Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 73–124

GEOMETRIC INTERSECTION NUMBERS ON A FIVE-PUNCTURED SPHERE

Yungyen Chiang

National Sun Yat-Sen University, Department of Applied Mathematics Kaohsiung, Taiwan 80424, R.O.C.; chiangyy@ibm7.math.nsysu.edu.tw

Abstract. Let 𝒢 be the set of all simple closed geodesics on a five-punctured sphere Σ_{5} . In this article, we associate to each γ ∈ 𝒢 four integers which are read off topologically from γ itself. These integers have three remarkable applications. First, the geometric intersection number of any two geodesics in 𝒢 can be written explicitly in terms of the corresponding integers. Secondly, there is a homeomorphism of the completion of 𝒢 onto a 3-sphere lying in R^{4} whose restriction to 𝒢 is written explicitly in terms of these integers. Finally, these integers are related to trace polynomials of the corresponding transformations in a representation of π_{1}(Σ_{5}) into PSL(2, C).

Introduction

According to Thurston, the set of all complete simple geodesics on a Riemann surface can be made into a topological space homeomorphic to a sphere whose dimension depends on the topology of the surface. By Thurston’s result, the space \overline{𝒢} _{n} of complete simple geodesics on an n-punctured sphere Σ_{n} with n ≥ 4 is homeomorphic to a sphere of dimension 2n − 7.

In [4], the author introduced to each simple closed geodesic γ on Σ_{4} a pair of integers I_{X}(γ) ≥ 0 and N(γ) whose absolute values are geometric intersection numbers of γ with a fixed pair of simple curves on Σ_{4} . With these integers, the author proved that the geometric intersection number of any two simple closed geodesics γ and δ on Σ_{4} is

2|I_{X}(γ)N(δ) − I_{X}(δ)N(γ)|.

The geometric intersection formula above was used to prove the injectivity of a homeomorphism Ψ of \overline{𝒢} _{4} onto the circle R ∪ \{∞\} with Ψ(γ) = N(γ)/I_{X}(γ) for all simple closed geodesics γ. Moreover, if G is a Maskit four-punctured sphere group, and if g ∈ G represents a simple closed geodesic γ on Σ_{4} , then the first two high-order terms of the trace polynomial of g are written explicitly in terms of I_{X}(γ) and N(γ).

The aim of this article is to generalize the results in [4] to the case of a five-punctured sphere.

2000 Mathematics Subject Classification: Primary 30F; Secondary 57.

Similar trace formulas for once and twice punctured tori are proved using different methods in [6] and [7] respectively. However, the methods adopted in [4], [6], [7] and in this article are all based on the cutting sequence technique developed by Birman and Series [2].

In [7], the trace formulas are obtained by factoring a representation of the first fundamental group of a twice punctured torus 𝒮 in SL(2, C) as a representation of the fundamental groupoid π_{1,2}(𝒮 , p_{1}, p_{2}) on 𝒮 with two basepoints p_{1} and p_{2} , where one basepoint is chosen on each of the two cyclindrical subsurfaces obtained by cutting along a pair of disjoint curves, one passing through each of the punctures. The fundamental groupoid π_{1,2}(𝒮 , p_{1}, p_{2}) is the groupoid of homotopy classes of paths in 𝒮 with endpoints in the set {p_{1}, p_{2}}.

In addition to trace formulas, in [7] Keen, Parker and Series also provide a set of projective coordinates for the set of all simple closed geodesics on 𝒮 , called the π_{1,2} -coordinates. For every simple loop γ on 𝒮 , they consider the restriction of the integral weighted π_{1} -train track associated with γ to each cylinder, and call the restricted train track the integral weighted π_{1,2} -train track associated with γ by relating it to π_{1,2}(𝒮 , p_{1}, p_{2}). The π_{1,2} -coordinates are integer functions of the integral weighted π_{1,2} -train tracks.

In this article, we shall give a set of projective coordinates to the set 𝒢 of all simple closed geodesics on a five punctured sphere Σ_{5} equipped with a hyperbolic metric. By using the coordinates, we provide a 3-sphere structure for the set \overline{𝒢} of all complete simple geodesics on Σ_{5} .

To enumerate the set 𝒢 , we start with a Fuchsian representation G of the first fundamental group of Σ_{5} acting on the upper half plane 𝒰 . The Fuchsian group G is generated by two parabolic transformations X and Y , and two hyperbolic transformations S and T.

In Section 2, we introduce four integer functions I_{X}, I_{Y} , N_{S} and N_{T} on 𝒢 . The integer functions I_{X} and I_{Y} are analogues of the integer function I_{X} defined in [4], and N_{S} and N_{T} are analogues of the integer function N defined in [4]. The values of I_{X} and I_{Y} are non-negative. The sign of N_{S} and that of N_{T} are determined by the symmetry of 𝒟, where 𝒟 is a fundamental domain for G acting on 𝒰 with Γ = {S, S^{−1}, T, T^{−1}, X, X^{−1}, Y, Y^{−1}} the set of side pairings.

For every γ ∈ 𝒢 , the integers I_{X}(γ), I_{Y} (γ), N_{S}(γ) and N_{T} (γ) are read off from the lift of γ to 𝒟. The lift of γ to 𝒟 also determines words in elements of Γ representing γ, which are called Γ-words. We shall write Γ-words representing geodesics in 𝒢 in a specific way, and call them cyclic semi-reduced Γ-words. In Section 2, we shall also relate these cyclic semi-reduced Γ-words to the integer functions I_{X}, I_{Y} , N_{S} and N_{T} .

By use of the integer functions I_{X}, I_{Y} , N_{S} and N_{T} , we prove a geometric intersection formula in Theorem 3.1. The geometric intersection formula says that if γ and δ are two geodesics in 𝒢 , then the geometric intersection number of γ with δ is

\begin{aligned}2|I_{X}(γ)N_{T} (δ) & − I_{X}(δ)N_{T} (γ)| + 2|I_{Y} (γ)N_{S}(δ) \\ & − I_{Y} (δ)N_{S}(γ)| + |I_{XY} (γ, δ)| − I_{XY} (γ, δ), \end{aligned}

where I_{XY} (γ, δ) = {I_{X}(γ) − I_{Y} (γ)} · {I_{X}(δ) − I_{Y} (δ)}.

As a consequence of the geometric intersection formula, we obtain the geometric intersection numbers of six fixed geodesics in 𝒢 with an arbitrary geodesic γ ∈ 𝒢 . These geometric intersection numbers will be called the elementary intersection numbers of γ.

The elementary intersection numbers are used to construct a homeomorphism Ψ of \overline{𝒢} onto a 3-sphere Δ lying in R^{6} (Theorem 4.3). We start with a function of 𝒢 into Δ which maps each γ ∈ 𝒢 to the point whose coordinates are the elementary intersection numbers of γ. Then, by a continuity argument, we extend the function to obtain a continuous map Ψ from \overline{𝒢} onto Δ. The injectivity of Ψ is proved by the geometric intersection formula.

By post composing Ψ by a map from R^{6} into R^{4} , we obtain an embedding Φ of \overline{𝒢} into R^{4} with

Φ(γ) = (\frac{I_{X}(γ)}{σ(γ)}, \frac{N_{T} (γ)}{σ(γ)}, \frac{I_{Y} (γ)}{σ(γ)}, \frac{N_{S}(γ)}{σ(γ)})

for every γ ∈ 𝒢 , where σ(γ) = I_{X}(γ) + |N_{T} (γ)| + I_{Y} (γ) + |N_{S}(γ)| (Theorem 4.4).

In the final section, we first find for each γ ∈ 𝒢 a cyclic semi-reduced Γ-word W (γ) to represent it, and write the word explicitly; see Theorem 5.1, Corollary 5.2 and Theorem 5.3. Then, we consider the Maskit embedding of the Teichmüller space of Σ_{5} , which is a holomorphic family of Kleinian groups G(μ, ν) parametrized by a subset ℳ_{5} of C^{2} . For every (μ, ν) ∈ ℳ_{5} , the group G(μ, ν) uniformizes a five-punctured sphere and three thrice punctured spheres.

For every γ ∈ 𝒢 , let W (γ; μ, ν) ∈ G(μ, ν) be the image of W (γ) under the canonical isomorphism of G onto G(μ, ν). The trace tr W (γ; μ, ν) of W (γ; μ, ν) is a polynomial in μ and ν. For γ ∈ 𝒢 with m = I_{X}(γ) > 0 or n = I_{Y} (γ) > 0, we prove in Theorem 5.5 that

tr W (γ; μ, ν) = ±{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯} whenever m ≥ n, and tr W (γ; μ, ν) = ±4^{n−m}{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯} whenever m ≤ n.

Together with the theory of pleating coordinates developed by Keen and Series [6], the trace formulas given above will be used to describe the shape of ℳ_{5} . The work will appear elsewhere.

Acknowledgement. The author would like to thank Professor L. Keen for her encouragement and for many informative communications. Thanks also go to the referee who read the original manuscript carefully and made many helpful comments.

1. Preliminaries

1.1. The space of complete simple geodesics. Let Σ_{5} be a 5-punctured sphere equipped with a hyperbolic metric. A loop on Σ_{5} with no self intersections will be called a simple loop. An essential simple loop on Σ_{5} is a simple loop which is neither homotopically trivial nor homotopically equivalent to a puncture of Σ_{5} . A finite union of pairwise disjoint essential simple loops on Σ_{5} will be called a multiple simple loop.

Let 𝒢 be the set of all free homotopy classes of non-oriented essential simple loops on Σ_{5} . Every element of 𝒢 contains a unique geodesic γ on Σ_{5} . By abuse of notation, we shall also use γ for the free homotopy class containing γ.

Let 𝒢 ℒ be the set of all free homotopy classes of non-oriented multiple simple loops on Σ_{5} . It is clear that 𝒢 is a subset of 𝒢 ℒ .

Let α be a multiple simple loop on Σ_{5} . All connected components of α fall into at most two distinct free homotopy classes. There are integers p ≥ 0 and q ≥ 0 with p + q > 0 such that α has exactly p connected components freely homotopic to a γ ∈ 𝒢 , and has exactly q connected components freely homotopic to a γ′ ∈ 𝒢 , where γ ≠ γ′. We shall write [α] = pγ ⊕ qγ′, where [α] is the free homotopy class represented by α. Similarly, the free homotopy class represented by a curve β on Σ_{5} will be denoted by [β].

Let [𝒢 , R_{+}] be the set of all functions from 𝒢 into the set R_{+} of all nonnegative real numbers. We provide 𝒢 with the discrete topology, and provide [𝒢 , R_{+}] with the compact-open topology. It is well known that [𝒢 , R_{+}] is homeomorphic to the product space ∏_{γ∈𝒢} Rγ_{+} , where each Rγ_{+} is a copy of R_{+} .

Two elements f and g of [𝒢 , R_{+}] − {0} are called projectively equivalent if there is a positive number t such that f = tg. Let P[𝒢 , R_{+}] be the set of all projective equivalence classes in [𝒢 , R_{+}]−{0} provided with the quotient topology. Let π be the quotient map of [𝒢 , R_{+}] − {0} onto P[𝒢 , R_{+}].

For any two curves α_{1} and α_{2} on Σ_{5} , let #(α_{1} ∩ α_{2}) denote the cardinality of the intersection α_{1} ∩α_{2} . The geometric intersection number i([α_{1}], [α_{2}]) of [α_{1}] with [α_{2}] is defined by

i([α_{1}], [α_{2}]) = min{#(α′_{1} ∩ α′_{2}) : [α′_{j}] = [α_{j}] for j = 1, 2}.

It follows immediately from the definition that if [α] = pγ ⊕ qγ′, then for any curve β on Σ_{5}

i([α], [β]) = pi(γ, [β]) + qiγ′, [β]),

where p and q are non-negative integers with p + q > 0, and where γ and γ′ are disjoint geodesics in 𝒢 .

Each α ∈ 𝒢 ℒ induces a function I_{α}: 𝒢 ⟶ R_{+} given by

I_{α}(γ) = i(α, γ) for all γ ∈ 𝒢 .

Let ℐ: 𝒢 ℒ ⟶ [𝒢 , R_{+}] be defined by

ℐ (α) = I_{α} for all α ∈ 𝒢 ℒ .

It is a well-known fact that the composition πℐ is injective; see [5]. This allows us to identify 𝒢 ℒ with πℐ (𝒢 ℒ ).

Let \overline{πℐ (𝒢 ℒ )} and \overline{πℐ (𝒢)} denote the closures of πℐ (𝒢 ℒ ) and πℐ (𝒢) in P[𝒢 , R_{+}], respectively. Poénaru proved that \overline{πℐ (𝒢 ℒ )} = \overline{πℐ (𝒢)} , (Theorem 4 of [5] Exposé 4).

Note that an element ℒ of P[𝒢 , R_{+}] is in \overline{πℐ (𝒢)} if and only if for any l in [𝒢 , R_{+}]−{0} with π(l) = ℒ there is a sequence {t_{k}}_{k=1}^{∞} of positive numbers, and there is a sequence {γ_{k}}_{k=1}^{∞} of geodesics in 𝒢 such that the sequence {t_{k} I_{γ_{k}}}_{k=1}^{∞} converges to l. A sequence {l_{k}}_{k=1}^{∞} in [𝒢 , R_{+}] is called convergent to l ∈ [𝒢 , R_{+}] if for every γ ∈ 𝒢 the sequence {l_{k}(γ)}_{k=1}^{∞} converges in R to l(γ).

According to Thurston, \overline{πℐ (𝒢)} is homeomorphic to a 3-sphere. In Section 4, we shall construct a homeomorphism of \overline{πℐ (𝒢)} onto a 3-sphere lying in R^{4} (see Theorem 4.4).

1.2. Cyclic reduced words. To enumerate free homotopy classes in 𝒢 ℒ , we consider the action of the fundamental group π_{1}(Σ_{5}) on the upper half plane 𝒰 = {z ∈ C : Im z > 0}.

Let G be the subgroup of PSL(2, R) generated by the transformations:

X = ( \begin{matrix}1 & 6 \\ 1 & 0 \end{matrix} ) , Y = ( \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ) , S = ( \begin{matrix}3 & 4 \\ 2 & 3 \end{matrix} ) and T = ( \begin{matrix}5 & 12 \\ 2 & 5 \end{matrix} ) .

For j = 1, 2, 3, let

C′_{j} = {z ∈ C : |2z + 2j − 1| = 1} and C_{j} = {z ∈ C : |2z − (2j − 1)| = 1},

and let

C′_{4} = {z ∈ C : Re z = −3} and C_{4} = {z ∈ C : Re z = 3}.

It is clear that 𝒰 /G = Σ_{5} , and that the domain 𝒟 ⊂ 𝒰 bounded by C_{j} and C′_{j}, 1 ≤ j ≤ 4, is a fundamental domain for G acting on 𝒰 . We shall schematically draw 𝒟 as a rectangular region shown in Figure 1, where the points on the boundary of 𝒟 marked by “×” correspond to punctures of Σ_{5} .

It is well known that every free homotopy class in 𝒢 corresponds to a unique conjugacy class in G. We shall find a representative for each conjugacy class in G by using Birman and Series’ cutting sequence technique [2].

