ℋ ^{∗}(p_{1}, …, p_{n}) translates of 𝒟^{∗}. Let φ: H^{2} → H^{2}/Γ be the canonical projection. Since Γ preserves 𝒯 ^{∗}, we have an induced tessellation 𝒯_{Γ}^{∗} of H^{2}/Γ. The φ-images of c_{j} -vertices, b_{j} -vertices, c_{j} -edges, b_{j} -edges, e_{j} - and f_{j} -edges will again be called c_{j} -vertices, b_{j} -vertices, c_{j} -edges, b_{j} -edges, e_{j} - and f_{j} -edges, respectively, in H^{2}/Γ. Let ℰ be the union of the e_{j} - and f_{j} -edges in H^{2}/Γ. Consider ℰ as a graph whose vertices are the c_{j} -vertices and b_{j} -vertices in H^{2}/Γ, and whose edges are the e_{j} - and f_{j} - edges in H^{2}/Γ. Note that each c_{j} -vertex is of valence 2, and each b_{j} -vertex is of valence 1 or k (respectively 2 or 2k), where k | p_{j}, if j = 1, n, (respectively j = 2, …, n − 1).

Since the union of the e_{j} - and f_{j} -edges in H^{2} is connected, so is ℰ . Let T be a maximal tree in ℰ . Let A be the union of all the c_{j} -edges in H^{2}/Γ at the c_{j} -vertices of valence 1 and all the b_{j} -edges at the b_{j} -vertices of valence k and 2k, where k | p_{j}, k ≠ p_{j}, in T. Make H^{2}/Γ into a polygon P in H^{2} by cutting A such that a_{1}, …, a_{n−1}, and ∞ are among the vertices of P. For each c_{j} -vertex u and each b_{j} -vertex v in A, there is a pair of c_{j} -lines and a pair of b_{j} -edges adjacent to u and v, respectively. Correspondingly, we obtain a pair of c_{j} -lines (respectively b_{j} -edges) on ∂P which are paired. Hence P is a Hecke polygon which is a fundamental domain for Γ.

3. A new proof of an extension of Kurosh’s theorem

We now give a new proof of an extension of Kurosh’s theorem to the groups ℋ (p_{1}, …, p_{n}). We mean by a (k + 2)-gon (respectively 2k + 2-gon) centered at a b_{j} -vertex a (k + 2)-gon (respectively (2k + 2)-gon) consisting of k Ω_{j} -polygons with a common b_{j} -vertex which attach to each other along the b_{j} -edges, where k | p_{j}, j = 1, n (respectively j = 2, …, n − 1), provided that p_{1}, p_{n} ≠ 2 (see Figure 3).

Proof of Theorem 1.4. Let Γ be a subgroup of index d in ℋ (p_{1}, …, p_{n}). Let P be the Hecke polygon for Γ, and let s_{j} be the number of ideal p_{j} -gons or 2p_{j} -gons centered at b_{j} -vertices in P. Then s_{j}p_{j} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i} is the total number of Ω_{j} -polygons in P, for j = 1, …, n. Hence the conditions (i) and (ii)

Figure 3. A 5-gon centered at e^{πi/6} for a group ℋ ^{∗}(6, 3).

follow directly from the Gauss–Bonnet theorem and the geometric interpretation of the Hecke polygon P.

Conversely, suppose that conditions (i) and (ii) hold. Substituting s_{j} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i}/p_{j} for d/p_{j}, j = 1, …, n in condition (i), we have

r = (n − 1)d − k_{n} − ∑_{j=1}^{n} s_{j} + 1.

Without loss of generality, we may assume that s_{1} ≥ s_{n} and s_{j} ≥ 2s_{n} − 1, for all j ≠ 1, n. We shall find a Hecke polygon P for Γ which consists of s_{j} ideal p_{j} -gons or 2p_{j} -gons (an ideal polygon in H^{2} is a hyperbolic polygon with vertices at the circle at infinity R∪{∞}), and the (m_{i} +2)-gon or (2m_{i} +2)-gon centered at a b_{j} -vertex, for i = k_{j−1} + 1, …, k_{j}, and j = 1, …, n, such that a subgroup of ℋ (p_{1}, …, p_{n}) generated by the side pairing transformations of P is Γ.

Start with an ideal p_{1} -gon Q_{1} centered at b_{1} . Then attach an ideal 2p_{2} -gon to Q_{1} along the c_{1} -line through ∞ and obtain a new polygon Q_{2} . Next attach an ideal 2p_{3} -gon to Q_{2} along the c_{2} -line through ∞. Continuing in this way, after 2(n − 1)s_{n} − n + 2 steps we obtain a polygon P_{0} whose boundary consists of c_{j} -lines and which contains s_{n} Ω_{1} -polygons, s_{n} Ω_{n} -polygons, and (2s_{n} − 1) Ω_{j} -polygons, for j = 2, …, n − 1.

Now there are s_{1} − s_{n} ideal p_{1} -gons, s_{j} − 2s_{n} + 1 ideal 2p_{j} -gons, for j = 2, …, n − 1, and the (m_{i} + 2)-gon or (2m_{i} + 2)-gon centered at a b_{j} -vertex, for i = k_{j−1} + 1, …, k_{j}, j = 1, …, n, to be attached. For each j = 1, …, n − 1, the number of c_{j} -lines on the boundary of those polygons and P_{0} that are sides of Ω_{j} -polygons or Ω_{j+1} -polygons is

\{ \begin{matrix}s_{n}(p_{j} − 2) + 1 + (s_{j} − s_{n})p_{1} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i}, & j = 1, n, \\ (2s_{n} − 1)(p_{j} − 1) + (s_{j} − 2s_{n} + 1)p_{j} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i}, & j ≠ 1, n, \end{matrix}

= \{ \begin{matrix}d − 2s_{n} + 1, & j = 1, n, \\ d − 2s_{n} + 1, & j ≠ 1, n. \end{matrix}

Hence, after attaching those

s_{1} − s_{n} + ∑_{j=2}^{n−1}(s_{j} − 2s_{n} + 1) + ∑_{j=1}^{n}(k_{j} − k_{j−1}) = k_{n} + ∑_{j=1}^{n} s_{j} − 2(n − 1)s_{n} + n − 2

polygons to P_{0} , we have

(n−1)(d−2s_{n}+1)−[k_{n}+∑_{j=1}^{n} s_{j} −2(n−1)s_{n}+n−2] = (n−1)d−k_{n}−∑_{j=1}^{n} s_{j} +1 = r

pairs of c_{j} -lines, and each pair consists of a side of an Ω_{j} -polygon and a side of an Ω_{j+1} -polygon on the boundary. Therefore we obtain a convex polygon P whose boundary is the union of k_{j} − k_{j−1} pairs of b_{j} -edges making an interior angle 2m_{i}π/p_{j}, where i = k_{j−1} + 1, …, k_{j}, j = 1, …, n, and r pairs of c_{j} -lines.

Each pair of b_{j} -edges of an interior angle 2m_{i}π/p_{j} are identified. Each pair of 2r c_{j} -lines are identified. Now P becomes a Hecke polygon. Then a subgroup of ℋ (p_{1}, …, p_{n}) generated by the side pairing transformations of P is isomorphic to Γ.

For the case n = 2 in Theorem 1.4, one can pair the r pairs of c_{j} -lines on ∂P as in the proof with the desired patterns. We state this result as a corollary of Theorem 1.4.

Theorem 3.1. Let k_{0} = 0, k_{1} , k_{2} , g, t, r be nonnegative integers, where k_{1} ≤ k_{2} , t ≥ 1, and r = 2g +t−1. Let Γ = F_{r} ∗^{∏∗} _{j=1}^{2}^{∏∗} _{i=k_{j−1}+1}^{k_{j}}Z_{p_{j}/m_{i}}, where m_{i} | p_{j}, i = k_{j−1} + 1, …, k_{j}, j = 1, 2. Then Γ can be embedded in ℋ (p_{1}, p_{2}) as a subgroup of index d and with a signature

(g; \frac{p_{1}}{m_{1}} , …, \frac{p_{1}}{m_{k_{1}}} , \frac{p_{2}}{m_{k_{1}+1}} , …, \frac{p_{2}}{m_{k_{2}}} ; t)

if and only if the following conditions hold:

(i) (The Riemann–Hurwitz condition)

∑_{j=1}^{2} ∑_{i=k_{j−1}+1}^{k_{j}} \frac{m_{i}}{p_{j}} − (k_{2} + r) + 1 = d(\frac{1}{p_{1}} + \frac{1}{p_{2}} − 1).