Let Γ denote the set of all side pairings of 𝒟, i.e.,

Γ = {X, X^{−1}, Y, Y^{−1}, S , S^{−1}, T, T^{−1}}.

Figure 1. The fundamental domain 𝒟.

For every E ∈ Γ, we label the common side s of 𝒟 and E(𝒟) by E^{−1} on the side inside 𝒟, and by E on the side inside E(𝒟); see Figure 1. The side s will be called the E -side of 𝒟.

For every g ∈ G, the image g(𝒟) will be called a G-translate of 𝒟. We transport the above side labelling to all G-translates of 𝒟.

Let γ be an arbitrary closed curve on Σ_{5} . Let γ̃ be a lift of γ to 𝒰 which projects to γ bijectively, and let z_{0} ∈ 𝒰 be an endpoint of γ̃. Without loss of generality, assume that there is a g_{0} ∈ G and there is a ξ_{0} ∈ 𝒟 such that z_{0} = g_{0}(ξ_{0}).

We orient γ̃ so that its initial point is z_{0} . The arc γ̃ cuts in order the Gtranslates g_{0}(𝒟), g_{1}(𝒟), …, g_{k}(𝒟) of 𝒟. Then the terminal point of g_{0}^{−1}(γ̃) is g_{0}^{−1} ∘ g_{k}(ξ_{0}), and γ is represented by g = g_{0}^{−1} ∘ g_{k}.

For every integer j with 1 ≤ j ≤ k, assume that the common side of g_{j−1}(𝒟) and g_{j}(𝒟) on the side inside g_{j}(𝒟) is labelled by E_{j} ∈ Γ. Then

E_{j}(D) = g_{j−1}^{−1}(g_{j}(D)),

or equivalently E_{j} = g_{j−1}^{−1} ∘ g_{j}. Thus

g = g_{0}^{−1} ∘ g_{k} = (g_{0}^{−1} ∘ g_{1}) ∘ (g_{1}^{−1} ∘ g_{2}) ∘ ⋯ ∘ (g_{k−1}^{−1} ∘ g_{k}) = E_{1} ∘ E_{2} ∘ ⋯ ∘ E_{k}.

We call E_{1} ∘ E_{2} ∘ ⋯ ∘ E_{k} a Γ-word representing γ.

From now on, we shall simply write the composition of a function f followed by the other function g as gf. Thus, we write

E_{1} ∘ E_{2} ∘ ⋯ ∘ E_{k} = ∏_{j=1}^{k} E_{j}.

A Γ-word ∏_{j=1}^{k} E_{j} will be called reduced if E_{j} ≠ E_{j+1}^{−1} for 1 ≤ j ≤ k − 1. It is called cyclically reduced if in addition E_{1} ≠ E_{k}^{−1} .

Let γ be a simple loop on Σ_{5} . Using the above notation, for every integer j with 0 ≤ j ≤ k, let l_{j} be the image of the intersection of γ̃ with g_{j}(\overline{𝒟} ) mapped by g_{j}^{−1} , where \overline{𝒟} is the relative closure of 𝒟 in 𝒰 . The union l_{0} ∪ l_{k} forms a simple arc in \overline{𝒟} connecting the E_{k}^{−1} -side to the E_{1} -side. We shall simply write the simple arc as l_{k}. If k > 1 and if 1 ≤ j ≤ k − 1, then l_{j} is a simple arc in \overline{𝒟} connecting the E_{j}^{−1} -side to the E_{j+1} -side. Each of these simple arcs l_{1} , …, l_{k} will be called a strand of γ.

Let α be a multiple simple loop on Σ_{5} . A strand of a connected component of α will be also called a strand of α.

A loop on Σ_{5} will be called reduced if it is represented by a reduced Γ-word. A multiple simple loop α on Σ_{5} will be called reduced if every connected component of α is reduced. It is easy to see that a simple loop or a multiple simple loop on Σ_{5} is reduced if and only if every strand of the loop connects two different sides of 𝒟.

If γ ∈ 𝒢 is a geodesic, then every strand of γ is a hyperbolic geodesic arc, and thus every strand of γ must connect two different sides of 𝒟 since 𝒟 is a geodesic polygon. This proves that every simple closed geodesic on Σ_{5} is a reduced loop. Thus every free homotopy class of multiple simple loops on Σ_{5} contains a reduced one.

If γ ∈ 𝒢 is a geodesic represented by a reduced Γ-word W, then γ is also represented by an arbitrary cyclic permutation of W. If γ′ ∈ 𝒢 is a geodesic which has the same underlying set as γ but with opposite orientation, then γ′ is represented by W^{−1} . Because we are only interested in non-oriented simple loops, we shall identify all reduced Γ-words which are cyclic permutations of W or cyclic permutations of W^{−1} , and call any one of them a cyclic reduced Γword representing γ and its free homotopy class. Every cyclic reduced Γ-word is cyclically reduced.

Figure 2. From the left to the right: γ_{11} , γ_{12} , γ_{13} , γ_{21} , γ_{22} , γ_{23} .

As examples, let γ_{jk} ∈ 𝒢 be the geodesics given in Figure 2. Each γ_{jk} is represented by a cyclic reduced Γ-word W_{jk} as follows:

\begin{aligned}W_{11} & = T, \\ W_{21} & = S, \end{aligned} \begin{aligned}W_{12} & = X^{−1}S, \\ W_{22} & = Y^{−1}T, \end{aligned} \begin{aligned}W_{13} & = XT^{−1}S, \\ W_{23} & = S^{−1}Y T. \end{aligned}

For simplicity, we shall also write γ_{11} = γ_{T} and γ_{21} = γ_{S}.

1.3. Subwords and admissible subarcs. The purpose of this subsection is to find some necessary conditions for cyclic reduced Γ-words representing geodesics in 𝒢̂ = 𝒢 − {γ_{S}, γ_{T} } from the geometry of the corresponding geodesics.

Let γ ∈ 𝒢̂ be a geodesic represented by a cyclic reduced Γ-word W (γ) given by

W (γ) = ∏_{j=1}^{k} E_{j}.

Note that k > 1 since γ ∈ 𝒢̂ . For any two integers j, l with 1 ≤ j ≤ k and 1 ≤ l ≤ k, the reduced Γ-word

(1)  W′ = E_{j} ⋯ E_{j+l−1}

will be called a subword of W (γ), where E_{j+i} = E_{j+i−k} whenever 1 ≤ i ≤ l and i + j > k.

Now, we shall relate W′ to γ geometrically. For every i, let l_{i} be the strand of γ connecting the E_{i−1}^{−1} -side to the E_{i} -side, where E_{i−1} = E_{k} if i = 1. Assume that 1 ≤ l < k, i.e., W′ ≠ W (γ). We think that W′ “represents” a subarc γ′ of γ. We choose γ′ to be the projection of the union ⋃_{i=j}^{j+l−1} l_{i} to Σ_{5} . Each of the arcs l_{j}, …, l_{j+l−1} is called a strand of γ′.

The subarc γ′ has two distinct endpoints. One of the two endpoints is the projection of the endpoint of l_{j} on the E_{j−1}^{−1} -side, and the other endpoint is the projection of the endpoint of l_{j+l−1} on the E_{j+l−1} -side.

The word given in equation (1) is not clear enough to indicate that γ′ has an endpoint which is the projection of a point lying on the E_{j−1}^{−1} -side. Also, to be different from cyclic reduced words representing simple closed geodesics, we shall write the reduced Γ-word representing γ′ as

(2)  E⃗_{j−1}W′ = E⃗_{j−1}E_{j} ⋯ E_{j+l−1},

where E⃗_{j−1} is to indicate that E⃗_{j−1}W′ is not cyclic, and one of the endpoints of γ′ is the projection of a point on the E_{j−1}^{−1} -side.

A subarc of a geodesic γ ∈ 𝒢 will be called admissible if either it is γ itself, or it is represented by a reduced Γ-word as given in equation (2).

Remark 1.1. Let γ ∈ 𝒢̂ be a geodesic represented by a cyclic reduced Γword W (γ). From now on, for ε = ±1, E ∈ Γ, E_{1}, E_{2} ∈ Γ − {E^{±1}}, and an integer k > 1, we shall write

E_{1} \underbrace{E^{ε} ⋯ E^{ε}}_{k times} E_{2} = E_{1}E^{kε}E_{2}

if above word is a subword of W (γ).

By the same reasoning as that in [4, Section 3], there are no admissible subarcs of γ represented by any one of the following words:

\begin{matrix}X⃗^{ε}X^{ε}, \\ X⃗^{ε}T^{k}X^{δ}, \end{matrix} \begin{matrix}Y⃗^{ε}Y^{ε}, \\ Y⃗^{ε}S^{k}Y^{δ}, \end{matrix} \begin{matrix}T⃗^{δ}X^{ε}T^{δ}, \\ T⃗^{ε}S^{δ}T^{ε}, \end{matrix} \begin{matrix}S⃗^{δ}Y^{ε}S^{δ}, \\ S⃗^{ε}T^{δ}S^{ε}, \end{matrix}

where ε, δ ∈ {1, −1}, and k ≠ 0 is an integer. Thus none of the following is a subword of W (γ):

\begin{matrix}X^{ε}X^{ε}, \\ X^{ε}T^{k}X^{δ}, \end{matrix} \begin{matrix}Y^{ε}Y^{ε}, \\ Y^{ε}S^{k}Y^{δ}, \end{matrix} \begin{matrix}T^{δ}X^{ε}T^{δ}, \\ T^{ε}S^{δ}T^{ε}, \end{matrix} \begin{matrix}S^{δ}Y^{ε}S^{δ}, \\ S^{ε}T^{δ}S^{ε}. \end{matrix}

Figure 3.

Proposition 1.1. Let γ ∈ 𝒢 be a geodesic represented by a cyclic reduced Γ-word W, and let k ≠ 0 be an integer.

(i) If E_{1}, E_{2} ∈ {T^{±1}, X^{±1}}, and if E_{1}S^{k}E_{2} is a subword of W, then |k| = 1. (ii) If E_{1}, E_{2} ∈ {S^{±1}, Y^{±1}}, and if E_{1}T^{k}E_{2} is a subword of W, then |k| = 1.

Proof. We shall prove the statement (i). The statement (ii) will follow by a similar argument.

Assume that k > 0. We choose once for all an orientation on the S^{−1} - side. Let ζ be the fixed point of the transformation S^{−1}T. If P and P′ are two distinct points lying on the S^{−1} -side, and if P lies between P′ and ζ, then we write P ≺ P′. This gives an orientation to the S -side as well. For any two distinct points Q and Q′ lying on the S -side, if S^{−1}(Q) ≺ S^{−1}(Q′), then we write Q ≺ Q′.

Let γ′ be the admissible subarc of γ represented by E⃗_{1}S^{k}E_{2} . Let l_{1} be the strand of γ′ joining the E_{1}^{−1} -side to the S -side with the endpoint Q_{1} on the S -side. Let l_{2} be the strand of γ′ joining the S^{−1} -side to the E_{2} -side with the endpoint P_{2} on the S^{−1} -side.

Suppose that k > 1. Then γ′ has a strand l joining the S^{−1} -side to the S -side with the endpoint P_{1} = S^{−1}(Q_{1}) on the S^{−1} -side. Let Q be the endpoint of l on the S -side. Since γ is simple, we have Q_{1} ≺ Q (see Figure 3). But, now, we have P_{1} ≺ P_{2} . This implies that l_{2} intersects l which is a contradiction. Hence, k = 1.

By the same reasoning as above, one proves that k = −1 if k < 0.

1.4. π_{1} -train tracks. In Section 3, we shall need π_{1} -train tracks introduced by Birman and Series (see [1]). A π_{1} -train track τ on 𝒟 is a collection of mutually disjoint simple arcs l_{j} in \overline{𝒟} with endpoints lying on the sides of 𝒟 such that

(i) except endpoints each l_{j} is contained in 𝒟, (ii) each l_{j} joins two distinct sides of 𝒟, and (iii) each pair of distinct sides of 𝒟 are connected by at most one l_{j}. A π_{1} -train track τ on 𝒟 is called integral weighted if every arc in τ is assigned

a non-negative integer.

Every reduced multiple simple loop α on Σ_{5} can be associated with an integral weighted π_{1} -train track as described below.

We choose for each E ∈ Γ a point P (E) on the E -side of 𝒟 so that P (E^{−1}) and P (E) are identified by the transformation E.

For any two distinct E_{1}, E_{2} ∈ Γ, let n_{α}(E_{1}, E_{2}) be the number of strands of α connecting the E_{1} -side to the E_{2} -side of 𝒟. If n_{α}(E_{1}, E_{2}) > 0, then we collapse all strands of α which connect the E_{1} -side to the E_{2} -side into a single arc from P (E_{1}) to P (E_{2}) weighted by the integer n_{α}(E_{1}, E_{2}). These weighted arcs form the required integral weighted π_{1} -train track τ(α) on 𝒟 (see [1, Theorem 1.3]).

It is clear that if α and β are freely homotopic reduced multiple simple loops on Σ_{5} , then n_{α}(E_{1}, E_{2}) = n_{β}(E_{1}, E_{2}) whenever E_{1}, E_{2} ∈ Γ are distinct, and thus τ(α) = τ(β). Since every free homotopy class of multiple simple loops on Σ_{5} contains a reduced one, we may write

n_{[α]}(E_{1}, E_{2}) = n_{α}(E_{1}, E_{2})

whenever α is a reduced multiple simple loop on Σ_{5} , and call n_{[α]}(E_{1}, E_{2}) the number of strands of [α] connecting the E_{1} -side to the E_{2} -side. Similarly, we write

τ([α]) = τ(α).

Let [α], [α_{1}] and [α_{2}] be any three elements of 𝒢 ℒ . If, as subsets of \overline{𝒟} , τ([α]) is the union of τ([α_{1}]) and τ([α_{2}]), and if there are two fixed non-negative integers p and q with p + q > 0 satisfying

n_{[α]}(E_{1}, E_{2}) = pn_{[α_{1}]}(E_{1}, E_{2}) + qn_{[α_{2}]}(E_{1}, E_{2})

for any two distinct E_{1}, E_{2} ∈ Γ, then we shall write

[α] = p[α_{1}] + q[α_{2}].

From the definition, we see that [α] = pγ + qγ′ if [α] = pγ ⊕ qγ′, where p ≥ 0, q ≥ 0 are integers with p + q > 0, and where γ, γ′ ∈ 𝒢 are disjoint geodesics.

2. Four integer functions

In Section 4, we shall construct a homeomorphism Φ of \overline{πℐ (𝒢 ℒ )} onto a 3-sphere lying in R^{4} . For α ∈ 𝒢 ℒ , the value Φ(α) is written in terms of four integers I_{X}(α) ≥ 0, I_{Y} (α) ≥ 0, N_{S}(α) and N_{T} (α). The sign of N_{S}(α) and that of N_{T} (α) are determined by the geometry of α. The integers I_{X}(α), I_{Y} (α), |N_{S}(α)| and |N_{T} (α)| are numbers of strands of α.

The integer functions I_{X} and I_{Y} are analogues of the integer function I_{X} given in [4], and the integer functions N_{S} and N_{T} are analogues of the integer function N given in [4]. In this section, we shall define the integer functions I_{X}, I_{Y} , N_{S} and N_{T} , and discuss their properties.