(ii) (The integrality condition) The numbers s_{1} , s_{2} satisfying

s_{j}p_{j} + ∑_{i=k_{j−1}+1}^{k_{j}} m_{i} = d, j = 1, 2,

are nonnegative integers.

4. Subgroups of finite index in ℋ ^{∗}(p_{1}, …, p_{n})

In this section we determine the necessary and sufficient conditions for the existence of a subgroup of finite index of a given type in ℋ ^{∗}(p_{1}, …, p_{n}).

Suppose that Γ^{∗} is a subgroup of finite index in ℋ ^{∗}(p_{1}, …, p_{n}) containing a reflection. Then S_{Γ^{∗}} = H^{2}/Γ^{∗} is a (possibly nonorientable) surface with boundary (see Figure 4). The boundary ∂S_{Γ^{∗}} is formed by the projection of the fixed lines of reflections in Γ^{∗}. Also, ∂S_{Γ^{∗}} contains a corner when the fixed lines of two reflections in Γ^{∗} intersect. Each component C of ∂S_{Γ^{∗}} is the projection of a simple curve C̃ in H^{2} which is either a finite union of e_{j} - and f_{j} -edges or the union of two of the b_{j} -edges and a finite number of the e_{j} - and f_{j} -edges, where any two consecutive edges intersect at a b_{j} -vertex v, and make an angle kπ/p_{j}, where k | p_{j}. If v is a center of a rotation which is the product of two reflections in Γ^{∗}, the stabilizer of v is isomorphic to a dihedral group D_{p_{j}/k}.

Figure 4. The marked point is an elliptic fixed point if it is in the interior, and a center of a rotation which is a product of two reflections if it is on the boundary.

We generalize the construction of Hecke polygons to extended Hecke polygons.

Definition. An extended Hecke polygon is a convex hyperbolic polygon P^{∗} of finite area containing a_{1} and ∞ as vertices such that each component of ∂P^{∗} is of one of the following forms:

(i) a c_{j} -line; (ii) a pair of b_{j} -edges making an interior angle 2kπ/p_{j}, where k | p_{j}; (iii) a simple curve which is the union of two of the b_{j} -edges and a finite number

of the e_{j} - and f_{j} -edges, satisfying the following conditions:

S_{1}^{∗}. Each c_{j} -line which is a side of an Ω_{j} -polygon in P^{∗} is paired to another c_{j} -line which is a side of an Ω_{j} -polygon in P^{∗} by an orientation-preserving or reversing transformation in ℋ ^{∗}(p_{1}, …, p_{n}).

S_{2}^{∗}. The b_{j} -edges of each pair as in (ii) are paired by a transformation in ℋ (p_{1}, …, p_{n}).

S_{3}^{∗}. Each of the e_{j} - and f_{j} -edges as in (iii) is paired to itself by a reflection in ℋ ^{∗}(p_{1}, …, p_{n}).

S_{4}^{∗}. Each of the b_{j} -edges as in (iii) is paired to itself by a reflection or to the other b_{j} -edge on the same component of ∂P by an orientation-preserving transformation in ℋ ^{∗}(p_{1}, …, p_{n}).

Note that the group ℋ ^{∗}(p_{1}, …, p_{n}) may contain a subgroup with a fundamental domain whose boundary has only one component, and contains only one cusp ∞ as a vertex. Such a fundamental domain is not an extended Hecke polygon. In this case, this subgroup is isomorphic to D_{m_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{m_{k}}, where each m_{i} divides some p_{j} and Z_{2} ’s are generated by reflections.

Theorem 4.1. Let P^{∗} be an extended Hecke polygon, and let Γ_{P^{∗}} be the subgroup of ℋ ^{∗}(p_{1}, …, p_{n}) generated by the side pairing transformations of P^{∗}. Then P^{∗} is a fundamental domain for Γ_{P^{∗}}, and Γ_{P^{∗}} is a subgroup of finite index in ℋ ^{∗}(p_{1}, …, p_{n}) which is isomorphic to a free product of the groups Z, Z_{r}, and D_{m_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{m_{k}}, where r, and each m_{i} divide some p_{j}. Conversely, every subgroup of finite index in ℋ ^{∗}(p_{1}, …, p_{n}) but ≠ D_{m_{1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{m_{k}}, where each m_{i} divides some p_{j}, admits an extended Hecke polygon.

Proof. The proof is similar to that of Theorem 2.1. The first assertion follows from the Poincaré polygon theorem.

Suppose that Γ is a subgroup of finite index in ℋ ^{∗}(p_{1}, …, p_{n}). Let ℰ^{∗} be the union of e_{j} - and f_{j} -edges in H^{2}/Γ. Let T^{∗} be the maximal tree in ℰ^{∗}. Let A^{∗} be the union of all the c_{j} -edges in H^{2}/Γ at the c_{j} -vertices of valence 1 and all the b_{j} -edges at the b_{j} -vertices of valence k and 2k in T^{∗}, where k | p_{j}, k ≠ p_{j}. Now as in the argument of Theorem 2.1, cut H^{2}/Γ open along the edges in A^{∗} into a set which is isometric to a simply connected convex hyperbolic polygon P and then obtain an extended Hecke polygon which is a fundamental domain for Γ.

We take a positive orientation on H^{2} to be the usual counterclockwise orientation on H^{2} . Suppose that P^{∗} is an extended Hecke polygon for Γ^{∗}. Let C be a boundary component of S_{Γ^{∗}} = H^{2}/Γ^{∗} which is the projection of a simple curve C̃ on ∂P^{∗}. Suppose that {w_{1}, w_{2}, …, w_{k}} is a set of the b_{j} -vertices on C̃ in positive order on ∂P^{∗} such that w_{1} and w_{k} are on the infinite edges. Note that for each j, no two b_{j} -vertices are adjacent along C̃. Let π: H^{2} → H^{2} be the projection map. Suppose that the corresponding stabilizer of w_{j} is D_{m_{j}}. If π(w_{1}) ≠ π(w_{k}), the ordered set (m_{1}, m_{2}, …, m_{k}) is called a boundary cycle on C̃ for P^{∗}. If π(w_{1}) = π(w_{k}), the ordered set (m_{1}, m_{2}, …, m_{k}) is called a closed boundary cycle on C̃ for P^{∗}. Each m_{j} is called a branching number on the boundary. If w_{i} is a b_{j} -vertex, the integer p_{j}/m_{i} is the number of Ω_{j}^{∗} -polygons in P^{∗} with a vertex at w_{i}.

Suppose that (y_{1}/x_{1}, y_{2}/x_{2}, …, y_{k}/x_{k}) is a boundary cycle of Γ^{∗}, where x_{i} | y_{i} and y_{i} ∈ {p_{1}, …, p_{n}}. Let y_{i} = p_{j}, for some j. Then from the property of a Hecke polygon for Γ^{∗} we have the following results.

(i) y_{1}, y_{k} ∈ {p_{1}, p_{n}}.

(ii) If k = 1, then x_{1} is an even number, and if k > 1, then x_{1} and x_{k} are odd numbers.

(iii) If x_{i} is an odd number, then y_{i−1} = p_{j−1} , y_{i+1} = p_{j+1} , or y_{i−1} = p_{j+1} , y_{i+1} = p_{j−1} .

(iv) If x_{i} is an even number, then y_{i−1} = y_{i+1} = p_{j−1} , or y_{i−1} = y_{i+1} = p_{j+1} . The above results (i)–(iv) are also true for a close boundary cycle (y_{1}/x_{1} ,

y_{2}/x_{2}, …, y_{k}/x_{k}) except for (i) which now becomes y_{1} = y_{k} ∈ {p_{1}, p_{n}}. Suppose that

\begin{aligned}Γ & = F_{r} ∗^{∏∗}_{α=1}^{n}^{∏∗}_{i=k_{α−1}}^{k_{α}}Z_{p_{α}/m_{i}} \\ & ∗^{∏∗}_{i=1}^{h_{0}}^{∏∗}_{j=1}^{u_{i}}(D_{y_{ij1}/x_{ij1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{y_{ija_{ij}} /x_{ija_{ij}}}) ∗^{∏∗}_{i=h_{0}+1}^{h}E_{i} \end{aligned}

is a subgroup of index d in ℋ ^{∗}(p_{1}, …, p_{n}), where E_{i} has a presentation

\begin{matrix}⟨f_{i}, v_{i1}, …, v_{ia_{i1}} | & (v_{i1}f_{i}v_{ia_{i1}}f_{i}^{−1})^{y_{i1a_{i1}} /x_{i11}+x_{i1a_{i1}}} = v_{il}^{2} \\ = v_{il+1}^{2} = (v_{il}v_{il+1})^{y_{i1l}/x_{i1l}}, l = 2, …, a_{i1} − 1⟩, \end{matrix}

x_{ijl} | y_{ijl}, y_{ijl} ∈ {p_{1}, …, p_{n}}, for all i, j, l, and (x_{i11} + x_{i1a_{i1}}) | y_{i11} , for i = h_{0} + 1, …, h.

Suppose that P is a fundamental polygon for Γ which is an extended Hecke polygon. Let B_{ij} = (y_{ij1}/x_{ij1}, y_{ij2}/x_{ij2}, …, y_{ija_{ij}}/x_{ija_{ij}}), for i = 1, …, h, and j = 1, …, u_{i}. For each i, j, we will construct a polygon R_{ij} whose boundary contains a corresponding boundary component for B_{ij}. For instance, assume that B_{ij} = (p_{1}/x_{ij1}, p_{2}/x_{ij2}, …, p_{n}/x_{ijn}). Start with an (x_{ij1} + 2)-gon Q_{1} centered at a b_{1} -vertex \overline{w}_{1} such that ∂Q_{1} has precisely one component C_{1} consisting of two b_{1} -edges (respectively one b_{1} -edge, one e_{1} -edge and one c_{1} -edge) if n = 1 (respectively n > 1). If n > 1, attach a (2x_{ij} + 2)-gon Q_{2} centered at a b_{2} - vertex \overline{w}_{2} to Q_{1} along a c_{1} -edge on C_{1} such that the vertices \overline{w}_{1} and \overline{w}_{2} are on ∂(Q_{1} ∪ Q_{2}) in positive order. Continuing in this way, we obtain a polygon R_{ij} such that the vertices w_{ij}, \overline{w}_{1}, \overline{w}_{2}, …, \overline{w}_{n}, w′_{ij} are on ∂R_{ij} in positive order, where w_{ij} and w′_{ij} are the cusps on the b_{1} -edge through \overline{w}_{1} and the b_{n} -edge through \overline{w}_{n}, respectively. Note that ∂R_{ij} contains c_{j} -lines through w_{ij} and w′_{ij}. Hence we can think of P as a polygon P − ⋃_{i=1}^{h} ⋃_{j=1}^{u_{i}} R_{ij} attached to R_{ij} along the

c_{j} -line through w_{ij} or wij^{′}, i = 1, …, h, j = 1, …, u_{i}.

Let k_{0} = 0. Apply the Gauss–Bonnet theorem to P. It follows that

\begin{aligned}\frac{d}{2}(∑_{α=1}^{n} \frac{1}{p_{α}} − n + 1) & = ∑_{α=1}^{n} ∑_{i=k_{α−1}+1}^{k_{α}} \frac{m_{i}}{p_{α}} + ∑_{i,j,l} \frac{x_{ijl}}{2y_{ijl}} \\ & − (k_{n} + \frac{1}{2} ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} a_{ij} + \frac{1}{2} ∑_{i=1}^{h} u_{i} + r) + h − h_{0} + 1. \end{aligned}

On the other hand, for each α = 1, …, n, since the number of Ω_{α}^{∗} -polygons is equal to d, there exists a nonnegative integer s_{α} such that

2s_{α}p_{α} + ∑_{i=k_{α−1}+1}^{k_{α}} 2m_{i} + ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} ∑_{l∈Φ_{ij}(α)} x_{ijl} = d

where Φ_{ij}(α) = {l | y_{ijl} = p_{α}, 1 ≤ l ≤ a_{ij}}.

Conversely, all the above equalities are also sufficient for a subgroup to exist in ℋ ^{∗}(p_{1}, …, p_{n}). We will use the previous notations to describe and prove this result.

Theorem 4.2. Let k_{0} = 0, k_{1}, …, k_{n}, h_{0}, h, r be nonnegative integers, where k_{i} ≤ k_{i+1} , for i = 1, …, n, and h_{0} ≤ h. Suppose that

\begin{aligned}Γ & = F_{r} ∗^{∏∗}_{α=1}^{n}^{∏∗}_{i=k_{α−1}}^{k_{α}}Z_{p_{α}/m_{i}} \\ & ∗^{∏∗}_{i=1}^{h_{0}}^{∏∗}_{j=1}^{u_{i}}(D_{y_{ij1}/x_{ij1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{y_{ija_{ij}} /x_{ija_{ij}}}) ∗^{∏∗}_{i=h_{0}+1}^{h}E_{i}, \end{aligned}

where E_{i} has a presentation

\begin{matrix}⟨f_{i}, v_{i1}, …, v_{ia_{i1}} | & (v_{i1}f_{i}v_{ia_{i1}}f_{i}^{−1})^{y_{i1a_{i1}} /x_{i11}+x_{i1a_{i1}}} = v_{il}^{2} \\ = v_{il+1}^{2} = (v_{il}v_{il+1})^{y_{i1l}/x_{i1l}}, l = 2, …, a_{i1} − 1⟩, \end{matrix}

x_{ijl} | y_{ijl}, y_{ijl} ∈ {p_{1}, …, p_{n}}, for all i, j, l, and (x_{i11} + x_{i1a_{i1}}) | y_{i11} , for i = h_{0} + 1, …, h. Then Γ can be embedded in ℋ ^{∗}(p_{1}, …, p_{n}) as a subgroup of index d if and only if the following conditions are satisfied:

(i) (The Riemann–Hurwitz condition)

\begin{aligned}\frac{d}{2}(∑_{α=1}^{n} \frac{1}{p_{α}} − n + 1) & = ∑_{α=1}^{n} ∑_{i=k_{α−1}+1}^{k_{α}} \frac{m_{i}}{p_{α}} + ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} ∑_{l=1}^{a_{ij}} \frac{x_{ijl}}{2y_{ijl}} \\ & − (k_{n} + \frac{1}{2} ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} a_{ij} + \frac{1}{2} ∑_{i=1}^{h} u_{i} + r) + h − h_{0} + 1. \end{aligned}

(ii) (The integrality condition) The numbers s_{1}, …, s_{n} satisfying

2s_{α}p_{α} + ∑_{i=k_{α−1}+1}^{k_{α}} 2m_{i} + ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} ∑_{l∈Φ_{ij}(α)} x_{ijl} = d, α = 1, …, n,

are nonnegative integers.

Proof. The proof of the necessity is in the above argument. Conversely, if (i) and (ii) are satisfied, it follows that

(1) r = (n − 1)\frac{d}{2} − k_{n} − \frac{1}{2} ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} a_{ij} − \frac{1}{2} ∑_{i=1}^{h} u_{i} − ∑_{α=1}^{n} s_{α} + h − h_{0} + 1.

We will construct an extended Hecke polygon P which consists of s_{α} ideal p_{α} - or 2p_{α} -polygons, the (m_{α} + 2)- or (2m_{α} + 2)-gons, for i = 1, …, k_{n}, α = 1, …, n, and the polygons R_{ij}, for i = 1, …, h, j = 1, …, u_{i}, such that a subgroup generated by the side pairing transformations of P is isomorphic to Γ.

Let P_{0} be a polygon in H^{2} whose boundary consists of c_{j} -lines as we constructed in the proof of Theorem 1.4. For each α = 1, …, n − 2, let μ_{α} and μ′_{α} be the numbers of polygons among those s_{1} − s_{n} ideal p_{1} -gons and (m_{i} + 2)-gons if α = 1, and the s_{α} − 2s_{n} + 1 ideal 2p_{α} -gons and (2m_{i} + 2)-gons if α ≠ 1, which are attached to P_{0} or any other polygon along the c_{α} -lines and the c_{α+1} -lines, respectively, where i = k_{α} + 1, …, k_{α+1} . Call this new polygon P_{1} . We will prove that after attaching the polygons R_{ij} to P_{1} to obtain a polygon P, the numbers of c_{α} -lines on ∂P which are sides of Ω_{α} -polygons and Ω_{α+1} -polygons, respectively, are the same, where α = 1, …, n − 1, and there are r pairs of such c_{j} -lines on ∂P.