2.1. Elementary intersection numbers. For the construction of the homeomorphism Φ, we shall start with a homeomorphism Ψ of \overline{πℐ (𝒢 ℒ )} onto a 3-sphere lying in R^{6} whose value at every α ∈ 𝒢 ℒ is written in terms of the geometric intersection numbers of α with the six geodesics γ_{jk} given in Figure 2. These six geometric intersection numbers i(α, γ_{jk}) will be called the elementary intersection numbers of α.

To compute elementary intersection numbers, we consider the projections of the sides of 𝒟 to Σ_{5} . For E ∈ {S, T, X, Y }, the E -side of 𝒟 projects to Σ_{5} a simple curve β_{E} connecting exactly two punctures. Write

I_{E}(α) = i(α, [β_{E}])

for all α ∈ 𝒢 ℒ . Note that

I_{E}(α) = #{strands of α which meet the E -side (or the E^{−1} -side)}.

Thus, we have

(3)  \begin{aligned}i(α, γ_{11}) & = 2I_{X}(α), \\ i(α, γ_{12}) & = 2I_{T} (α), \end{aligned} \begin{aligned}i(α, γ_{21}) & = 2I_{Y} (α), \\ i(α, γ_{22}) & = 2I_{S}(α). \end{aligned}

We shall prove later that the elementary intersection numbers of α can be written in terms of I_{X}(α), I_{Y} (α), N_{S}(α) and N_{T} (α) (see Corollary 3.4). This allows us to construct the homeomorphism Ψ by use of the functions I_{X}, I_{Y} , N_{S} and N_{T} .

For later use, we extend the integer functions I_{E} to admissible subarcs of geodesics in 𝒢 as follows. For E ∈ Γ, and for an arbitrary admissible subarc γ′ of a geodesic γ ∈ 𝒢 , let

I_{E}(γ′) = #(strands of γ′ which meet the E -side of 𝒟 ).

Note that I_{E}(γ) = I_{E^{−1}}(γ) for γ ∈ 𝒢 and for E ∈ Γ.

2.2. Cyclic semi-reduced Γ-words. Let γ ∈ 𝒢̂ = 𝒢 − {γ_{T} , γ_{S}} be represented by a cyclic reduced Γ-word W (γ). We have known that for E ∈ {S, T, X, Y } the integer I_{E}(γ) is the number of strands of γ which meet the E -side. We may also relate the number I_{E}(γ) to W (γ) as follows

I_{E}(γ) = the total number of the letters E and E^{−1} appearing in W (γ).

Therefore, to compute the elementary intersection numbers of γ ∈ 𝒢̂ is equivalent to finding a cyclic reduced Γ-word representing γ.

In general, it is not easy to write cyclic reduced Γ-words representing geodesics in 𝒢 explicitly. Therefore, we shall introduce cyclic semi-reduced Γ-words. Cyclic semi-reduced Γ-words also work for our purposes. To compute geometric intersection numbers, we only need a partial description of cyclic semi-reduced Γ-words, which will be given in Section 2.5. The complete description is given in Section 5.

Figure 4. From the left to the right: α_{1} , α_{2} , α_{3} , α_{4} , α_{5} , α_{6} .

To motivate the definition of cyclic semi-reduced Γ-words, we consider the geodesics represented by the following cyclic reduced Γ-words:

\begin{aligned}W_{1} & = XS^{−1}Y, \\ W_{4} & = X^{−1}SY^{−1}, \end{aligned} \begin{aligned}W_{2} & = T X^{−1}S^{−1}Y^{−1}S, \\ W_{5} & = T^{−1}XSY S^{−1}, \end{aligned} \begin{aligned}W_{3} & = T XT^{−1}S^{−2}Y S, \\ W_{6} & = T^{−1}X^{−1}T S^{2}Y^{−1}S^{−1}. \end{aligned}

Let α_{j} be the geodesic represented by W_{j} for 1 ≤ j ≤ 6 (see Figure 4). By defining the zero power E^{0} of the transformation E to be the identity transformation for E = S or T, we may rewrite above words as

(4)  W_{j} = T^{r_{j}}X^{ω_{j}}T^{t_{j}}S^{p_{j}}Y^{ε_{j}}S^{q_{j}},

where χ_{i} = (r_{i}, ω_{i}, t_{i}, p_{i}, ε_{i}, q_{i}) are given below:

\begin{aligned}χ_{1} & = (0, 1, 0, −1, 1, 0), \\ χ_{3} & = (1, 1, −1, −2, 1, 1), \\ χ_{5} & = (−1, 1, 0, 1, 1, −1), \end{aligned} \begin{aligned}χ_{2} & = (1, −1, 0, −1, −1, 1), \\ χ_{4} & = (0, −1, 0, 1, −1, 0), \\ χ_{6} & = (−1, −1, 1, 2, −1, −1). \end{aligned}

From the word given in (4), we have

I_{X}(α_{j}) = 1, I_{Y} (α_{j}) = 1, I_{S}(α_{j}) = |p_{j}| + |q_{j}| and I_{T} (α_{j}) = |r_{j}| + |t_{j}|.

Now, we define the cyclic semi-reduced Γ-words representing geodesics in 𝒢̂ as follows. Let γ ∈ 𝒢̂ be a geodesic represented by a cyclic reduced Γ-word W (γ). If Y^{ε}E or EY^{ε} is a subword of W (γ) with ε = ±1 and E ∈ {X^{±1}, T^{±1}}, we shall write

Y^{ε}E = Y^{ε}S^{0}E and EY^{ε} = ES^{0}Y^{ε}.

Similarly, if E ∈ {Y^{±1}, S^{±1}}, and if X^{ε}E or EX^{ε} is a subword of W (γ), then we write

X^{ε}E = X^{ε}T^{0}E and EX^{ε} = ET^{0}X^{ε}.

The resulting cyclic Γ-word will be called semi-reduced, still denoted by W (γ).

2.3. Four automorphisms of 𝒢 ℒ . Let α_{j} be the geodesics given in Section 2.2, and let W_{j} be the corresponding cyclic semi-reduced Γ-words. By considering the symmetry of the fundamental domain 𝒟, we realize that for 1 ≤ j ≤ 3 the words W_{j+3} are the images of W_{j} under the automorphism Θ_{1} of G defined by

Θ_{1}(E) = E^{−1} for E ∈ {S, T, X, Y }.

There is another automorphism Θ_{2} of G obtained from the symmetry of 𝒟 defined by

Θ_{2}(S) = T, Θ_{2}(T) = S, Θ_{2}(X) = Y, Θ_{2}(Y) = X.

For j = 1 or 2, the automorphism Θ_{j} induces an orientation reversing homeomorphism of Σ_{5} onto itself which is also denoted by Θ_{j}. If γ ∈ 𝒢 is a geodesic, let Θ_{j}(γ) denote the free homotopy class in 𝒢 represented by the image of γ mapped by Θ_{j}. This defines an injective function, still denoted by Θ_{j}, of 𝒢 onto itself such that if W is a cyclic reduced (or semi-reduced) Γ-word representing γ ∈ 𝒢 , then Θ_{j}(γ) is represented by Θ_{j}(W).

For instance, we have Θ_{1}(α_{j}) = α_{j+3} for 1 ≤ j ≤ 3. For every integer j with 1 ≤ j ≤ 6, the geodesic Θ_{2}(α_{j}) is represented by the word

Θ_{2}(W_{j}) = S^{r_{j}}Y^{ω_{j}}S^{t_{j}}T^{p_{j}}X^{ε_{j}}T^{q_{j}},

where W_{j} is the cyclic semi-reduced Γ-word given in (4).

Now, we extend the functions Θ_{1} and Θ_{2} to 𝒢 ℒ by defining

Θ_{j}(aγ ⊕ bγ′) = aΘ_{j}(γ) ⊕ bΘ_{j}(γ′)

for j = 1, 2, where a ≥ 0 and b ≥ 0 are integers with a + b > 0, and where γ and γ′ are disjoint geodesics in 𝒢 .

With the two maps Θ_{1} and Θ_{2} , we may simplify the argument on finding cyclic semi-reduced Γ-words by considering subsets of 𝒢 which are related by Θ_{1} and Θ_{2} . Let

𝒢 ℒ _{S}^{+} = {α ∈ 𝒢 ℒ : α has no strands joining the S^{−1} -side to the Y^{ε} -side, ε = ±1}, and let

𝒢 ℒ S^{−} = Θ_{1}(𝒢 ℒ S^{+}), 𝒢 ℒ T^{+} = Θ_{2}(𝒢 ℒ S^{−}) and 𝒢 ℒ T^{−} = Θ_{1}(𝒢 ℒ T^{+}) = Θ_{2}(𝒢 ℒ S^{+}).

For E = S or T, let 𝒢E^{+} = 𝒢 ℒ E^{+} ∩ 𝒢 and 𝒢E^{−} = 𝒢 ℒ E^{−} ∩ 𝒢 .

Note that for E = S or T the sets 𝒢 ℒ E^{+} and 𝒢 ℒ E^{−} are not disjoint since

aγ_{S} ⊕ bγ_{T} ∈ 𝒢 ℒ E^{+} ∩ 𝒢 ℒ E^{−},

where a ≥ 0 and b ≥ 0 are integers with a + b > 0.

The following proposition is an immediate consequence of the definition.

Proposition 2.1. If α ∈ 𝒢 ℒ , then I_{E}(Θ_{1}(α)) = I_{E}(α) for E ∈ {S, T, X, Y } and

\begin{aligned}I_{X}(Θ_{2}(α)) & = I_{Y} (α), \\ I_{S}(Θ_{2}(α)) & = I_{T} (α), \end{aligned} \begin{aligned}I_{Y} (Θ_{2}(α)) & = I_{X}(α), \\ I_{T} (Θ_{2}(α)) & = I_{S}(α). \end{aligned}

Taking a further step to investigate the relations among the geodesics α_{1} , α_{2} and α_{3} , we found that the geodesics α_{1} , α_{2} and α_{3} are related by the automorphisms 𝒯_{1} and 𝒯_{2} of G defined by

\begin{matrix}𝒯_{1}: \\ 𝒯_{2}: \end{matrix} \begin{matrix}S ⟶ S, \\ S ⟶ S, \end{matrix} \begin{matrix}T ⟶ T, \\ T ⟶ T, \end{matrix} \begin{matrix}X ⟶ X, \\ X ⟶ X^{−1}T, \end{matrix} \begin{matrix}Y ⟶ Y^{−1}S, \\ Y ⟶ Y. \end{matrix}

From the definition, we obtain

Θ_{2}𝒯_{1}Θ_{2} = 𝒯_{2} and Θ_{1}𝒯_{j}Θ_{1} = 𝒯_{j}^{−1} for j = 1, 2.

For j = 1 or 2, the automorphism 𝒯_{j} induces an orientation preserving homeomorphism of Σ_{5} onto itself, denoted by 𝒯_{j} as well. The homeomorphism 𝒯_{1} interchanges the two punctures on Σ_{5} corresponding to the fixed point of Y and the fixed point of Y^{−1}S, and leaves the other punctures invariant. The homeomorphism 𝒯_{2} interchanges the two punctures on Σ_{5} corresponding to the fixed point of X and the fixed point of X^{−1}T, and leaves the other punctures invariant.

Each 𝒯_{j} also induces an injective function of 𝒢 onto itself so that if W is a cyclic reduced (or semi-reduced) Γ-word representing γ ∈ 𝒢 , then 𝒯_{j}(γ) is represented by 𝒯_{j}(W). Now, α_{1} , α_{2} and α_{3} are related by 𝒯_{1} and 𝒯_{2} as follows:

𝒯_{1}𝒯_{2}^{−1}(α_{1}) = α_{2} and 𝒯_{1}𝒯_{2}^{−1}(α_{2}) = α_{3}.

Like Θ_{1} and Θ_{2} , the functions 𝒯_{1} and 𝒯_{2} extend to 𝒢 ℒ defined by

𝒯_{j}(aγ ⊕ bγ′) = a𝒯_{j}(γ) ⊕ b𝒯_{j}(γ′), j = 1, 2,

where a ≥ 0 and b ≥ 0 are integers with a + b > 0, and where γ and γ′ are disjoint geodesics in 𝒢 .

Proposition 2.2. Let α ∈ 𝒢 ℒ .

(i) If I_{Y} (α) = 0, then 𝒯_{1}(α) = α. (ii) If I_{X}(α) = 0, then 𝒯_{2}(α) = α. (iii) If k is an integer, and if E = X or Y , then I_{E}(𝒯_{1}^{k}(α)) = I_{E}(α) =

I_{E}(𝒯_{2}^{k}(α)).

Proof. For the proof of (i) and (ii), it suffices to consider the case where α ∈ 𝒢 . Let W be a cyclic semi-reduced Γ-word representing α. If I_{Y} (α) = 0, then Y and Y^{−1} are not subwords of W, and 𝒯_{1}(W) = W. This proves that α is invariant under 𝒯_{1} . Similarly, α is invariant under 𝒯_{2} if I_{X}(α) = 0.

Since γ_{11} and γ_{21} are invariant under 𝒯_{1} and 𝒯_{2} , we have

i(𝒯_{j}^{k}(α), γ_{m1}) = i(α, 𝒯_{j}^{−k}(γ_{m1})) = i(α, γ_{m1})

for j, m ∈ {1, 2}. Now, the statement (iii) follows from equation (3).

2.4. Definition of the integer functions N_{S} and N_{T} . Let γ ∈ 𝒢 be a geodesic. If γ ∈ 𝒢S^{+} , let

N_{S}(γ) = #(strands of γ joining the S-side and the S^{−1}-side)

+ #(strands of α joining the S-side and the Y^{ε}-side)

for ε = ±1. If γ ∈ 𝒢T^{+} , let

N_{T} (γ) = #(strands of γ joining the T -side and the T^{−1}-side)

+ #(strands of α joining the T^{−1}-side and the X^{ε}-side)

for ε = ±1. For E = S or T, if γ ∈ 𝒢E^{−}, let N_{E}(γ) = −N_{E}(Θ_{1}(γ)).

From the definition, we have

Proposition 2.3. If γ ∈ 𝒢̂ , then N_{S}(γ) = −N_{T} (Θ_{2}(γ)) and N_{T} (γ) = −N_{S}(Θ_{2}(γ)).

For two integers a ≥ 0 and b ≥ 0 with a + b > 0, let

N_{S}(aγ_{S} ⊕ bγ_{T}) = a and N_{T} (aγ_{S} ⊕ bγ_{T}) = b.

Next, if γ ∈ 𝒢̂ is a geodesic disjoint from γ_{S}, let

N_{S}(aγ_{S} ⊕ bγ) = a and N_{T} (aγ_{S} ⊕ bγ) = bN_{T} (γ).

If γ ∈ 𝒢̂ is a geodesic disjoint from γ_{T} , let

N_{S}(aγ_{T} ⊕ bγ) = bN_{S}(γ) and N_{T} (aγ_{T} ⊕ bγ) = a.

Finally, if γ_{1} and γ_{2} are disjoint geodesics in 𝒢̂ , we define

N_{E}(aγ_{1} ⊕ bγ_{2}) = aN_{E}(γ_{1}) + bN_{E}(γ_{2}) for E = S, T.

To interpret N_{S}(α) and N_{T} (α) geometrically for α ∈ 𝒢 ℒ , we need Lemma 2.4. If γ_{1} and γ_{2} are disjoint geodesics in 𝒢̂ , then

N_{S}(γ_{1})N_{S}(γ_{2}) ≥ 0 and N_{T} (γ_{1})N_{T} (γ_{2}) ≥ 0.

Proof. We shall prove N_{T} (γ_{1})N_{T} (γ_{2}) ≥ 0. This implies, by Proposition 2.3, that N_{S}(γ_{1})N_{S}(γ_{2}) ≥ 0. First, note that if γ ∈ 𝒢̂ with N_{T} (γ) ≠ 0, then I_{X}(γ) > 0.

Suppose that N_{T} (γ_{1}) > 0 and N_{T} (γ_{2}) < 0. Then γ_{1} has a strand l_{1} joining the T^{−1} -side to the X^{ε} -side with ε = ±1, and has a strand l′_{1} joining the X^{−ε} - side to some E -side with E ∈ {T^{−1}, S^{±1}, Y^{±1}} so that its endpoint on X^{−ε} -side is identified with that of l_{1} on the X^{ε} -side by the transformation X^{ε}.

Similarly, γ_{2} has a strand l_{2} joining the T -side to the X^{δ} -side with δ = ±1, and has a strand l′_{2} joining the X^{−δ} -side to some E′ -side with E′ ∈ {T, S^{±1}, Y^{±1}} so that its endpoint on the X^{−δ} -side is identified with that of l_{2} on the X^{δ} -side by the transformation X^{δ}.

Since l_{1} ∪ l′_{1} must intersect l_{2} ∪ l′_{2} , then i(γ_{1}, γ_{2}) > 0. Contradiction!

Now, for α ∈ 𝒢 ℒ we have

|N_{S}(α)| = #(strands of α joining the S-side and the S^{−1}-side)

+ #(strands of α joining the S^{δ}-side and the Y^{ε}-side);

|N_{T} (α)| = #(strands of α joining the T -side and the T^{−1}-side)

+ #(strands of α joining the T^{δ}-side and the X^{ε}-side),

where δ, ε = ±1.

Proposition 2.5. Let α ∈ 𝒢 ℒ .

(i) If I_{X}(α) > 0, then N_{T} (α) ≥ 0 whenever α ∈ 𝒢 ℒ T^{+} , and N_{T} (α) ≤ 0 whenever α ∈ 𝒢 ℒ T^{−}. Thus, N_{T} (Θ_{1}(α)) = −N_{T} (α).

(ii) If I_{Y} (α) > 0, then N_{S}(α) ≥ 0 whenever α ∈ 𝒢 ℒ S^{+} , and N_{S}(α) ≤ 0 whenever α ∈ 𝒢 ℒ S^{−}. Thus, N_{S}(Θ_{1}(α)) = −N_{S}(α).

(iii) If I_{X}(α)I_{Y} (α) > 0, then

N_{S}(α) = −N_{T} (Θ_{2}(α)) and N_{T} (α) = −N_{S}(Θ_{2}(α)).

Proof. The statement (ii) will follow from (i) by considering Θ_{2}(α). The statement (iii) is a consequence of (i) and (ii). It remains to prove the statement (i).

Write α = aγ_{1} ⊕ bγ_{2} , where a ≥ 0 and b ≥ 0 are integers with a + b > 0, and where γ_{1} and γ_{2} are disjoint geodesics in 𝒢 . If ab = 0, then the statement (i) holds trivially since I_{X}(α) > 0.

Assume that ab > 0. Since I_{X}(α) > 0, then γ_{1} ≠ γ_{T} and γ_{2} ≠ γ_{T} . If γ_{1} = γ_{S}, then I_{X}(γ_{2}) > 0, and N_{T} (α) = bN_{T} (γ_{2}). Now, the assertion follows from the definition of the function N_{T} on 𝒢̂ .

Similarly, the statement (i) is true if γ_{2} = γ_{S}. If γ_{1} ≠ γ_{S} and γ_{2} ≠ γ_{S}, the proof is completed by Lemma 2.4.

2.5. Relating N_{S} and N_{T} to cyclic semi-reduced Γ-words. Now, we shall explain how to determine N_{S}(γ) and N_{T} (γ) from a cyclic semi-reduced Γ-word W representing γ ∈ 𝒢̂ . Note that I_{X}(γ) > 0 or I_{Y} (γ) > 0.

If I_{Y} (γ) = n > 0, then there are exactly n triples of integers (p_{i}, ε_{i}, q_{i}) with ε_{i} = ±1 such that E_{i}S^{p_{i}}Y^{ε_{i}}S^{q_{i}}E′_{i} is a subword of W for every integer i ∈ {1, …, n}, where E_{i}, E′_{i} ∈ {T^{±1}, X^{±1}, Y^{±1}}. From Remark 1.1, we have E_{i}, E′_{i} ∈ {T^{±1}, X^{±1}} for every i. Thus W must be of the form

(5)  W = ∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}W_{i},

where each W_{i} is a semi-reduced Γ-word of the form

W_{i} = ∏_{i=1}^{m_{i}} E_{ij}

with E_{i1}, E_{im_{i}} ∈ {T^{±1}, X^{±1}}, and E_{ij} ≠ Y^{±1} whenever 1 < j < m_{i}.

If I_{X}(γ) = n > 0, then I_{Y} (Θ_{2}(γ)) = n, and γ is represented by a cyclic semi-reduced Γ-word as given in equation (5). Thus γ is represented by a cyclic semi-reduced Γ-word W of the form

(6)  W = ∏_{i=1}^{n} T^{p_{i}}X^{ε_{i}}T^{q_{i}}W_{i},

where ε = ±1, where p_{i} and q_{i} are integers, and where each W_{i} is a semi-reduced Γ-word of the form

W_{i} = ∏_{i=1}^{m_{i}} E_{ij}

with E_{i1}, E_{im_{i}} ∈ {S^{±1}, Y^{±1}}, and E_{ij} ≠ X^{±1} whenever 1 < j < m_{i}.

Before continuing our discussion, we shall find necessary conditions for the integers p_{i} and q_{i} given in (5) and (6).

Lemma 2.6. Let ε = ±1, let p and q be integers, let γ ∈ 𝒢̂ , and let W be a cyclic semi-reduced Γ-word representing γ.

(i) If W′ = ES^{p}Y^{ε}S^{q}E′ is a subword of W with E, E′ ∈ {X^{±}, T^{±}}, then

−1 ≤ (p + q)ε ≤ 0.

Moreover, p ≤ 0 and q ≥ 0 when γ ∈ 𝒢S^{+} , and p ≥ 0 and q ≤ 0 when γ ∈ 𝒢S^{−}. (ii) If W′ = ET^{p}X^{ε}T^{q}E′ is a subword of W with E, E′ ∈ {Y^{±}, S^{±}}, then

−1 ≤ (p + q)ε ≤ 0.

Moreover, p ≥ 0 and q ≤ 0 when γ ∈ 𝒢T^{+} , and p ≤ 0 and q ≥ 0 when γ ∈ 𝒢T^{−}.

Proof. For the proof of (i), we may assume that ε = 1 and γ ∈ 𝒢S^{+} . By the definition of 𝒢S^{+} , we have p ≤ 0 and q ≥ 0.

We rewrite W′ as W′ = ES^{−p}Y^{ε}S^{q}E′ = ES^{−p}Y S^{q}E′, where p ≥ 0 and q ≥ 0. If q > p, then 𝒯_{1}^{−2p}(W′) = EY S^{q−p}E′ is a subword of 𝒯_{1}^{−2p}(W), and 𝒯_{1}^{−2p}(γ) is not simple. Contradiction!

If p > q +1, then 𝒯_{1}^{−2q}(W′) = ES^{−p+q}Y E′. This implies that 𝒯_{1}^{−2q}(γ) has a strand joining the S -side to the S^{−1} -side, and has a strand joining the Y^{−1} -side to the E′ -side with E′ ∈ {T^{±}, X^{±}}. This is impossible. Therefore, q ≤ p ≤ q +1.

By considering 𝒯_{2} , the statement (ii) will follow by a similar argument.

Proposition 2.7. Let γ ∈ 𝒢̂ be a geodesic, and let W be a cyclic semireduced Γ-word representing γ.

(i) If W is of the form given in equation (5), then N_{S}(γ) = ∑_{i=1}^{n}(q_{i} − p_{i}). (ii) If W is of the form given in equation (6), then N_{T} (γ) = ∑_{i=1}^{n}(p_{i} − q_{i}).

Proof. From Proposition 2.3, the statement (ii) follows from the statement (i). On the other hand, since N_{S}(Θ_{1}(γ)) = −N_{S}(γ), we may assume that γ ∈ 𝒢S^{+} . Thus p_{i} ≤ 0 and q_{i} ≥ 0 for all i by Lemma 2.6.

For every i, let γ_{i} be the admissible subarc of γ represented by E⃗_{i}W_{i}E′_{i}, where

E_{i} = \{ \begin{matrix}S & if q_{i} > 0, \\ Y^{ε_{i}} & if q_{i} = 0, \end{matrix} and E′_{i} = \{ \begin{matrix}S^{−1} & if p_{i} < 0, \\ Y^{ε_{i+1}} & if p_{i} = 0. \end{matrix}

From the definition of W_{i}, we know that each γ_{i} neither has strands connecting the S -side to the Y -side, nor has strands connecting the S -side to the Y^{−1} -side. From Proposition 1.1, each γ_{i} has no strands joining the S -side and the S^{−1} -side. Thus N_{S}(γ) is completely determined by the subwords S^{p_{i}}Y^{ε_{i}}S^{q_{i}}, 1 ≤ i ≤ n.

Using notation given in equation (5), for every i let γ′_{i} be the admissible subarc represented by E⃗_{(i−1)m_{i−1}}S^{p_{i}}Y^{ε_{i}}S^{q_{i}}, and let

N_{i}^{(1)} = #(strands of γ′_{i} connecting the S-side and the S^{−1}-side),
N_{i}^{(2)} = #(strands of γ′_{i} connecting the S-side and the Y -side)

+ #(strands of γ′_{i} connecting the S-side and the Y^{−1}-side).

Since −1 ≤ (p_{i} + q_{i})ε_{i} ≤ 0 for every i, then

(N_{i}^{(1)}, N_{i}^{(2)}) = \{ \begin{matrix}(q_{i} − p_{i} − 2, 2) & if q_{i} − p_{i} > 2, \\ (0, q_{i} − p_{i}) & if q_{i} − p_{i} ≤ 2. \end{matrix}

Thus

N_{S}(γ) = ∑_{i=1}^{n}(N_{i}^{(1)} + N_{i}^{(2)}) = ∑_{i=1}^{n}(q_{i} − p_{i}).

At the end of this section, we shall investigate how the integers N_{S}(𝒯_{j}^{k}(γ)) and N_{T} (𝒯_{j}^{k}(γ)) relate to the integers N_{S}(γ) and N_{T} (γ) for j = 1 or 2, where k ≠ 0 is an integer.

Proposition 2.8. Let γ ∈ 𝒢 , and let k be an arbitrary integer. Then

(i) N_{S}(𝒯_{1}^{k}(γ)) = N_{S}(γ) + kI_{Y} (γ) and N_{S}(𝒯_{2}^{k}(γ)) = N_{S}(γ); (ii) N_{T} (𝒯_{1}^{k}(γ)) = N_{T} (γ) and N_{T} (𝒯_{2}^{k}(γ)) = N_{T} (γ) − kI_{X}(γ).

Proof. The proposition holds trivially for γ = γ_{T} and for γ = γ_{S}. In the following, we assume that γ ∈ 𝒢̂ .

Since Θ_{2}𝒯_{1}Θ_{2} = 𝒯_{2} , then the equations in (ii) follow from that given in (i) by Proposition 2.1 and Proposition 2.3.

Now, we shall only prove the equations given in (i) for k = ±1. Then the proof of the proposition is completed by applying mathematical induction to |k|.

If I_{Y} (γ) = 0, then N_{S}(γ) = 0. From Proposition 2.2, we have I_{Y} (𝒯_{j}^{k}(γ)) = 0 for j = 1, 2. Thus N_{S}(𝒯_{j}^{k}(γ)) = 0, and the equations in (i) hold.

Let I_{Y} (γ) = n > 0. Assume that γ ∈ 𝒢S^{+} . Then γ is represented by a cyclic semi-reduced Γ-word W of the form

W = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}W_{i},

where ε = ±1, p_{i} ≥ 0, q_{i} ≥ 0 are integers, and where each W_{i} is a semi-reduced Γ-word as given in equation (5). Since

𝒯_{1}(W) = ∏_{i=1}^{n} S^{−p′_{i}}Y^{−ε_{i}}S^{q′_{i}}W_{i} and 𝒯_{1}^{−1}(W) = ∏_{i=1}^{n} S^{−p′′_{i}}Y^{−ε_{i}}S^{q′′_{i}}W_{i},

with p′_{i} + q′_{i} = p_{i} + q_{i} + 1 and p′′_{i} + q′′_{i} = p_{i} + q_{i} − 1, from Proposition 2.7 we have

\begin{aligned}N_{S}(𝒯_{1}(γ)) & = ∑_{i=1}^{n}(p′_{i} + q′_{i}) = n + ∑_{i=1}^{n}(p_{i} + q_{i}) = N_{S}(γ) + I_{Y} (γ) and \\ N_{S}(𝒯_{1}^{−1}(γ)) & = ∑_{i=1}^{n}(p′′_{i} + q′′_{i}) = −n + ∑_{i=1}^{n}(p_{i} + q_{i}) = N_{S}(γ) − I_{Y} (γ). \end{aligned}

Let W′_{i} = 𝒯_{2}(W_{i}) and W′′_{i} = 𝒯_{2}^{−1}(W_{i}) for every i. By the definition of W_{i} and that of 𝒯_{2} , we easily see that W′_{i} and W′′_{i} have the same form as W_{i} has. Since

𝒯_{2}(W) = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}W′_{i} and 𝒯_{2}^{−1}(W) = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}W′′_{i},

then

N_{S}(𝒯_{2}(γ)) = N_{S}(𝒯_{2}^{−1}(γ)) = ∑_{i=1}^{n}(p_{i} + q_{i}) = N_{S}(γ).

If γ ∈ 𝒢S^{−}, then Θ_{1}(γ) ∈ 𝒢S^{+} , and
N_{S}(𝒯_{1}(γ)) = −N_{S}(Θ_{1}𝒯_{1}(γ)) = −N_{S}(𝒯_{1}^{−1}Θ_{1}(γ))

= −{N_{S}(Θ_{1}(γ)) − I_{Y} (Θ_{1}(γ))} = N_{S}(γ) + I_{Y} (γ);

N_{S}(𝒯_{1}^{−1}(γ)) = −N_{S}(Θ_{1}𝒯_{1}^{−1}(γ)) = −N_{S}(𝒯_{1}Θ_{1}(γ))

= −{N_{S}(Θ_{1}(γ)) + I_{Y} (Θ_{1}(γ))} = N_{S}(γ) − I_{Y} (γ);

N_{S}(𝒯_{2}^{k}(γ)) = −N_{S}(Θ_{1}𝒯_{2}^{k}(γ)) = −N_{S}(𝒯_{2}^{−k}Θ_{1}(γ))

= −N_{S}(Θ_{1}(γ)) = N_{S}(γ) for k = ±1.