Suppose that for i = 1, …, h, j = 1, …, u_{i}, there exist integers ξ_{ijα}, σ_{ijαβ_{1}} , η_{ijα}, τ_{ijαβ_{2}} , ζ_{ijα}, and ρ_{ijαβ_{3}} , where α = 1, …, n − 1, β_{1} = 1, …, ξ_{ijα}, β_{2} = 1, …, ξ_{ijη_{ijα}}, and β_{3} = 1, …, ζ_{ijα}, such that in each boundary cycle B_{ij}, there are ξ_{ijα} collections of branching numbers on the boundary of type (I_{α}):

\frac{p_{α}}{l_{1}} , \frac{p_{α+1}}{l_{2}} , …, \frac{p_{α}}{l_{2σ_{ijαβ}−1}} , \frac{p_{α+1}}{l_{2σ_{ijαβ}}} or \frac{p_{α+1}}{l_{1}} , \frac{p_{α}}{l_{2}} , …, \frac{p_{α+1}}{l_{2σ_{ijαβ}−1}} , \frac{p_{α}}{l_{2σ_{ijαβ}}} ,

η_{ijα} collections of branching numbers on the boundary of type (II_{α}):

\frac{p_{α}}{l_{1}} , \frac{p_{α+1}}{l_{2}} , …, \frac{p_{α+1}}{l_{2τ_{ijαβ}−1}} , \frac{p_{α}}{l_{2τ_{ijαβ}}} ,

and ζ_{ijα} collections of branching numbers on the boundary of type (III_{α}):

\frac{p_{α+1}}{l_{1}} , \frac{p_{α}}{l_{2}} , …, \frac{p_{α}}{l_{2ρ_{ijαβ}−1}} , \frac{p_{α+1}}{l_{2ρ_{ijαβ}}} .

Since each boundary cycle except for the types (II_{1} ) and (III_{n−1} ) starts and ends up with branching numbers on the boundary of types (I_{1} ) or (I_{n−1} ), we have the following equation:

(2)  u_{i} = ∑_{j=1}^{u_{i}}[\frac{1}{2}(ξ_{ij1} + ξ_{ijn−1}) + η_{ij1} + ζ_{ijn−1}], i = 1, …, h.

Note that for each α = 1, …, n − 1, l_{1}, l_{2σijαβ}, l_{2τijαβ}, and l_{2ρijαβ} are odd numbers, and any other l_{2}, …, l_{2σijαβ−1} (or l_{2τijαβ−1} , or l_{2ρijαβ−1} ) are even numbers. Also, the number of the branching numbers p_{α+1}/l_{2σijαβ} or p_{α+1}/l_{1} as in type (I_{α}) and p_{α+1}/l_{1} , p_{α+1}/l_{2ρijαβ} as in type (II_{α}) is equal to the number of the same branching numbers as in type (I_{α+1} ) and (II_{α+1} ). Then we have

(3)  ξ_{ijα} + 2ζ_{ijα} = ξ_{ijα+1} + 2η_{ijα+1}, α = 1, …, n − 2,

and

(4)  \begin{aligned}a_{ij} & = ∑_{α=1}^{n−1}[∑_{β=1}^{ξ_{ijα}} 2σ_{ijαβ} + ∑_{β=1}^{η_{ijα}}(2τ_{ijαβ} − 1) + ∑_{β=1}^{ζ_{ijα}}(2ρ_{ijαβ} − 1)] \\ & − ∑_{α=1}^{n−2}(ξ_{ijα} + 2ζ_{ijα}) \\ & = 2 ∑_{α=1}^{n−1}(∑_{β} σ_{ijαβ} + ∑_{β} τ_{ijαβ} + ∑_{β} ρ_{ijαβ}) \\ & − ∑_{α=1}^{n−1}(ξ_{ijα} + η_{ijα} + 3ζ_{ijα}) \\ & + ξ_{ijn−1} + 2ζ_{ijn−1}, \end{aligned}

where i = 1, …, h, j = 1, …, u_{i}.

To compute the numbers of c_{j} -lines, let ε(α) and δ(α) be the numbers of c_{α} -lines on ∂P and ∂R_{ij}, for i = 1, …, h, j = 1, …, u_{i}, which are sides of Ω_{α} -polygons and Ω_{α+1} -polygons, respectively, where α = 1, …, n − 1. First, we have

ε(1) = (s_{1}p_{1} + ∑_{i=1}^{k_{1}} m_{i} − 2s_{n} + 1) − (s_{1} − s_{n}) − k_{1} − μ_{1} − μ′_{1} + ∑_{i,j} ∑_{l∈Φ_{ij}(1)} \frac{1}{2}x_{ijl}

− ∑_{i,j} ∑_{l∈Φ_{ij}(1)} #(x_{ijl} is even) − \frac{1}{2} ∑_{i,j} ∑_{l∈Φ_{ij}(1)} #(x_{ijl} is odd)

= (s_{1}p_{1} + ∑_{i=1}^{k_{1}} m_{i} − 2s_{n} + 1) − (s_{1} − s_{n}) − k_{1} − μ_{1} − μ′_{1} + ∑_{i,j} ∑_{l∈Φ_{ij}(1)} \frac{1}{2}x_{ijl}

− ∑_{i,j}[∑_{β=1}^{ξ_{ij1}}(σ_{ij1β} − 1) + ∑_{β=1}^{η_{ij1}}(τ_{ij1β} − 2) + ∑_{β=1}^{ζ_{ij1}}(ρ_{ij1β} − 1)]

− \frac{1}{2} ∑_{i,j}(ξ_{ij1} + 2η_{ij1})

= \frac{1}{2}d − 2s_{n} + 1 − (s_{1} − s_{n}) − k_{1} − μ_{1} − μ′_{1}

− ∑_{i,j}(∑_{β} σ_{ij1β} + ∑_{β} τ_{ij1β} + ∑_{β} ρ_{ij1β}) + ∑_{i,j}(\frac{1}{2}ξ_{ij1} + η_{ij1} + ζ_{ij1}),

and

δ(1) = (s_{2}p_{2} + ∑_{i=k_{1}+1}^{k_{2}} m_{i} − 2s_{n} + 1) − (s_{1} − s_{n})

− k_{1} − μ_{1} − μ′_{1} + ∑_{i,j} ∑_{l∈Φ_{ij}(2)} \frac{1}{2}x_{ijl}

− ∑_{i,j} ∑_{l∈Φ_{ij}(2)} #(x_{ijl} is even) − \frac{1}{2} ∑_{i,j} ∑_{l∈Φ_{ij}(2)} #(x_{ijl} is odd)

= (s_{2}p_{2} + ∑_{i=k_{1}+1}^{k_{2}} m_{i} − 2s_{n} + 1) − (s_{1} − s_{n})

− k_{1} − μ_{1} − μ′_{1} + ∑_{i,j} ∑_{l∈Φ_{ij}(2)} \frac{1}{2}x_{ijl}

− ∑_{i,j}[∑_{β=1}^{ξ_{ij1}}(σ_{ij1β} − 1) + ∑_{β=1}^{η_{ij1}}(τ_{ij1β} − 1) + ∑_{β=1}^{ζ_{ij1}}(ρ_{ij1β} − 2)]

− \frac{1}{2} ∑_{i,j}(ξ_{ij1} + 2ζ_{ij1})

= \frac{1}{2}d − 2s_{n} + 1 − (s_{1} − s_{n}) − k_{1} − μ_{1} − μ1^{′}

− ∑_{i,j}(∑_{β} σ_{ij1β} + ∑_{β} τ_{ij1β} + ∑_{β} ρ_{ij1β})

+ ∑_{i,j}(\frac{1}{2}ξ_{ij1} + η_{ij1} + ζ_{ij1}) = ε(1).

Similarly, for α = 2, …, n − 2,

ε(α) = \frac{1}{2}d − 2s_{n} + 1 − (s_{α} − 2s_{n} + 1 − μ_{α−1}) − (k_{α} − k_{α−1} − μα−1^{′}) − μ_{α} − μ′_{α}

− ∑_{i,j}(∑_{β} σ_{ijαβ} + ∑_{β} τ_{ijαβ} + ∑_{β} ρ_{ijαβ})

+ ∑_{i,j}(\frac{1}{2}ξ_{ijα} + η_{ijα} + ζ_{ijα}) = δ(α),

and

ε(n − 1) = \frac{1}{2}d − 2s_{n} + 1 − (s_{n−1} − 2s_{n} + 1 − μ_{n−1}) − (k_{n−1} − k_{n−2} − μ′_{n−1})

− (k_{n} − k_{n−1}) − ∑_{i,j}(∑_{β} σ_{ijn−1β} + ∑_{β} τ_{ijn−1β} + ∑_{β} ρ_{ijn−1β})

+ ∑_{i,j}(\frac{1}{2}ξ_{ijn−1} + η_{ijn−1} + ζ_{ijn−1}) = δ(n − 1).

This implies that after attaching those polygons R_{ij}, for i = 1, …, h, j = 1, …, u_{i}, to P_{1} , which is called a polygon P, the numbers of c_{α} -lines on ∂P which are sides of Ω_{α} -polygons and sides of Ω_{α+1} -polygons, respectively, where α = 1, …, n − 1, are the same.

On the other hand, from equations (1), (2), (3), and (4), it follows that

∑_{α=1}^{n−1} ε(α) = ∑_{α=1}^{n−1} δ(α) = r + ∑_{i=1}^{h_{0}} u_{i}.

This proves that on ∂P, there are r − (h − h_{0}) pairs of c_{j} -lines, and each pair of them are sides of an Ω_{α} -polygon and an Ω_{α+1} -polygon. Therefore we obtain a convex polygon P whose boundary is the union of k_{j} − k_{j−1} pairs of b_{j} -edges making an interior angle 2m_{i}π/p_{j}, where i = k_{j−1} + 1, …, k_{j}, j = 1, …, n, r pairs of c_{j} -lines, and the e_{j} - and f_{j} -edges on R_{ij} corresponding to B_{ij}.

Let each pair of those c_{j} -lines be identified, and each pair of b_{j} -edges in an interior angle 2m_{i}π/p_{j} be identified. For i = 1, …, h, let each of an e_{j} -edge and an f_{j} -edge on R_{ij} ∩∂P be identified with itself by a reflection. Let each b_{j} -edge on R_{ij} ∩ ∂P be identified with itself by a reflection if i = 1, …, h_{0} , and be identified with the other b_{j} -edge on R_{ij} ∩∂P by an orientation-preserving transformation if i = h_{0} + 1, …, h. Now P becomes an extended Hecke polygon. Hence a subgroup generated by the side pairings of P is isomorphic to Γ.

5. Special cases

Theorem 5.1. Suppose that p_{1}, …, p_{n} are distinct primes. Let

\begin{aligned}Γ & = F_{r} ∗^{∏∗}_{α=1}^{n} \underbrace{(Z_{p_{α}} ∗ ⋯ ∗ Z_{p_{α}})}_{k_{α}} \\ & ∗^{∏∗}_{i=1}^{h_{0}}^{∏∗}_{j=1}^{u_{i}}(D_{y_{ij1}/x_{ij1}} ∗_{Z_{2}} ⋯ ∗_{Z_{2}} D_{y_{ija_{ij}} /x_{ija_{ij}}}) ∗^{∏∗}_{i=h_{0}+1}^{h}E_{i}, \end{aligned}

where each E_{i} has a presentation as in Theorem 4.2. Then Γ can be embedded in ℋ ^{∗}(p_{1}, …, p_{n}) as a subgroup of finite index d if and only if the Riemann–Hurwitz condition holds, i.e. (5)

\begin{aligned}\frac{d}{2}(∑_{α=1}^{n} \frac{1}{p_{α}} − n + 1) & = ∑_{α=1}^{n} \frac{k_{α}}{p_{α}} + ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} ∑_{l=1}^{a_{ij}} \frac{x_{ijl}}{2y_{ijl}} \\ & − (∑_{α=1}^{n} k_{α} + \frac{1}{2} ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} a_{ij} + \frac{1}{2} ∑_{i=1}^{h} u_{i} + r) + h − h_{0} + 1, \end{aligned}

and for each α = 1, …, n, d − 2k_{α} − ∑_{i=1}^{h} ∑_{j=1}^{u_{i}} ∑_{l∈Φ_{ij}(α)} x_{ijl} is a nonnegative even integer, where Φ_{ij}(α) = {l | y_{ijl} = p_{α}, 1 ≤ l ≤ a_{ij}}.

Proof. It is sufficient to prove that the integrality condition follows from the two conditions in the theorem. Let β ∈ {1, …, n}. Multiplying ∏_{α=1}^{n} p_{α} to equation (5), we have

( ∏_{α≠β} p_{α})(d − 2k_{β} − ∑_{i,j} ∑_{l∈Φ_{ij}(β)} x_{ijl})

= (∏_{α=1}^{n} p_{α})[(n − 1)d − d ∑_{α≠β} \frac{1}{p_{α}} + 2 ∑_{α≠β} \frac{k_{α}}{p_{α}} + ∑_{i,j} ∑_{l∉Φ_{ij}(β)} \frac{x_{ijl}}{y_{ijl}}

− (2 ∑_{α} k_{α} + ∑_{i,j} a_{ij} + ∑_{i} u_{i} + 2r) + 2(h − h_{0}) + 2].

Note that the right-hand side of this equation is a nonnegative even integer divisible by p_{β}. Hence there is a nonnegative integer s_{β} such that

2s_{β}p_{β} + 2k_{β} + ∑_{i,j} ∑_{Φ_{ij}(β)} x_{ijl} = d.

In particular, if Γ as in Theorem 5.1 contains only orientation-preserving transformations, then h = 0 and the index d is an even number. Therefore we have the following corollary.

Corollary 5.2. Let p_{1}, …, p_{n} be distinct primes, and

Γ = F_{r} ∗ ^{∏∗} _{j=1}^{n}(\underbrace{Z_{p_{j}} ∗ ⋯ ∗ Z_{p_{j}}}_{k_{j}}).

Then Γ can be embedded in ℋ (p_{1}, …, p_{n}) as a subgroup of finite index d if and only if the Riemann–Hurwitz condition

d(∑_{j=1}^{n} \frac{1}{p_{j}} − n + 1) = ∑_{j=1}^{n} \frac{k_{j}}{p_{j}} − (∑_{j=1}^{n} k_{j} + r) + 1

holds and d ≥ k_{j}, for j = 1, …, n.

6. Hecke polygons with associated permutations

We will show how to associate a collection of permutations to a Hecke polygon. Suppose that Γ is a subgroup of index d in ℋ (p_{1}, …, p_{n}). By Theorem 2.1,

Γ has a fundamental domain P which is a Hecke polygon. Suppose that P consists of s_{j} ideal p_{j} -gons or 2p_{j} -gons Q_{j1}, …, Q_{js_{j}}, which are a union of p_{j} Ω_{j} - polygons, and the m_{i} + 2-gon or 2m_{i} + 2-gon Q_{ji} centered at a b_{j} -vertex, which is a union of m_{i} Ω_{j} -polygons, for i = s_{j} + 1, …, s_{j} + k_{j} − k_{j−1} , j = 1, …, n.

Figure 5. For the case of ℋ (4, 2, 4), a Hecke polygon P = Q_{11} ∪ Q_{21} ∪ Q_{22} ∪ Q_{31} with the indicated side pairings, where A_{1}(Q_{11}) = (1 2 3 4), A_{2}(Q_{21}) = (1 3), A_{2}(Q_{22}) = (2 4), A_{3}(Q_{31}) = (1 4 3 2).

We assign an element in {1, …, d} to each of Ω_{j} -polygons in P for each j as follows (see also Figure 5). For j = 1, …, n, let A_{j} be a function of a collection ℳ_{j} of Ω_{j} -polygons onto {1, …, d} such that

(1) A_{j}(R_{1}) ≠ A_{j}(R_{2}), for any two elements R_{1}, R_{2} ∈ ℳ_{j};

(2) A_{j}(R_{1}) = A_{j+1}(R_{2}), if R_{1} ∈ ℳ_{j}, R_{2} ∈ ℳ_{j+1} , and they have an identified c_{j} -line.

Write all the elements of A_{j}(Q_{ji}) in counterclockwise order, say {l_{1}, …, l_{r}}. An element (l_{1}, …, l_{r}) of a symmetric group S_{d} is called a permutation associated to Q_{ji}. Let α_{j} be a product of the permutations associated to Q_{ji}, i = 1, …, s_{j} + k_{j} −k_{j−1} , j = 1, …, n. Note that α_{j} is a product of disjoint s_{j} p_{j} -cycles and m_{i} - cycles, i = k_{j−1} + 1, …, k_{j}. Then {α_{1}, …, α_{n}} is called a system of permutations associated to P or Γ with respect to p_{j} ’s and m_{i} ’s.