3. Geometric intersection numbers

In this section, we shall prove the geometric intersection formula (see Theorem 3.1). The geometric intersection formula will be used to prove the injectivity of a homeomorphism Ψ of \overline{πℐ (𝒢 ℒ )} onto a 3-sphere. The homeomorphism Ψ will be constructed with elementary intersection numbers. From the geometric intersection formula, we obtain the elementary intersection numbers of geodesics in 𝒢 . Then we will get elementary intersection numbers of α ∈ 𝒢 ℒ .

3.1. The geometric intersection formula. The main work of this subsection is to prove the following theorem:

Theorem 3.1 (Geometric intersection formula). If γ_{1} and γ_{2} are two simple closed geodesics on Σ_{5} , then

\begin{aligned}i(γ_{1}, γ_{2}) & = 2|I_{X}(γ_{1})N_{T} (γ_{2}) − I_{X}(γ_{2})N_{T} (γ_{1})| + 2|I_{Y} (γ_{1})N_{S}(γ_{2}) − I_{Y} (γ_{2})N_{S}(γ_{1})| \\ & + |I_{XY} (γ_{1}, γ_{2})| − I_{XY} (γ_{1}, γ_{2}), \end{aligned}

where I_{XY} (γ_{1}, γ_{2}) = {I_{X}(γ_{1}) − I_{Y} (γ_{1})} · {I_{X}(γ_{2}) − I_{Y} (γ_{2})}.

As a consequence of the geometric intersection formula, we obtain the elementary intersection numbers of geodesics in 𝒢 as follows.

Corollary 3.2. If γ ∈ 𝒢 , then

\begin{aligned}i(γ, γ_{12}) & = 2|N_{T} (γ)| + |I_{Y} (γ) − I_{X}(γ)| + I_{Y} (γ) − I_{X}(γ), \\ i(γ, γ_{13}) & = 2|N_{T} (γ) − I_{X}(γ)| + |I_{Y} (γ) − I_{X}(γ)| + I_{Y} (γ) − I_{X}(γ), \\ i(γ, γ_{22}) & = 2|N_{S}(γ)| + |I_{X}(γ) − I_{Y} (γ)| + I_{X}(γ) − I_{Y} (γ), and \\ i(γ, γ_{23}) & = 2|N_{S}(γ) − I_{Y} (γ)| + |I_{X}(γ) − I_{Y} (γ)| + I_{X}(γ) − I_{Y} (γ). \end{aligned}

Proof of the geometric intersection formula. It is easy to see that the geometric intersection formula is valid if γ_{1} or γ_{2} is in {γ_{T} , γ_{S}}. It remains to prove the formula for γ_{1}, γ_{2} ∈ 𝒢̂ .

For every integer k, write F_{k} = 𝒯_{2}^{−k}𝒯_{1}^{k}. From Proposition 2.8, we obtain

\begin{aligned}I_{XY} (γ_{1}, γ_{2}) & = I_{XY} (F_{k}(γ_{1}), F_{k}(γ_{2})), \\ I_{X}(γ_{1})N_{T} (γ_{2})−I_{X}(γ_{2})N_{T} (γ_{1}) & = I_{X}(F_{k}(γ_{1}))N_{T} (F_{k}(γ_{2}))−I_{X}(F_{k}(γ_{2}))N_{T} (F_{k}(γ_{1})), \\ I_{Y} (γ_{1})N_{S}(γ_{2})−I_{Y} (γ_{2})N_{S}(γ_{1}) & = I_{Y} (F_{k}(γ_{1}))N_{S}(F_{k}(γ_{2}))−I_{Y} (F_{k}(γ_{2}))N_{S}(F_{k}(γ_{1})) \end{aligned}

for all integers k. From Proposition 2.2 and Proposition 2.8, there is an integer k > 0 such that

N_{T} (F_{k}(γ_{j})) ≥ 2I_{X}(γ_{j}) = 2I_{X}(F_{k}(γ_{j})) and N_{S}(F_{k}(γ_{j})) ≥ 2I_{Y} (γ_{j}) = 2I_{Y} (F_{k}(γ_{j}))

for j = 1, 2; thus we may assume that

N_{T} (γ_{j}) ≥ 2I_{X}(γ_{j}) and N_{S}(γ_{j}) ≥ 2I_{Y} (γ_{j}).

Figure 5. From the left to the right: τ_{1} , τ_{2} , τ_{3} .

If β ∈ 𝒢̂ is a geodesic with N_{T} (β) ≥ 2I_{X}(β) and N_{S}(β) ≥ 2I_{Y} (β), then β lies in 𝒢S^{+} ∩ 𝒢T^{+} , and β can be written as

β = pγ_{S} + qγ_{T} + rτ_{1} + sτ_{2} or β = pγ_{S} + qγ_{T} + rτ_{1} + sτ_{3},

where p, q, r and s are non-negative integers with p + q + r + s > 0, and where τ_{1} , τ_{2} and τ_{3} are geodesics represented by the following cyclic reduced Γ-words (see Figure 5):

W (τ_{1}) = S^{−1}Y^{−1}ST XT^{−1}, W (τ_{2}) = S^{−1}T XT^{−1} and W (τ_{3}) = S^{−1}Y^{−1}ST.

Let 𝒢 ℒ _{1} be the set of all elements of 𝒢 ℒ of the form pγ_{S} +qγ_{T} +rτ_{1} +sτ_{2} , and let 𝒢 ℒ _{2} be the set of all elements of 𝒢 ℒ of the form pγ_{S} + qγ_{T} + rτ_{1} + sτ_{3} , where p, q, r and s are non-negative integers with p + q + r + s > 0.

Let 𝒟 be the fundamental domain for G given in Section 1.2. Let ℛ denote the reflection in the imaginary axis. Let l^{∗} be the semi-circle contained in 𝒟 joining the fixed point of S^{−1}T to the fixed point of T S^{−1} . Note that l^{∗} is invariant under ℛ. Let

P^{∗} be the point of intersection of l^{∗} with the imaginary axis, 𝒟^{+} be the connected component of 𝒟 − l^{∗} lying above l^{∗}, 𝒟^{−} be the connected component of 𝒟 − l^{∗} lying below l^{∗}, Σ5^{+} and Σ5^{−} be the projections of 𝒟^{+} and 𝒟^{−} to Σ_{5} , respectively, 𝒮4^{+} be the four-punctured sphere obtained from 𝒟^{+} − {P^{∗}} by identifying the boundary points of 𝒟^{+} − {P^{∗}} via X, T and ℛ, 𝒮4^{−} be the four-punctured sphere obtained from 𝒟^{−} − {P^{∗}} by identifying the boundary points of 𝒟^{−} − {P^{∗}} via Y , S and ℛ, and γ^{∗} be the projection of l^{∗} to Σ_{5} , which is the common boundary of Σ5^{+} and Σ5^{−}. The free homotopy class containing γ^{∗} is also denoted by γ^{∗}.

The fixed point ζ of S^{−1}T projects to a puncture ζ^{+} on 𝒮4^{+} , and projects to a puncture ζ^{−} on 𝒮4^{−}. Let [ζ^{+}] denote the free homotopy class of simple loops on 𝒮4^{+} enclosing ζ^{+} , and let [ζ^{−}] denote the free homotopy class of simple loops on 𝒮4^{−} enclosing ζ^{−}. It is obvious that i([ζ^{+}], α) = 0 for all free homotopy classes α of multiple simple loops on 𝒮4^{+} , and that i([ζ^{−}], β) = 0 for all free homotopy classes β of multiple simple loops on 𝒮4^{−}.

Figure 6. τj^{+} and τj^{−} for j = 1, 2, 3.

For any reduced simple loop α in the free homotopy class γ ∈ 𝒢 , let

α^{+} = α ∩ Σ5^{+} and α^{−} = α ∩ Σ5^{−}.

We shall call a connected component of the lift of α^{+} to 𝒟 a strand of α^{+} , and call a connected component of the lift of α^{−} to 𝒟 a strand of α^{−}. Let

γ^{+} = {α^{+} : α is a reduced simple loop in the free homotopy class γ} and
γ^{−} = {α^{−} : α is a reduced simple loop in the free homotopy class γ}.

See Figure 6 for examples of γ^{+} and γ^{−}. When there is no risk of confusion, we shall also use γ^{+} and γ^{−} to represent any curve in them. Since the geodesic γ_{T} is disjoint from Σ5^{−}, we shall also write γT^{+} = γ_{T} . Similarly, write γS^{−} = γ_{S}.

If γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2} is an arbitrary geodesic in 𝒢̂ ∩ 𝒢 ℒ _{1} , then γ^{−} has 2d strands whose union is homotopic to d copies of τ2^{−}. We shall call such strands τ2^{−} -type strands of γ^{−}.

If γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3} is an arbitrary geodesic in 𝒢̂ ∩ 𝒢 ℒ _{2} , then γ^{+} has 2d strands whose union is homotopic to d copies of τ3^{+} . We shall call such strands τ3^{+} -type strands of γ^{+} .

Let γ ∈ 𝒢̂ ∩ (𝒢 ℒ _{1} ∪ 𝒢 ℒ _{2}) be a geodesic, and write

γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2} or γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3}.

Then i(γ, γ^{∗}) = 2(c + d) since

i(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2}, γ^{∗}) = 2(c + d) = i(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3}, γ^{∗}).

Set k = c + d. Every simple closed curve α in the homotopy class γ is homotopic to a simple loop α̂ with the following properties:

(i) The lift of α̂ to 𝒟 intersects l^{∗} − {P^{∗}} at P_{1}, …, P_{k}, P′_{1}, …, P′_{k} with P′_{j} = ℛ(P_{j}).

(ii) The endpoints of strands of α̂ coincide with that of α. Then α̂^{+} projects to 𝒮4^{+} a multiple simple loop α̃^{+} , and α̂^{−} projects to 𝒮4^{−} a multiple simple loop α̃^{−}. Let γ̃^{+} denote the free homotopy class of multiple simple loops on 𝒮4^{+} represented by α̃^{+} , and let γ̃^{−} denote the free homotopy class of multiple simple loops on 𝒮4^{−} represented by α̃^{−}.

If γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2} with c + d > 0, then

γ̃^{+} = aγ_{T} + (c + d)τ̃1^{+} and γ̃^{−} = {bγ_{S} + cτ̃1^{−}} ⊕ d[ζ^{−}].

If γ = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3} with c + d > 0, then

γ̃^{+} = {aγ_{T} + cτ̃1^{+}} ⊕ d[ζ^{+}] and γ̃^{−} = bγ_{S} + (c + d)τ̃1^{−}.

Now we are in the position to compute i(γ_{1}, γ_{2}) for γ_{1}, γ_{2} ∈ 𝒢̂ ∩ (𝒢 ℒ _{1} ∪ 𝒢 ℒ _{2}). Without loss of generality, we may assume that all points of intersection of γ_{1} and γ_{2} are not on γ^{∗}.

Case 1. Assume that γ_{1}, γ_{2} ∈ 𝒢̂ ∩ 𝒢 ℒ _{1} . Clearly, I_{XY} (γ_{1}, γ_{2}) ≥ 0 and |I_{XY} (γ_{1}, γ_{2})| − I_{XY} (γ_{1}, γ_{2}) = 0. By applying suitable homotopy maps to γ_{1} and γ_{2} , we may assume that τ2^{−} -type strands of γ1^{−} are disjoint from γ_{2} , and that τ2^{−} -type strands of γ2^{−} are disjoint from γ_{1} . Then by Theorem 2.6 of [4] we obtain

i(γ_{1}, γ_{2}) = i(γ1^{+}, γ2^{+}) + i(γ1^{−}, γ2^{−}) = i(γ̃1^{+}, γ̃2^{+}) + iγ̃1^{−}, γ̃2^{−})

= 2|I_{X}(γ_{1})N_{T} (γ_{2}) − I_{X}(γ_{2})N_{T} (γ_{1})| + 2|I_{Y} (γ_{1})N_{S}(γ_{2}) − I_{Y} (γ_{2})N_{S}(γ_{1})| = 2|I_{X}(γ_{1})N_{T} (γ_{2}) − I_{X}(γ_{2})N_{T} (γ_{1})| + 2|I_{Y} (γ_{1})N_{S}(γ_{2}) − I_{Y} (γ_{2})N_{S}(γ_{1})|

+ |I_{XY} (γ_{1}, γ_{2})| − I_{XY} (γ_{1}, γ_{2}).

Case 2. If γ_{1}, γ_{2} ∈ 𝒢̂ ∩ 𝒢 ℒ _{2} , then Θ_{1}Θ_{2}(γ_{1}) and Θ_{1}Θ_{2}(γ_{2}) are both in 𝒢̂ ∩ 𝒢 ℒ _{1} , and the geometric intersection formula is valid for this case by Proposition 2.1.

Case 3. Assume that γ_{1} ∈ 𝒢̂ ∩ 𝒢 ℒ _{1} and γ_{2} ∈ 𝒢̂ ∩ 𝒢 ℒ _{2} . Write
γ_{1} = aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2} and γ_{2} = a′γ_{T} + b′γ_{S} + c′τ_{1} + d′τ_{3},

where dd′ > 0. Clearly, I_{XY} (γ_{1}, γ_{2}) < 0 and

|I_{XY} (γ_{1}, γ_{2})| − I_{XY} (γ_{1}, γ_{2}) = 2dd′.

Write the union of τ2^{−} -type strands of γ1^{−} as dτ2^{−}, and write the union of τ3^{+} -type strands of γ2^{+} as d′τ3^{+} .

To compute i(dτ2^{−}, γ2^{−}) + i(γ1^{+}, d′τ3^{+}), we need the orientation on the S -side and that on the S^{−1} -side (see the proof of Proposition 1.1). Also, we need an orientation to the T -side and an orientation to the T^{−1} -side.

Recall that ζ is the fixed point of the transformation S^{−1}T. If P and P′ are two distinct points on the T^{−1} -side, and if P lies between ζ and P′, then we write P ≺ P′. For any two distinct points Q and Q′ on the T -side, if T^{−1}(Q) ≺ T^{−1}(Q′), then we write Q ≺ Q′.

Let m = a′ + 2c′ + d′ and n = b′ + 2c′ + 2d′. Let

P_{1} ≺ ⋯ ≺ P_{m} be the endpoints of strands of γ_{2} on the T -side, Q_{1} ≺ ⋯ ≺ Q_{n} be the endpoints of the strands of γ_{2} on the S -side, L_{j}^{(2)} be the strand of γ_{2} with P_{j} an endpoint, 1 ≤ j ≤ d′,

l_{j}^{(2)} be the strand of γ_{2} with Q_{j} an endpoint, 1 ≤ j ≤ d′,

A_{1} ≺ ⋯ ≺ A_{d} be the first d points on the S -side where the lift of γ_{1} meets, A′_{j} be the point on the S^{−1} -side identified with A_{j} by S^{−1} , 1 ≤ j ≤ d,

L_{j}^{(1)} be the strand of γ_{1} with A′_{j} an endpoint, 1 ≤ j ≤ d, and

l_{j}^{(1)} be the strand of γ_{1} with A_{j} an endpoint, 1 ≤ j ≤ d.

Note that L_{j}^{(1)} connects the S^{−1} -side to the T -side, and each l_{j}^{(1)} connects the S -side to the T -side. Let B_{j} be the endpoint of l_{j}^{(1)} on the T -side. It is clear that B_{1} ≺ ⋯ ≺ B_{d}.

Figure 7.

Geometric intersection numbers on a five-punctured sphere 97 Without loss of generality, we assume that i(γ1^{+}, d′τ3^{+}) = 0, and that the

union L of all L_{j}^{(1)} is disjoint from γ_{2} (see Figure 7). Then

P_{d′} ≺ B_{1} ≺ ⋯ ≺ B_{d} ≺ P_{d′+1} and Q_{d′} ≺ A_{1} ≺ ⋯ ≺ A_{d} ≺ Q_{d′+1}.

This implies that each l_{j}^{(1)} intersects all L_{i}^{(2)} and all l_{i}^{(2)} transversally. Then

i(dτ2^{−}, γ2^{−}) = 2dd′.

By Theorem 2.6 of [4] again, we complete the proof of Theorem 3.1 as follows:

i(γ_{1}, γ_{2}) = i(γ1^{+}, γ2^{+}) + i(γ1^{−}, γ2^{−})

= i(γ̃1^{+}, γ̃2^{+}) + i(γ̃1^{−}, γ̃2^{−}) + i(dτ2^{−}, γ2^{−}) + i(γ1^{+}, d′τ3^{+})

= i(aγ_{T} + (c + d)τ̃1^{+}, a′γ_{T} + c′τ̃1^{+})

+ i(bγ_{S} + cτ̃1^{−}, b′γ_{S} + (c′ + d′)τ̃1^{−}) + 2dd′

= 2|I_{X}(γ_{1})N_{T} (γ_{2}) − I_{X}(γ_{2})N_{T} (γ_{1})| + 2|I_{Y} (γ_{1})N_{S}(γ_{2}) − I_{Y} (γ_{2})N_{S}(γ_{1})| + |I_{XY} (γ_{1}, γ_{2})| − I_{XY} (γ_{1}, γ_{2}).

3.2. Elementary intersection numbers of multiple simple loops. In the rest of this section, we shall prove the following proposition.

Proposition 3.3. If α ∈ 𝒢 ℒ , and if k is an integer, then

i(𝒯_{j}^{k}(α), γ_{11}) = i(α, γ_{11}), i(𝒯_{j}^{k}(α), γ_{21}) = i(α, γ_{21}) for j = 1, 2,
i(𝒯_{1}^{k}(α), γ_{1j}) = i(α, γ_{1j}), i(𝒯_{2}^{k}(α), γ_{2j}) = i(α, γ_{2j}) for j = 2, 3,
i(𝒯_{2}^{k}(α), γ_{12}) = 2|N_{T} (α) − kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α),
i(𝒯_{2}^{k}(α), γ_{13}) = 2|N_{T} (α) − (k + 1)I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α),
i(𝒯_{1}^{k}(α), γ_{22}) = 2|N_{S}(α) + kI_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α), and
i(𝒯_{1}^{k}(α), γ_{23}) = 2|N_{S}(α) + (k − 1)I_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α).

By letting k = 0 in the last four equations of the above proposition, we have Corollary 3.4 (Elementary intersection numbers). If α ∈ 𝒢 ℒ , then

\begin{aligned}i(α, γ_{12}) & = 2|N_{T} (α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α), \\ i(α, γ_{13}) & = 2|N_{T} (α) − I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α), \\ i(α, γ_{22}) & = 2|N_{S}(α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α), and \\ i(α, γ_{23}) & = 2|N_{S}(α) − I_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α). \end{aligned}

Lemma 3.5. Let γ and γ′ ∈ 𝒢̂ be disjoint geodesics, and let α = aγ ⊕ bγ′, where a ≥ 0 and b ≥ 0 are integers with a + b > 0. Then for all integers k

\begin{aligned}N_{T} (𝒯_{1}^{k}(α)) & = N_{T} (α), \\ N_{S}(𝒯_{2}^{k}(α)) & = N_{S}(α), \end{aligned} \begin{aligned}N_{T} (𝒯_{2}^{k}(α)) & = N_{T} (α) − kI_{X}(α), \\ N_{S}(𝒯_{1}^{k}(α)) & = N_{S}(α) + kI_{Y} (α). \end{aligned}

Proof. Since N_{E}(α) = aN_{E}(γ) + bN_{E}(γ′) for E = S or T, from Proposition 2.8 we obtain

N_{T} (𝒯_{1}^{k}(α)) = aN_{T} (𝒯_{1}^{k}(γ)) + bN_{T} (𝒯_{1}^{k}(γ′))
= aN_{T} (γ) + bN_{T} (γ′) = N_{T} (α) and
N_{T} (𝒯_{2}^{k}(α)) = aN_{T} (𝒯_{2}^{k}(γ)) + bN_{T} (𝒯_{2}^{k}(γ′))

= a{N_{T} (γ) − kI_{X}(γ)} + b{N_{T} (γ′) − kI_{X}(γ′)} = N_{T} (α) − kI_{X}(α).

Similarly, N_{S}(𝒯_{2}^{k}(α)) = N_{S}(α) and N_{S}(𝒯_{1}^{k}(α)) = N_{S}(α) + kI_{Y} (α).

Lemma 3.5. If γ and γ′ are two disjoint geodesics in 𝒢̂ , then

\begin{aligned}(N_{T} (γ) − I_{X}(γ))(N_{T} (γ′) − I_{X}(γ′)) & ≥ 0, \\ (N_{T} (γ) + I_{X}(γ))(N_{T} (γ′) + I_{X}(γ′)) & ≥ 0, \\ (N_{S}(γ) − I_{Y} (γ))(N_{S}(γ′) − I_{Y} (γ′)) & ≥ 0, \\ (N_{S}(γ) + I_{Y} (γ))(N_{S}(γ′) + I_{Y} (γ′)) & ≥ 0. \end{aligned}

Proof. We shall prove that (N_{T} (γ) − I_{X}(γ))(N_{T} (γ′) − I_{X}(γ′)) ≥ 0. The other three inequalities will follow by a similar argument.

From Lemma 2.4, we have N_{T} (γ)N_{T} (γ′) ≥ 0, then

(N_{T} (γ) − I_{X}(γ))(N_{T} (γ′) − I_{X}(γ′)) ≥ 0 when N_{T} (γ) ≤ 0.

Now, consider the case where N_{T} (γ) ≥ 0, and suppose that

(N_{T} (γ) − I_{X}(γ))(N_{T} (γ′) − I_{X}(γ′)) < 0.

Without loss of generality, we assume that

N_{T} (γ) > I_{X}(γ) and 0 ≤ N_{T} (γ′) < I_{X}(γ′).

There is a strand l_{1} of γ joining the X -side to the T^{−1} -side, and there is a strand l_{2} of γ joining the X^{−1} -side to the T^{−1} -side.

Let m = I_{X}(γ′) > 0. There exist m strands L_{1}, …, L_{m} of γ′ with endpoints on the X^{−1} -side.

If every L_{j} connects the X^{−1} -side to the T^{−1} -side, then N_{T} (γ′) ≥ m = I_{X}(γ′). This is a contradiction to the assumption. Therefore, there is an integer j such that L_{j} connects the X^{−1} -side to the E -side with E ≠ T^{−1} . This implies L_{j} ∩ (l_{1} ∪ l_{2}) ≠ ∅. This is impossible since γ and γ′ are disjoint.

Lemma 3.7. Let γ, γ′ ∈ 𝒢 be two disjoint geodesics, and let α = aγ ⊕ bγ′, where a ≥ 0 and b ≥ 0 are integers with a + b > 0. Then

{I_{X}(γ) − I_{Y} (γ)} · {I_{X}(γ′) − I_{Y} (γ′)} ≥ 0,

and thus

|I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α) = a{|I_{X}(γ) − I_{Y} (γ)| + I_{X}(γ) − I_{Y} (γ)}
+ b{|I_{X}(γ′) − I_{Y} (γ′)| + I_{X}(γ′) − I_{Y} (γ′)};
|I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α) = a{|I_{Y} (γ) − I_{X}(γ)| + I_{Y} (γ) − I_{X}(γ)}
+ b{|I_{Y} (γ′) − I_{X}(γ′)| + I_{Y} (γ′) − I_{X}(γ′)}.
Proof. If γ ∈ {γ_{T} , γ_{S}} or γ′ ∈ {γ_{T} , γ_{S}}, then
{I_{X}(γ) − I_{Y} (γ)} · {I_{X}(γ′) − I_{Y} (γ′)} = 0.

In the following, we assume that γ, γ′ ∈ 𝒢̂ .

Now, choose an integer k > 0 such that

\begin{aligned}N_{T} (𝒯_{2}^{−k}𝒯_{1}^{k}(γ)) & ≥ 2I_{X}(γ) = 2I_{X}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ)), \\ N_{S}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ)) & ≥ 2I_{Y} (γ) = 2I_{Y} (𝒯_{2}^{−k}𝒯_{1}^{k}(γ)), \\ N_{T} (𝒯_{2}^{−k}𝒯_{1}^{k}(γ′)) & ≥ 2I_{X}(γ′) = 2I_{X}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ′)), \\ N_{S}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ′)) & ≥ 2I_{Y} (γ′) = 2I_{Y} (𝒯_{2}^{−k}𝒯_{1}^{k}(γ′)). \end{aligned}

Since for E = X or Y

\begin{aligned}I_{E}(𝒯_{2}^{−k}𝒯_{1}^{k}(α)) & = aI_{E}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ)) + bI_{E}(𝒯_{2}^{−k}𝒯_{1}^{k}(γ′)) \\ & = aI_{E}(γ) + bI_{E}(γ′) = I_{E}(α), \end{aligned}

we may assume that

N_{T} (γ) ≥ 2I_{X}(γ), N_{S}(γ) ≥ 2I_{Y} (γ), N_{T} (γ′) ≥ 2I_{X}(γ′), N_{S}(γ′) ≥ 2I_{Y} (γ′).

Let 𝒢 ℒ _{1} and 𝒢 ℒ _{2} be the subsets of 𝒢 ℒ given in the proof of Theorem 3.1. If γ and γ′ both are in 𝒢 ℒ _{1} , write

γ = pγ_{S} + qγ_{T} + rτ_{1} + sτ_{2} and γ′ = p′γ_{S} + q′γ_{T} + r′τ_{1} + s′τ_{2}.

Then

{I_{X}(γ) − I_{Y} (γ)} · {I_{X}(γ′) − I_{Y} (γ′)} = ss′ ≥ 0.

Similarly,

{I_{X}(γ) − I_{Y} (γ)} · {I_{X}(γ′) − I_{Y} (γ′)} ≥ 0

if γ and γ′ both are in 𝒢 ℒ _{2} .

Finally, assume that γ ∈ 𝒢 ℒ _{1} and γ′ ∈ 𝒢 ℒ _{2} , and write

γ = pγ_{S} + qγ_{T} + rτ_{1} + sτ_{2} and γ′ = p′γ_{S} + q′γ_{T} + r′τ_{1} + s′τ_{3}.

If ss′ > 0, then i(γ, γ′) > 0. This is impossible. Thus ss′ = 0. This implies that both γ and γ′ are either in 𝒢 ℒ _{1} or in 𝒢 ℒ _{2} , and completes the proof.

Proof of Proposition 3.3. It follows from equation (3) and Proposition 2.2, we have

i(𝒯_{j}^{k}(α), γ_{11}) = i(α, γ_{11}), i(𝒯_{j}^{k}(α), γ_{21}) = i(α, γ_{21}) for j = 1, 2.

Since γ_{1j} is invariant under 𝒯_{1} , and since γ_{2j} is invariant under 𝒯_{2} for j = 2, 3, then

\begin{aligned}i(𝒯_{1}^{k}(α), γ_{1j}) & = i(α, 𝒯_{1}^{−k}(γ_{1j})) = i(α, γ_{1j}), and \\ i(𝒯_{2}^{k}(α), γ_{2j}) & = i(α, 𝒯_{2}^{−k}(γ_{2j})) = i(α, γ_{2j}). \end{aligned}

It remains to prove the last four equations given in the proposition. In the following, a and b are assumed to be non-negative integers with a + b > 0.

If α = aγ_{S} ⊕ bγ_{T} , then α is invariant under 𝒯_{j} for j = 1, 2, and I_{E}(α) = 0 for E = X, Y . Thus the equations hold trivially.

Let γ ∈ 𝒢̂ be a geodesic disjoint from γ_{S}. If α = aγ ⊕ bγ_{S}, then

I_{Y} (γ) = 0 = I_{Y} (α), N_{S}(γ) = 0 and N_{S}(α) = b.

Since I_{Y} (γ) = 0, then γ is invariant under 𝒯_{1} , and so is α. From Corollary 3.2 and Lemma 3.7, we have

|I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α) = 2aI_{X}(γ)

and

i(𝒯_{1}^{k}(α), γ_{22}) = ai(γ, γ_{22}) + bi(γ_{S}, γ_{22}) = 2aI_{X}(γ) + 2b
= 2|N_{S}(α) + kI_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α);
i(𝒯_{1}^{k}(α), γ_{23}) = ai(γ, γ_{23}) + bi(γ_{S}, γ_{23}) = 2aI_{X}(γ) + 2b
= 2|N_{S}(α) + (k − 1)I_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α).
Since γ_{S} is invariant under 𝒯_{2} , and i(γ_{S}, γ_{1j}) = 0 for j = 1, 2, then
i(𝒯_{2}^{k}(α), γ_{1j}) = ai(𝒯_{2}^{k}(γ), γ_{1j}).

Since I_{X}(α) = aI_{X}(γ), and since N_{T} (α) = aN_{T} (γ), from Corollary 3.2 and Lemma 3.5 we have

\begin{aligned}i(𝒯_{2}^{k}(α), γ_{12}) & = 2|N_{T} (α) − kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α); \\ i(𝒯_{2}^{k}(α), γ_{13}) & = 2|N_{T} (α) − (k + 1)I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α). \end{aligned}

By a similar argument as above, one proves that the last four equations hold for α = aγ ⊕ bγ_{T} , where γ ∈ 𝒢̂ is a geodesic disjoint from γ_{T} .

Finally, we consider the free homotopy classes α = aγ ⊕ bγ′, where γ and γ′ are disjoint geodesics in 𝒢̂ . If ab = 0, the equations hold trivially by Corollary 3.2.

Assume that a > 0 and b > 0. Then I_{X}(α)I_{Y} (α) > 0. Otherwise, say I_{Y} (α) = 0, we have I_{Y} (γ) = I_{Y} (γ′) = 0. This is impossible since any two distinct simple closed geodesics on a four-punctured sphere must meet (see [4, Theorem 2.5] and [4, Theorem 2.6]).