In fact, the group ⟨α_{1}, …, α_{n}⟩ acts transitively on {1, …, d}. For, if two elements a, b of {1, …, d} are in disjoint cycles of α_{1} , say a ∈ A_{1}(Q_{11}), b ∈ A_{1}(Q_{12}), by the connectivity of the set ℰ of the union of the e_{j} - and f_{j} -edges in P there is a path of e_{j} - and f_{j} -edges in ℰ , for some j = j_{1}, …, j_{r}, which connects the b_{1} -vertex in Q_{11} to the b_{1} -vertex in Q_{12} . Note that any of the e_{j} - and f_{j} - edges in the same Q_{ji} can be mapped to an e_{j} - or f_{j} -edge under some power of α_{j}. Then a gets mapped to b through some powers of α_{1}, α_{j_{1}}, …, α_{j_{r}}, α_{1} , respectively. Hence any Hecke polygon gives us a group of permutations in S_{d} acting transitively on {1, …, d}.

On the other hand, we will show that the number of cusps on P/Γ is the number of disjoint cycles of σ = α_{n} ⋯ α_{1} . If x is a cusp on P/Γ, then there is a sequence of c_{j} -lines and b_{j} -edges around x as follows. Start with a c_{1} -line or a b_{1} -edge L_{1} through x on an Ω_{1} -polygon R_{1} in P such that R_{1} remains on the left when we walk along L_{1} toward x. Suppose that A_{1}(R_{1}) = 1. Let R_{2} be an Ω_{1} -polygon, possibly R_{1} = R_{2} , with A_{1}(R_{2}) = α_{1}(1). Let L_{2} be a c_{1} -line of R_{2} which contains a cusp equivalent to x. Then there is a c_{1} -line M_{2} on an Ω_{2} -polygon R_{3} with A_{2}(R_{3}) = α_{1}(1). Again there is a c_{2} -line L_{3} on an Ω_{2} -polygon R_{4} which contains a cusp equivalent to x with A_{2}(R_{4}) = α_{2}α_{1}(1). Continuing this way, we generate a sequence of edges {L_{1}, L_{2}, M_{2}, …, L_{n}, M_{n}} each of which contains a cusp equivalent to x, and a sequence of Ω_{j} -polygons {R_{1}, R_{2}, …, R_{2n−1}, R_{2n}} such that A_{j}(R_{2j−1}) = α_{j−1} ⋯ α_{1}(1), A_{j}(R_{2j}) = α_{j} ⋯ α_{1}(1), where j = 1, …, n, and α_{0} = identity.

Next there is an Ω_{n−1} -polygon R_{2n+1} , an Ω_{n−2} -polygon R_{2n+2} , …, and an Ω_{2} -polygon R_{3n−2} attached to M_{n} cyclically, where A_{2}(R_{3n−2}) = ⋯ = A_{n−1}(R_{2n+1}) = A_{n}(R_{2n}). If A_{n}(R_{2n}) ≠ 1, then repeat the same argument for an edge on an Ω_{1} -polygon R_{3n−1} with A_{1}(R_{3n−1}) = A_{n}(R_{2n}). This will stop at the l-th step when A_{n}(R_{2ln}) = 1 (see Figure 6).

Figure 6. The subgroup generated by the side pairings of this Hecke polygon is of index 4 in ℋ (3, 2, 4).

From the above observation we see that the number of cusps on P/Γ is exactly the number of disjoint cycles of σ.

Conversely, given p_{j} ’s and m_{i} ’s satisfying the conditions (i) and (ii) in Theorem 1.4, let α_{j} be a permutation of disjoint s_{j} p_{j} -cycles and m_{i} -cycles, i = k_{j−1} + 1, …, k_{j}, j = 1, …, n. Suppose that ⟨α_{1}, …, α_{n}⟩ is acting transitively on {1, …, d}, and σ = α_{n} ⋯ α_{1} has t cycles. We will construct a Hecke polygon P such that a group generated by the side pairings of P has a signature (g; p_{1}/m_{1}, …, p_{n}/m_{k_{n}}; t).

For each j, let Q_{ji}, be an ideal p_{j} -gon or an 2p_{j} -gon, for i = 1, …, s_{j}, and an m_{i} +2-gon or a 2m_{i} +2-gon, for i = s_{j} +1, …, s_{j} +k_{j} −k_{j−1} . If (i_{1}, …, i_{r}) is a cycle of α_{j}, assign those elements i_{1}, …, i_{r} cyclically in counterclockwise order to Ω_{j} -polygons of some Q_{ji} in which the number of Ω_{j} -polygons is r. Then this Q_{ji} with those assigned numbers is called a polygon associated to (i_{1}, …, i_{r}).

Let a, b be any two elements of {1, …, d}. Since ⟨α_{1}, …, α_{n}⟩ is acting transitively on {1, …, d}, there are some permutations, say, α_{1}, …, α_{q} such that

α_{q}^{l_{q}} ⋯ α_{1}^{l_{1}}(a) = b,

for some powers l_{1}, …, l_{q}. Then there is a cycle β_{j} in α_{j}, for each j = 1, …, q, such that β_{q}^{l_{q}} ⋯ β_{1}^{l_{1}}(a) = b. Let z_{j} = β_{j}^{l_{j}} ⋯ β_{1}^{l_{1}}(a), j = 1, …, q. Then z_{1} = β_{1}^{l_{1}}(a) ∈ β_{1} ∩ β_{2} , z_{2} ∈ β_{2} ∩ β_{3} , …, z_{q−1} ∈ β_{q−1} ∩ β_{q}, b = z_{q} ∈ β_{q}.

Suppose that Q_{j1} is a polygon associated to β_{j}, j = 1, …, q. Then there is an Ω_{j} -polygon R_{j} ⊂ Q_{j1} and an Ω_{j+1} -polygon R′_{j} ⊂ Q_{j+11} with A_{j}(R_{j}) = A_{j+1}(R′_{j}) = z_{j}, for j = 1, …, q − 1. Hence Q_{j+11} can be attached to Q_{j1} along the c_{j} -lines which are sides of R_{j} and R′_{j}, j = 1, …, q − 1. Call this polygon P_{0} . Suppose that Q_{q+11}, …, Q_{n1} are the polygons whose associated permutations contain b. Let W_{j} be an Ω_{j} -polygon contained in Q_{j1} with A_{j}(W_{j}) = b, for j = q + 1, …, n. Now attach Q_{q+11}, …, Q_{n1} to P_{0} along the c_{j} -lines which are sides of W_{q+1}, …, W_{n}. Call this polygon P_{1} . Similarly, the Q_{ji} ’s whose associated permutations contain a can be attached to P_{1} . Call this polygon P_{2} . Hence all the polygons Q_{ji} ’s whose associated permutations contain a and b are attached together.

Since any element in {1, …, d} is mapped to a under some permutation in ⟨α_{1}, …, α_{n}⟩, all the rest of the Q_{ji} ’s can be attached to P_{2} in the same way as above. Call this polygon P. The boundary of P consists of c_{j} -lines and pairs of b_{j} -edges making an angle 2m_{i}π/p_{j}.

Before the Q_{ji} ’s are attached to each other, there are d c_{j} -lines on Q_{ji} ’s and Q′_{j+1i}s, respectively. When any two of the Q_{ji} ’s are attached along a c_{j} -line, we lose one c_{j} -line from each of Q_{ji} ’s and Q_{j+1i} ’s. Hence the number of c_{j} -lines on ∂P which are sides of Ω_{j} -polygons and Ω_{j+1} -polygons, respectively, is the same. Therefore, to find the number of c_{j} -lines on ∂P, it is sufficient to count the number of c_{j} -lines on ∂P which are sides of Ω_{j} -polygons, where j = 1, …, n − 1. There are (n − 1)d c_{j} -lines on Q_{ji} ’s, j = 1, …, n − 1, altogether. Then after attaching those ∑_{j=1}^{n} s_{j} + ∑_{j=1}^{n}(k_{j} − k_{j−1}) polygons Q_{ji} ’s, there are

r = (n − 1)d − ∑_{j=1}^{n} s_{j} − k_{n} + 1

c_{j} -lines on ∂P which are sides of Ω_{j} -polygons, j = 1, …, n − 1.

Now pair a c_{j} -line of an Ω_{j} -polygon U_{j} on ∂P to a c_{j} -line of an Ω_{j+1} - polygon U_{j+1} on ∂P with A_{j}(U_{j}) = A_{j+1}(U_{j+1}). Any two b_{j} -edges on ∂P making an interior angle 2m_{i}π/p_{j} are identified. Then P together with those side pairings becomes a Hecke polygon.