Note that the last three equations given in the proposition follow from the equation

i(𝒯_{2}^{k}(α), γ_{12}) = 2|N_{T} (α) − kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

Since

i(𝒯_{2}^{k}(α), γ_{13}) = i(𝒯_{2}^{k}(α), 𝒯_{2}^{−1}(γ_{12})) = i(𝒯_{2}^{k+1}(α), γ_{12}),

then

i(𝒯_{2}^{k}(α), γ_{13}) = 2|N_{T} (α) − (k + 1)I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

Because 𝒯_{1}^{k} = Θ_{2}𝒯_{2}^{k}Θ_{2} , from Propositions 2.1 and 2.4 we obtain

i(𝒯_{1}^{k}(α), γ_{22}) = i(Θ_{2}𝒯_{2}^{k}Θ_{2}(α), γ_{22}) = i(𝒯_{2}^{k}Θ_{2}(α), Θ_{2}(γ_{22})) = i(𝒯_{2}^{k}Θ_{2}(α), γ_{12})
= 2|N_{T} (Θ_{2}(α)) − kI_{X}(Θ_{2}(α))| + |I_{Y} (Θ_{2}(α)) − I_{X}(Θ_{2}(α))|
+ I_{Y} (Θ_{2}(α)) − I_{X}(Θ_{2}(α))
= 2| − N_{S}(α) − kI_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α)
= 2|N_{S}(α) + kI_{Y} (α)| + |I_{X}(α) − I_{Y} α)| + I_{X}(α) − I_{Y} (α)

and

\begin{aligned}i(𝒯_{1}^{k}(α), γ_{23}) & = i(𝒯_{1}^{k}(α), 𝒯_{1}(γ_{22})) = i(𝒯_{1}^{k−1}(α), γ_{22}) \\ & = 2|N_{S}(α) + (k − 1)I_{Y} (α)| + |I_{X}(α) − I_{Y} (α)| + I_{X}(α) − I_{Y} (α). \end{aligned}

Now, we shall prove the equation

i(𝒯_{2}^{k}(α), γ_{12}) = 2|N_{T} (α) − kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

From Proposition 2.8, Lemma 3.5 and Lemma 3.7, we obtain

i(𝒯_{2}(α), γ_{12}) = ai(𝒯_{2}(γ), γ_{12}) + bi(𝒯_{2}(γ′), γ_{12})
= 2a|N_{T} (𝒯_{2}(γ))| + 2b|N_{T} (𝒯_{2}(γ′))|
+ a{|I_{Y} (𝒯_{2}(γ)) − I_{X}(𝒯_{2}(γ))| + I_{Y} (𝒯_{2}(γ)) − I_{X}(𝒯_{2}(γ))}
+ b{|I_{Y} (𝒯_{2}(γ′)) − I_{X}(𝒯_{2}(γ′))| + I_{Y} (𝒯_{2}(γ′)) − I_{X}(𝒯_{2}(γ′))}
= 2a|N_{T} (γ) − I_{X}(γ)| + 2b|N_{T} (γ′) − I_{X}(γ′)|
+ a{|I_{Y} (γ) − I_{X}(γ)| + I_{Y} (γ) − I_{X}(γ)}
+ b{|I_{Y} (γ′) − I_{X}(γ′)| + I_{Y} (γ′) − I_{X}(γ′)}
= 2|a{N_{T} (γ) − I_{X}(γ)} + b{N_{T} (γ′) − I_{X}(γ′)}|
+ |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α)
= 2|N_{T} (α) − I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

If k > 1, by Lemma 3.5 we have

i(𝒯_{2}^{k}(α), γ_{12}) = 2|N_{T} (𝒯_{2}^{k−1}(α)) − I_{X}(𝒯_{2}^{k−1}(α))|
+ |I_{Y} (𝒯_{2}^{k−1}(α)) − I_{X}(𝒯_{2}^{k−1}(α))|
+ I_{Y} (𝒯_{2}^{k−1}(α)) − I_{X}(𝒯_{2}^{k−1}(α))
= 2|N_{T} (α) − kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

By the same reasoning as above, one shows

i(𝒯_{2}^{−1}(α), γ_{12}) = 2|N_{T} (α) + I_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).

Thus for k > 1

i(𝒯_{2}^{−k}(α), γ_{12}) = 2|N_{T} (𝒯_{2}^{−k+1}(α)) − I_{X}(𝒯_{2}^{−k+1}(α))|
+ |I_{Y} (𝒯_{2}^{−k+1}(α)) + I_{X}(𝒯_{2}^{−k+1}(α))|
+ I_{Y} (𝒯_{2}^{−k+1}(α)) − I_{X}(𝒯_{2}^{−k+1}(α))
= 2|N_{T} (α) + kI_{X}(α)| + |I_{Y} (α) − I_{X}(α)| + I_{Y} (α) − I_{X}(α).
4. A homeomorphism of \overline{πℐ (𝒢)} onto a 3-sphere

Now, we are ready to construct a homeomorphism of \overline{πℐ (𝒢)} onto a 3-sphere.

Let Π = {(r_{1}, r_{2}, …, r_{6}) ∈ R6_{+} : r_{1} + r_{2} + ⋯ + r_{6} = 1}, and let 𝒞 = Π_{1} ∪ Π_{2} ∪ Π_{3} ,

where

\begin{aligned}Π_{1} & = \{(r_{1}, r_{2}, r_{3}) ∈ R3_{+} : r_{2} + r_{3} = r_{1}\}, \\ Π_{2} & = \{(r_{1}, r_{2}, r_{3}) ∈ R3_{+} : r_{1} + r_{3} = r_{2}\}, \\ Π_{3} & = \{(r_{1}, r_{2}, r_{3}) ∈ R3_{+} : r_{1} + r_{2} = r_{3}\}. \end{aligned}

Following Poénaru ([5], Exposé 4), we shall first construct a function Ψ of ℐ (𝒢 ℒ ) into (𝒞 × 𝒞) ∩ Π so that its extension to π^{−1}πℐ (𝒢 ℒ ) satisfies

Ψ(t I_{α}) = Ψ(I_{α}) for α ∈ 𝒢 ℒ and for t > 0.

Thus Ψ induces a function on πℐ (𝒢 ℒ ), also denoted by Ψ.

By using a continuity argument, we extend Ψ to \overline{πℐ (𝒢)} , and prove that Ψ is a homeomorphism of \overline{πℐ (𝒢)} onto a 3-sphere lying in R^{6} (Theorem 4.3). Finally, by postcomposing Ψ by a function from R^{6} into R^{4} , we will get a homeomorphism of \overline{πℐ (𝒢)} into a 3-sphere lying in R^{4} (Theorem 4.4).

4.1. The definition of Ψ on 𝒢 ℒ . For integers i ∈ {1, 2} and j ∈ {1, 2, 3}, and for α ∈ 𝒢 ℒ , let

x_{ij}(α) = \frac{i(α, γ_{ij})}{λ(α)}, where λ(α) = ∑_{i=1}^{2} ∑_{j=1}^{3} i(α, γ_{ij}),

and let ψ_{1}: 𝒢 ℒ ⟶ R6_{+} be defined by

ψ_{1}(α) = (x_{11}(α), x_{12}(α), x_{13}(α), x_{21}(α), x_{22}(α), x_{23}(α)).

Note that the image of ψ_{1} lies in Π since ∑_{i=1}^{2} ∑_{j=1}^{3} x_{ij}(α) = 1 for all α ∈ 𝒢 ℒ . To construct a function of 𝒢 ℒ into (𝒞 × 𝒞) ∩ Π, we form the sum

ρ(α) = 2{I_{X}(α)+I_{Y} (α)+|N_{T} (α)|+|N_{T} (α)−I_{X}(α)|+|N_{S}(α)|+|N_{S}(α)−I_{Y} (α)|}.

From Corollary 3.4, we have 0 < ρ(α) ≤ λ(α) for all α ∈ 𝒢 ℒ , and

\frac{ρ(α)}{λ(α)} = 1 − \frac{4|I_{X}(α) − I_{Y} (α)|}{λ(α)} = 1 − 2|x_{11}(α) − X_{21}(α)|.

Thus |x_{11}(α) − x_{21}(α)| < \frac{1}{2} for all α ∈ 𝒢 ℒ , and the image of ψ_{1} is contained in the set ℰ = \{(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ Π : |r_{1} − r_{4}| < \frac{1}{2}\}.

Let

\begin{aligned}ℰ^{+} & = \{(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ Π : 0 ≤ r_{1} − r_{4} < \frac{1}{2}\} and \\ ℰ^{−} & = \{(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ Π : 0 ≤ r_{4} − r_{1} < \frac{1}{2}\}. \end{aligned}

Let ψ_{2}: ℰ ⟶ R^{6} be defined by ψ_{2}(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}), where

t_{j} = \{ \begin{matrix}\frac{r_{j}}{1 − 2(r_{1} − r_{4})} & for j = 1, 2, 3, 4 and (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ ℰ^{+}, \\ \frac{r_{j} − r_{1} + r_{4}}{1 − 2(r_{1} − r_{4})} & for j = 5, 6 and (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ ℰ^{+}, \\ \frac{r_{j}}{1 − 2(r_{4} − r_{1})} & for j = 1, 4, 5, 6 and (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ ℰ^{−}, \\ \frac{r_{j} + r_{1} − r_{4}}{1 − 2(r_{4} − r_{1})} & for j = 2, 3 and (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ ℰ^{−}. \end{matrix}

It is clear that ψ_{2} is continuous on ℰ with

\begin{aligned}ψ_{2}(ℰ^{+}) ⊂ Π^{+} & = \{(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π : t_{1} ≥ t_{4}\} and \\ ψ_{2}(ℰ^{−}) ⊂ Π^{−} & = \{(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π : t_{1} ≤ t_{4}\}. \end{aligned}

A direct computation proves that ψ_{2} is an injective function onto Π with the inverse ψ_{2}^{−1}(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ ℰ , where

r_{j} = \{ \begin{matrix}\frac{t_{j}}{1 + 2(t_{1} − t_{4})} & for j = 1, 2, 3, 4, and (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π^{+}, \\ \frac{t_{j} + t_{1} − t_{4}}{1 + 2(t_{1} − t_{4})} & for j = 5, 6, and (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π^{+}, \\ \frac{t_{j}}{1 + 2(t_{4} − t_{1})} & for j = 1, 4, 5, 6, and (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π^{−}, \\ \frac{t_{j} − t_{1} + t_{4}}{1 + 2(t_{4} − t_{1})} & for j = 2, 3, and (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) ∈ Π^{−}. \end{matrix}

This proves that ψ_{2} is a homeomorphism of ℰ onto Π.

Let Ψ be the composition of ψ_{1} followed by ψ_{2} . We shall prove that Ψ maps 𝒢 ℒ into Δ = (𝒞 × 𝒞) ∩ Π. For α ∈ 𝒢 ℒ , write

\begin{matrix}(ξ_{11}(α), ξ_{12}(α), ξ_{13}(α), ξ_{21}(α), ξ_{22}(α), ξ_{23}(α)) \\ = ψ_{2}(x_{11}(α), x_{12}(α), x_{13}(α), x_{21}(α), x_{22}(α), x_{23}(α)). \end{matrix}

From the definition of ρ(α), we have

\begin{aligned}ξ_{11}(α) & = \frac{2I_{X}(α)}{ρ(α)},  ξ_{12}(α) = \frac{2|N_{T} (α)|}{ρ(α)},  ξ_{13}(α) = \frac{2|N_{T} (α) − I_{X}(α)|}{ρ(α)}, \\ ξ_{21}(α) & = \frac{2I_{Y} (α)}{ρ(α)},  ξ_{22}(α) = \frac{2|N_{S}(α)|}{ρ(α)},  ξ_{23}(α) = \frac{2|N_{S}(α) − I_{Y} (α)|}{ρ(α)}. \end{aligned}

For simplicity, write N_{T} = N_{T} (α), N_{S} = N_{S}(α), I_{X} = I_{X}(α), I_{Y} = I_{Y} (α), and ξ_{ij} = ξ_{ij}(α) for all α ∈ 𝒢 ℒ . Then

\begin{aligned}N_{T} ≤ 0 & ⟹ ξ_{11} + ξ_{12} = ξ_{13}, \\ 0 ≤ N_{T} ≤ I_{X} & ⟹ ξ_{11} − ξ_{12} = ξ_{13}, \\ N_{T} ≥ I_{X} & ⟹ −ξ_{11} + ξ_{12} = ξ_{13}, \end{aligned} \begin{aligned}N_{S} ≤ 0 & ⟹ ξ_{21} + ξ_{22} = ξ_{23}, \\ 0 ≤ N_{S} ≤ I_{Y} & ⟹ ξ_{21} − ξ_{22} = ξ_{23}, \\ N_{S} ≥ I_{Y} & ⟹ −ξ_{21} + ξ_{22} = ξ_{23}. \end{aligned}

Therefore, Ψ(𝒢 ℒ ) ⊂ Δ.

4.2. A homeomorphism of Δ onto a 3-sphere. In this subsection, we shall prove that Δ = (𝒞 × 𝒞) ∩ Π is homeomorphic to a 3-sphere.

Let A be the invertible linear transformation of R^{3} onto itself carrying the vectors (1, 0, 1), (1, 1, 0) and (0, 1, 1) to the vectors (1, 0, 1), (−\frac{1}{2}, \frac{1}{2} \sqrt{3}, 1) and (−\frac{1}{2}, −\frac{1}{2} \sqrt{3}, 1) in this order. The matrix representation of A is

A = ( \begin{matrix}\frac{1}{2} & −1 & \frac{1}{2} \\ \sqrt[\frac{1}{2}]{3} & 0 & −\frac{1}{2} \sqrt{3} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix} ) with the inverse A^{−1} = ( \begin{matrix}\frac{1}{3} & \sqrt{3}^{−1} & \frac{2}{3} \\ \frac{−2}{3} & 0 & \frac{2}{3} \\ \frac{1}{3} & −\sqrt{3}^{−1} & \frac{2}{3} \end{matrix} ) .

Let 𝒞′ = A(𝒞). Note that if (x_{1}, x_{2}, x_{3}) = A(r_{1}, r_{2}, r_{3}) ∈ 𝒞′, then x_{3} ≥ 0. Let

\begin{aligned}L_{1} & = \{(t, 0, t) ∈ R^{3} : t ≥ 0\}, \\ L_{2} & = \{(−\frac{1}{2}t, \frac{1}{2} \sqrt{3} t, t) ∈ R^{3} : t ≥ 0\} and \\ L_{3} & = \{(−\frac{1}{2}t, −\frac{1}{2} \sqrt{3} t, t) ∈ R^{3} : t ≥ 0\}. \end{aligned}

By a direct computation, one proves easily that Π′_{1} = A(Π_{1}) lies on the plane x_{1}+\sqrt{3} x_{2} = x_{3} bounded by L_{1} and L_{2} , Π′_{2} = A(Π_{2}) lies on the plane 2x_{1}+x_{3} = 0 bounded by L_{2} and L_{3} , and Π′_{3} = A(Π_{3}) lies on the plane \sqrt{3} x_{2} + x_{3} = x_{1} bounded by L_{1} and L_{3} . By the definition, 𝒞′ = Π′_{1} ∪ Π′_{2} ∪ Π′_{3} . Let J be the linear transformation of R^{6} onto itself represented by the following matrix

( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) .