Moreover, if Γ is a subgroup of ℋ (p_{1}, …, p_{n}) generated by the side pairings of P, we see by using the previous argument that the number of cusps of the surface H^{2}/Γ is t, and Γ has a signature (g; p_{1}/m_{1}, …, p_{n}/m_{k_{n}}; t), where r = 2g +t−1. Therefore we proved the following theorem.

Theorem 6.1. Let k_{0} = 0, k_{1}, …, k_{n}, g, t, r be nonnegative integers, where k_{i} ≤ k_{i+1} , for i = 1, …, n − 1, t ≥ 1, and r = 2g + t − 1. Let m_{i} be positive integers, where m_{i} | p_{j}, i = k_{j−1} + 1, …, k_{j}, j = 1, …, n. Then ℋ (p_{1}, …, p_{n}) contains a subgroup of index d with a signature (g; p_{1}/m_{1}, …, p_{n}/m_{k_{n}}; t) if and only if

(i) The numbers r, p_{j} ’s m_{i} ’s and s_{j} ’s satisfy the Riemann–Hurwitz and integrality conditions as in Theorem 1.4.

(ii) For j = 1, …, n, there exists a permutation α_{j} in S_{d} such that

(a) α_{j} is a product of disjoint p_{j} -cycles (in all s_{j} of them) and m_{i} -cycles, i = k_{j−1} + 1, …, k_{j}.

(b) The group ⟨α_{1}, …, α_{n}⟩ acts transitively on {1, …, d}.

(c) The permutation σ = α_{n} ⋯ α_{1} has t disjoint cycles.

7. Branched coverings of punctured spheres

In this section we will construct branched coverings of a punctured sphere H^{2}/ℋ (p_{1}, …, p_{n}) by applying Theorem 6.1 or using a Hecke polygon.

First note that the Riemann–Hurwitz and integrality conditions are not sufficient for the existence of a subgroup with a given signature if n ≥ 3 (see the following example).

Example. Consider a torsion free subgroup isomorphic to F_{7} of index 6 with a signature (0; 8) in ℋ (2, 3, 6). Here, χ(F_{7}) = −6 and χ(ℋ (2, 3, 6)) = −1. Integrality conditions are also satisfied. However, by Proposition 7.1 below F_{7} cannot be regarded as a subgroup of ℋ (2, 3, 6) with a signature (0; 8).

Figure 7. The c_{1} - and c_{2} -lines marked by the same letters are identified.

Note that F_{7} can be regarded as a subgroup in ℋ (2, 3, 6) with a signature (1; 6), or (2; 4), or (3; 2). Indeed, take two ideal hexagons Q_{1} , Q_{2} which both consist of six Ω_{2} -polygons and one ideal hexagon R which consists of six Ω_{3} - polygons. Glue those hexagons together along the c_{2} -lines through ∞ to get a polygon P as in Figure 7. Let the c_{1} -lines through ∞ on ∂P be identified. Then the side pairings of P

abcded^{−1}b^{−1}fgc^{−1}g^{−1}e^{−1}f^{−1}a^{−1}
abcdeb^{−1}d^{−1}fgc^{−1}g^{−1}e^{−1}f^{−1}a^{−1}
abcdeb^{−1}d^{−1}fgc^{−1}f^{−1}e^{−1}g^{−1}a^{−1}

correspond to subgroups with signatures (1; 6) (see Figure 7), (2; 4) and (3; 2), respectively.

Proposition 7.1. Suppose that Γ is a subgroup of index d in ℋ (p_{1}, …, p_{n}) whose number of cusps is t. Then we have a partition d = ∑_{i=1}^{t} d_{i}. In particular t ≤ d.

Proof. The surface H^{2}/Γ is a branched cover of degree d of the oncepunctured sphere H^{2}/ℋ (p_{1}, …, p_{n}). Let {x̃_{1}, …, x̃_{t}} and x be the cusps in H^{2}/Γ and H^{2}/ℋ (p_{1}, …, p_{n}). Compactify H^{2}/Γ and H^{2}/ℋ (p_{1}, …, p_{n}) by filling with cusps. The original branched covering is extended to the one of degree d between the compactified surfaces. Then there are t points {x̃_{1}, …, x̃_{t}} in the fiber of x. Let γ̃_{i} be a simple closed curve around x̃_{i}. It projects in H^{2}/ℋ (p_{1}, …, p_{n}) as a (not necessarily simple) closed curve γ_{i} around x. Let d_{i} be the winding number of γ_{i} around x. Then d = ∑_{i=1}^{t} d_{i}. In particular t ≤ d.

The result in [2] implies that in our case, if n ≥ 3 and t | d, then t can be realized as the number of cusps of a subgroup of index d.

Theorem 7.2. Suppose that n ≥ 3, lcm(p_{1}, …, p_{n}) | d and t | d. Then there exists a torsion free subgroup of index d with a signature (g; t) in ℋ (p_{1}, …, p_{n}) if 2g + t − 2 = d(n − 1 − ∑_{j=1}^{n} 1/p_{j}).

Proof. Let d = kt, and let E, V_{1}, …, V_{n+1} be integers satisfying E = (n+1)d, p_{j}V_{j} = d, where j = 1, …, n + 1, and p_{n+1} = k. Then ∑_{j=1}^{n+1} V_{j} − E + 2d = 2 − 2g. From Theorem 1.3 in [2], there is a tessellation of a surface M of genus g into 2d (n + 1)-gons with E edges and ∑_{j=1}^{n+1} V_{j} vertices, V_{j} of valence 2p_{j}, j = 1, …, n + 1, such that each face has vertices of valence 2p_{1}, …, 2p_{n+1} , up to cyclic order. Remove the vertices of valence 2k from M to obtain a topological surface X of genus g with t cusps. Then X is a branched cover of a oncepunctured sphere S with n branch points P_{1}, …, P_{n} with branching numbers p_{1}, …, p_{n}.

Let π: X → S be the corresponding projection map. If π_{S}: H^{2} → S is the universal branched covering with branching numbers p_{1}, …, p_{n}, the covering group of π_{S}: H^{2} → S is isomorphic to ℋ (p_{1}, …, p_{n}). Then π_{S} factors through H^{2}⟶^{π_{X}}X⟶^{π}S. The set π^{−1}(P_{j}) is precisely the V_{j} vertices of the tessellation of X lying over P_{j}. The condition p_{j}V_{j} = d ensures that H^{2}⟶^{π_{X}}X is an unbranched covering. This corresponds to a torsion free subgroup Γ of the covering group of π_{S}: H^{2} → S. To realize Γ as a subgroup of the Fuchsian group ℋ (p_{1}, …, p_{n}), we proceed as follows.

First add a vertex to each compact edge of X as a “midpoint”. Next on each face of X, add all noncompact edges through a cusp and any other vertices. Now each face has n + 1 triangles. Let T be one of these triangles. Replace T by one of the triangles Δ_{j}^{∗} ’s and Δ̃_{j}^{∗} ’s, called T̃, as in Section 2. On the interior of T̃, we have a well-defined hyperbolic metric. These T̃’s can be glued along edges by uniquely defined isometries. So at the end, we get a complete Riemannian metric of constant curvature −1 on X. This extends uniquely to a universal covering H^{2} → X which is an isometry on each component of the inverse image on each face. Therefore X is homeomorphic to H^{2}/Γ, for some torsion free subgroup Γ of index d in ℋ (p_{1}, …, p_{n}).

We now discuss some special cases for the realizability of signatures by subgroups of finite index in ℋ (p_{1}, …, p_{n}). We suppose that n and p_{j} ’s satisfy any of the following conditions: (i) p_{1} = p_{n} = 2, p_{2} = ⋯ = p_{n−1} = p; (ii) n = 4, p_{j} ≥ 4, (j = 1, …, 4); (iii) n = 5, p_{j} ≥ 3, (j = 1, …, 5); (iv) n ≥ 6, p_{j} ≥ 2, (j = 1, …, n).

Theorem 7.3. Let g and t ≥ 1 be nonnegative integers. Suppose that p and d are positive integers, where p ≥ 2 and d is divisible by lcm(2, p). Then ℋ (2, \underbrace{p, …, p}_{n}, 2) contains a torsion free subgroup of index d with a signature (g; t)

if and only if 2g + t = (1 − 1/p)nd + 2 and t ≤ d.

Proof. The necessity of the conditions follow from the Riemann–Hurwitz condition and Proposition 7.1. We will prove the sufficiency by constructing a Hecke polygon.

First, take d/p ideal 2p-gons centered at b_{j} -vertices, for each j = 1, …, n. Glue these ideal polygons together along the c_{j} -lines through ∞ to obtain a polygon P. Then we can identify the c_{j} -lines on ∂P with the desired pattern to have a surface of genus g with t cusps.

Corollary 7.4. Let Γ be a torsion free subgroup of index d in ℋ (2, \underbrace{p, …, p}_{n}, 2), where p ≥ 2 and d is divisible by lcm(2, p). Then the sur-

face H^{2}/Γ of genus g with t cusps is a branched cover of degree d of the once-punctured sphere H^{2}/ℋ (2, p, …, p, 2) branched at all b_{j} -vertices to order

\{\underbrace{2, …, 2}_{d}, \underbrace{p, …, p}_{nd/p}\} if and only if 2g + t = (1 − 1/p)nd + 2 and t ≤ d.

Theorem 7.5. Suppose that n ≥ 3, p_{j} ≥ 2, j = 1, …, n, and d is divisible by lcm(p_{1}, …, p_{n}). Let g, t ≥ 1 be nonnegative integers, and let d = k_{j}p_{j}, for j = 1, …, n. If 2g + t = (n − 1)d − ∑_{j=1}^{n} k_{j} + 2 and t ≤ min{d, (n − 2)d − ∑_{j=1}^{n} k_{j} + 3}, then ℋ (p_{1}, …, p_{n}) contains a torsion free subgroup of index d with a signature (g; t).

Proof. To find a subgroup with a signature (g; t) amounts to choosing an appropriate α_{j} in S_{d}, j = 1, …, n, such that α_{n} ⋯ α_{1} has disjoint t cycles.

There will be two cases. First, suppose that d = 4. Then each p_{j} is either 2 or 4. If each p_{j} equals 2, then the result follows directly from Theorem 7.3. Suppose that there are n_{1} of those p_{j} ’s equal to 2 and n_{2} of them equal to 4. Then t = 2n + n_{2} − 2 − 2g.

If n_{2} is an odd number, t is also an odd number, namely 1 or 3. Consider two collections

ℰ_{1} = \{\underbrace{[2, 2], …, [2, 2]}_{n_{1}}, \underbrace{[4], …, [4]}_{n_{2}}, [4]\}

and

ℰ_{2} = \{\underbrace{[2, 2], …, [2, 2]}_{n_{1}}, \underbrace{[4], …, [4]}_{n_{2}}, [1, 1, 2]\}

of partitions of d. Then the total branchings [3] v(ℰ_{1}) = 2n_{1} + 3n_{2} + 3 and v(ℰ_{2}) = 2n_{1} + 3n_{2} + 1 are even numbers.

If n_{2} is an even number, t is also an even number, namely 2 or 4. Consider two collections

ℰ_{3} = \{\underbrace{[2, 2], …, [2, 2]}_{n_{1}}, \underbrace{[4], …, [4]}_{n_{2}}, [2, 2]\}

and

ℰ_{4} = \{\underbrace{[2, 2], …, [2, 2]}_{n_{1}}, \underbrace{[4], …, [4]}_{n_{2}}, [1, 1, 1, 1]\}

of partitions of d. Then the total branchings v(ℰ_{3}) = 2n_{1} + 3n_{2} + 2 and v(ℰ_{4}) = 2n_{1} +3n_{2} are also even numbers. Moreover, for j = 1, 2, 3, 4, v(ℰ_{j}) ≥ 2n_{1} +3n_{2} = 2n + n_{2} ≥ 2d − 2 = 6, because n ≥ 3. Therefore ℰ_{1} , ℰ_{2} , ℰ_{3} and ℰ_{4} are realizable by Complement 5.6 in [3].

Secondly, suppose that d ≠ 4. Let ℱ = {A_{1}, …, A_{n+1}} be a collection of partitions of d, where A_{j} = [p_{j}, …, p_{j}], for j = 1, …, n, and A_{n+1} = [m_{1}, …, m_{t}] with ∑_{i=1}^{t} m_{i} = d. The total branching is

v(ℱ) = ∑_{j=1}^{n}(p_{j} − 1)k_{j} + ∑_{i=1}^{t}(m_{i} − 1) = (n + 1)d − ∑_{j=1}^{n} k_{j} − t.

Since t ≤ (n−2)d−∑_{j=1}^{n} k_{j} +3, it follows that v(ℱ) ≥ 3(d−1). By Theorem 5.4 in [3], ℱ is realized as the branch data of a connected branched covering of a closed sphere S^{2} . Hence, by Lemma 2.1 in [3], for each j = 1, …, n, there exists α_{j} in S_{d} which is a product of k_{j} disjoint p_{j} -cycles such that α_{1} ⋯ α_{n} is a product of disjoint m_{i} -cycles, i = 1, …, t, and ⟨α_{1}, …, α_{n}⟩ acts transitively on {1, …, d}.

Corollary 7.6. Suppose that n ≥ 3, p_{j} ≥ 2, j = 1, …, n, and d is divisible by lcm(p_{1}, …, p_{n}). Let d = k_{j}p_{j}, for j = 1, …, n. If 2g + t = (n − 1)d − ∑_{j=1}^{n} k_{j} + 2 and t ≤ min{d, (n − 2)d − ∑_{j=1}^{n} k_{j} + 3}, then ℋ (p_{1}, …, p_{n}) contains a torsion free subgroup Γ of index d such that H^{2}/Γ is a surface of genus g with t cusps which is a branched cover of degree d of the once-punctured sphere H^{2}/ℋ (p_{1}, …, p_{n}), branched at all b_{j} -vertices to order

\{\underbrace{p_{1}, …, p_{1}}_{k_{1}}, …, \underbrace{p_{n}, …, p_{n}}_{k_{n}}\}.

We would like to know whether or not the Riemann–Hurwitz condition and the condition on t in Theorem 7.5 are also necessary for the existence of a subgroup of ℋ (p_{1}, …, p_{n}) with a prescribed signature. In fact, in each of the cases: (i) n = 4, p_{j} ≥ 4, (j = 1, …, 4); (ii) n = 5, p_{j} ≥ 3, (j = 1, …, 5); (iii) n ≥ 6, p_{j} ≥ 2, (j = 1, …, n), it follows that (n − 2)d − ∑_{j=1}^{n} k_{j} + 3 ≥ d. Then the sufficient condition on t in Theorem 7.5 (which is now reduced to t ≤ d) is the same as the necessary end-condition in Proposition 7.1. Those consequences are stated as the following theorem.

Theorem 7.7. Suppose that n and p_{j}, j = 1, …, n, satisfy any of the following conditions:

(i) n = 4, p_{j} ≥ 4, j = 1, …, 4; (ii) n = 5, p_{j} ≥ 3, j = 1, …, 5; (iii) n ≥ 6, p_{j} ≥ 2, j = 1, …, n, and that d is divisible by lcm(p_{1}, …, p_{n}). Let g, t ≥ 1 be nonnegative integers, and let d = k_{j}p_{j}, for j = 1, …, n. Then ℋ (p_{1}, …, p_{n}) contains a torsion free subgroup of index d with a signature (g; t) if and only if 2g + t = (n − 1)d − ∑_{j=1}^{n} k_{j} + 2 and t ≤ d.

Corollary 7.8. Suppose that n and p_{j}, j = 1, …, n, satisfy any of the following conditions: (i) n = 4, p_{j} ≥ 4, j = 1, …, 4; (ii) n = 5, p_{j} ≥ 3, j = 1, …, 5; (iii) n ≥ 6, p_{j} ≥ 2, j = 1, …, n, and that d is divisible by lcm(p_{1}, …, p_{n}). Let Γ be a torsion free subgroup of index d in ℋ (p_{1}, …, p_{n}). Then the surface H^{2}/Γ of genus g with t cusps is a branched cover of degree d of the once-punctured sphere H^{2}/ℋ (p_{1}, …, p_{n}), branched at all b_{j} -vertices to

order \{\underbrace{p_{1}, …, p_{1}}_{k_{1}}, …, \underbrace{p_{n}, …, p_{n}}_{k_{n}}\} if and only if 2g +t = (1−1/p)nd+2 and t ≤ d.

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214 Shuechin Huang

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Received 6 August 1997