Then J is a homeomorphism of R^{6} onto itself with

Π′ = J(Π) = \{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) ∈ R^{6} : x_{3} + x_{6} = \frac{1}{2}\},

and J(Δ) = (𝒞′ × 𝒞′) ∩ Π′ = Δ′.

It is clear that the orthogonal projection η: R^{3} ⟶ R^{2} defined by

η(x_{1}, x_{2}, x_{3}) = (x_{1}, x_{2})

restricted to 𝒞′ is a homeomorphism onto R^{2} . Then the projection ϕ: R^{6} ⟶ R^{4} defined by

ϕ(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) = (η(x_{1}, x_{2}, x_{3}), η(x_{4}, x_{5}, x_{6}))

restricted to 𝒞′ × 𝒞′ is a homeomorphism onto R^{2} × R^{2} ≅ R^{4} . Let

B = (𝒞′ × 𝒞′) ∩ \{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) ∈ R^{6} : x_{3} + x_{6} ≤ \frac{1}{2}\}.

Now, we shall prove that ϕ(B) is bounded and convex, and has non-empty interior. This implies that ϕ(B) is homeomorphic to the closed unit ball

{(x_{1}, x_{2}, x_{3}, x_{4}) ∈ R^{4} : x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} ≤ 1}.

By the definition of B, as a subspace of 𝒞′ × 𝒞′, the boundary of B is Δ′, then ϕ(Δ′) is homeomorphic to a 3-sphere, and so is Δ.

Let R be the rotation in R^{3} with the matrix representation

( \begin{matrix}cos \frac{2}{3}π & − sin \frac{2}{3}π & 0 \\ sin \frac{2}{3}π & cos \frac{2}{3}π & 0 \\ 0 & 0 & 1 \end{matrix} ) = ( \begin{matrix}−\frac{1}{2} & −\frac{1}{2} \sqrt{3} & 0 \\ \sqrt[\frac{1}{2}]{3} & −\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Then

Π′_{j} × Π′_{k} = R^{j−1}(Π′_{1}) × R^{k−1}(Π′_{1}) = (R^{j−1} × R^{k−1})(Π′_{1} × Π′_{1})

for j, k ∈ {1, 2, 3}, where R^{j−1} × R^{k−1} is the linear transformation of R^{6} onto itself represented by the following matrix

( \begin{matrix}R^{j−1} & 0 \\ 0 & R^{k−1} \end{matrix} ) .

It easy to see that

(R^{j−1} × R^{k−1})(0, 0, r, 0, 0, s) = (0, 0, r, 0, 0, s)

for any two real numbers r and s. Since the normal vector (0, 0, 1, 0, 0, 1) of Π′ is invariant under R^{j−1} × R^{k−1} , and since the point (0, 0, \frac{1}{4}, 0, 0, \frac{1}{4}) of Π′ is fixed by R^{j−1} × R^{k−1} , then Π′ is invariant under R^{j−1} × R^{k−1} , and thus

ϕ(B) = ⋃_{j=1}^{3} ⋃_{k=1}^{3} ϕ((R^{j−1} × R^{k−1})(V)),

where

\begin{aligned}V & = \{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) ∈ Π′_{1} × Π′_{1} : x_{3} + x_{6} ≤ \frac{1}{2}\} \\ & = \{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) ∈ Π′_{1} × Π′_{1} : x_{1} + \sqrt{3} x_{2} + x_{4} + \sqrt{3} x_{5} ≤ \frac{1}{2}\}. \end{aligned}

Clearly, V is bounded. This proves that ϕ(B) is bounded since R^{j−1} × R^{k−1} is a Euclidean isometry.

To prove the convexity of ϕ(B), we consider any two distinct points Q and Q′ of B with coordinates (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) and (x′_{1}, x′_{2}, x′_{3}, x′_{4}, x′_{5}, x′_{6}) respectively. Let

P_{1} = (x_{1}, x_{2}, x_{3}), P_{2} = (x_{4}, x_{5}, x_{6}), P′_{1} = (x′_{1}, x′_{2}, x′_{3}) and P′_{2} = (x′_{4}, x′_{5}, x′_{6}),

and let \overline{P_{j}P′_{j}} denote the line segment connecting P_{j} to P′_{j} for j = 1, 2. The vertical plane in R^{3} containing \overline{P_{j}P′_{j}} intersects 𝒞′ in a polygonal curve σ_{j} with parametric equation f_{j}(t), 0 ≤ t ≤ 1, so that f_{j}(0) = P_{j} and f_{j}(1) = P′_{j}. Note that η(σ_{j}) = η(\overline{P_{j}P′_{j}}). The curve

L = {(f_{1}(t), f_{2}(t)) ∈ R^{3} × R^{3} : 0 ≤ t ≤ 1}

lies on 𝒞′ × 𝒞′ connecting Q to Q′, and ϕ(L) is a line segment in ϕ(B) with ϕ(Q) and ϕ(Q′) as its endpoints. Therefore, ϕ(B) is convex.

Note that (Π′_{1} × Π′_{1}) ∩ Π′ is contained in the hyperplane in R^{6} of equation

x_{1} + \sqrt{3} x_{2} + x_{4} + \sqrt{3} x_{5} = \frac{1}{2},

then the distance from the origin to (Π′_{1} ×Π′_{1})∩Π′ is at least 1/4\sqrt{2}. This implies that ϕ(B) contains the closed ball centered at the origin with radius 1/4\sqrt{2}, and ϕ(B) has non-empty interior. The proof is complete.

4.3. The extension of Ψ to \overline{πℐ (𝒢)} . Now, we are going to extend the map Ψ to \overline{πℐ (𝒢)} = \overline{πℐ (𝒢 ℒ )} .

For every α ∈ 𝒢 ℒ , we define x_{ij}(I_{α}) = x_{ij}(α). Since each x_{ij} is homogeneous, then x_{ij} extends naturally to π^{−1}πℐ (𝒢 ℒ ) defined by x_{ij}(t I_{α}) = x_{ij}(I_{α}) for all t > 0 and for all α ∈ 𝒢 ℒ . Thus each x_{ij} induces a well-defined map, also denoted by x_{ij}, on πℐ (𝒢 ℒ ) defined by x_{ij}(π(I_{α})) = x_{ij}(I_{α}).

For an arbitrary ℒ ∈ π^{−1}\overline{πℐ (𝒢)} , there is a sequence \{t_{n}\}_{n=1}^{∞} of positive numbers, and there is a sequence {γ_{n}}_{n=1}^{∞} in 𝒢 such that {t_{n} I_{γ_{n}}}_{n=1}^{∞} converges to ℒ . Thus

t_{n}i(γ_{n}, γ_{ij}) = t_{n} I_{γ_{n}}(γ_{ij}) → ℒ (γ_{ij})

as n → ∞ for i = 1, 2 and for j = 1, 2, 3. This implies

lim_{n→∞} x_{ij}(t_{n} I_{γ_{n}}) = \frac{ℒ (γ_{ij})}{∑_{k=1}^{2} ∑_{l=1}^{3} ℒ (γ_{kl})}

for i = 1, 2 and for j = 1, 2, 3. Let λ: π^{−1}\overline{πℐ (𝒢)} ⟶ R_{+} be defined by

λ(ℒ) = ∑_{k=1}^{2} ∑_{l=1}^{3} ℒ (γ_{kl}) for all ℒ ∈ π^{−1}\overline{πℐ (𝒢)} ,

and let x_{ij}: π^{−1}\overline{πℐ (𝒢)} ⟶ R_{+} be defined by

x_{ij}(ℒ) = \frac{ℒ (γ_{ij})}{∑_{k=1}^{2} ∑_{l=1}^{3} ℒ (γ_{kl})} for all ℒ ∈ π^{−1}\overline{πℐ (𝒢)} .

It is easy to see that each x_{ij} is continuous on π^{−1}\overline{πℐ (𝒢)} with x_{ij}(tℒ) = x_{ij}(ℒ) for all t > 0 and for all ℒ ∈ π^{−1}\overline{πℐ (𝒢)} .

Since the restriction of π to π^{−1}\overline{πℐ 𝒢)} is a quotient map onto \overline{πℐ (𝒢)} , then each x_{ij} extends to \overline{πℐ (𝒢)} a continuos map given by x_{ij}(π(ℒ)) = x_{ij}(ℒ) for ℒ in π^{−1}\overline{πℐ (𝒢)} . This gives a continuous map of \overline{πℐ (𝒢)} into R6_{+} whose restriction to 𝒢 ℒ is ψ_{1} . We also use ψ_{1} for this continuous map on \overline{πℐ (𝒢)} , and let Ψ = ψ_{2}ψ_{1} as before.

Proposition 4.1. The function Ψ maps \overline{πℐ (𝒢)} continuously onto Δ.

Clearly, Ψ is a continuous map of \overline{πℐ (𝒢)} into Π. Since Ψ(𝒢) ⊂ Δ, and since Δ is closed in R^{6} , then Ψ(\overline{πℐ (𝒢)} ) ⊂ Δ.

To complete the proof of Proposition 4.1, we have to show that Ψ(πℐ (𝒢 ℒ )) is dense in Δ since Ψ is continuous and \overline{πℐ (𝒢)} = \overline{πℐ (𝒢 ℒ )} is compact.

A point (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) of Q^{6} will be called a rational point, where Q is the set of all rational numbers.

Lemma 4.2. Every rational point of Π ∩ (Π_{2} × Π_{2}) lies in Ψ(πℐ (𝒢 ℒ )).

Proof. Let (v_{1}/u, v_{2}/u, v_{3}/u, v_{4}/u, v_{5}/u, v_{6}/u) be any rational point of (Π_{2} × Π_{2}) ∩ Π, where u > 0 and all v_{j} ≥ 0 are even integers. Note that

2(v_{2} + v_{5}) = u, v_{1} + v_{3} = v_{2} and v_{4} + v_{6} = v_{5}.

We want to show that there are non-negative integers a, b, c and d with a + b + c + d > 0 such that

(\frac{v_{1}}{u} , \frac{v_{2}}{u} , \frac{v_{3}}{u} , \frac{v_{4}}{u} , \frac{v_{5}}{u} , \frac{v_{6}}{u} ) = \{ \begin{matrix}Ψ(𝒯_{1}^{−1}𝒯_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2})) & if v_{1} ≥ v_{4}, \\ Ψ(𝒯_{1}^{−1}𝒯_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3})) & if v_{1} ≤ v_{4}, \end{matrix}

where τ_{1} , τ_{2} and τ_{3} are the geodesics given in the proof of Theorem 3.1.

Let α = 𝒯_{1}^{−1}𝒯_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2}). From Proposition 3.3 and Corollary 3.4, we have

\begin{aligned}I_{X}(α) & = c + d, \\ N_{T} (α) & = a + c + d, \\ I_{Y} (α) & = c, \\ N_{S}(α) & = b + c, and \\ ρ(α) & = 2(2a + 2b + 4c + 2d). \end{aligned}

If v_{1} ≥ v_{4} , by solving the following equations for a, b, c and d

\begin{aligned}2(c + d) & = 2I_{X}(α) = v_{1}, \\ 2(a + c + d) & = 2N_{T} (α) = v_{2}, \\ 2c & = 2I_{Y} (α) = v_{4}, \\ 2(b + c) & = 2N_{S}(α) = v_{5}, \end{aligned}

we have

a = \frac{1}{2}(v_{2} − v_{1}), b = \frac{1}{2}(v_{5} − v_{4}), c = \frac{1}{2}v_{4} and d = \frac{1}{2}(v_{1} − v_{4}).

A direct computation gives ρ(α) = 2(v_{2} + v_{5}) = u,

2|N_{T} (α) − I_{X}(α)| = v_{2} − v_{1} = v_{3} and 2|N_{S}(α) − I_{Y} (α)| = v_{5} − v_{4} = v_{6}.

This proves

Ψ(α) = (\frac{v_{1}}{u} , \frac{v_{2}}{u} , \frac{v_{3}}{u} , \frac{v_{4}}{u} , \frac{v_{5}}{u} , \frac{v_{6}}{u} ).

Next, assume that v_{1} ≤ v_{4} . Let α be given as above such that

Ψ(α) = (\frac{v_{4}}{u} , \frac{v_{5}}{u} , \frac{v_{6}}{u} , \frac{v_{1}}{u} , \frac{v_{2}}{u} , \frac{v_{3}}{u} ).

Geometric intersection numbers on a five-punctured sphere 109

Since 𝒯_{2}Θ_{2} = Θ_{2}𝒯_{1} , Θ_{1}𝒯_{1}^{−1} = 𝒯_{1}Θ_{1} and Θ_{1}𝒯_{2}^{−1} = 𝒯_{2}Θ_{1} , then

𝒯_{1}^{−1}𝒯_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3}) = 𝒯_{1}^{−1}𝒯_{2}Θ_{1}Θ_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2})
= Θ_{1}Θ_{2}𝒯_{2}𝒯_{1}^{−1}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{2})
= Θ_{1}Θ_{2}𝒯_{1}^{−1}𝒯_{2}(aγ_{T} + bγ_{S} + cτ_{1} + dτ_{3})
= Θ_{1}Θ_{2}(α).

Let β = Θ_{1}Θ_{2}(α). It follows immediately from Proposition 2.1 that

I_{X}(β) = I_{Y} (α), I_{Y} (β) = I_{X}α), N_{T} (β) = N_{S}(α) and N_{S}(β) = N_{T} (α) and

Ψ(β) = (ξ_{21}(α), ξ_{22}(α), ξ_{23}(α), ξ_{11}(α), ξ_{12}(α), ξ_{13}(α)) = (\frac{v_{1}}{u} , \frac{v_{2}}{u} , \frac{v_{3}}{u} , \frac{v_{4}}{u} , \frac{v_{5}}{u} , \frac{v_{6}}{u} ).

Proof of Proposition 4.1. We shall prove that Ψ(πℐ (𝒢 ℒ )) is dense in Δ by showing that every rational point of Δ is in Ψ(πℐ (𝒢 ℒ )), and this completes the proof.

Let ζ = (v_{1}/u, v_{2}/u, v_{3}/u, v_{4}/u, v_{5}/u, v_{6}/u) be an arbitrary rational point of Δ, where u > 0 and all v_{j} ≥ 0 are even integers. There are non-negative integers m and n such that

mv_{1} ≤ v_{2} < (m + 1)v_{1} and nv_{4} ≤ v_{5} < (n + 1)v_{4}.

Let

ζ_{1} = (\frac{v_{1}}{u} , \frac{v_{2}}{u} , \frac{v_{3}}{u} ) and ζ_{2} = (\frac{v_{4}}{u} , \frac{v_{5}}{u} , \frac{v_{6}}{u} ).

Set v′_{j} = v_{j} for j = 1, 4, set

v′_{2} = \{ \begin{matrix}v_{2} + (m + 1)v_{1} & if ζ_{1} ∈ Π_{1} ∪ Π_{2}, \\ −v_{2} + (m + 1)v_{1} & if ζ_{1} ∈ Π_{3}, \end{matrix}
v′_{3} = \{ \begin{matrix}v_{2} + mv_{1} & if ζ_{1} ∈ Π_{1} ∪ Π_{2}, \\ −v_{2} + mv_{1} & if ζ_{1} ∈ Π_{3}, \end{matrix}
v′_{5} = \{ \begin{matrix}v_{5} + (n + 1)v_{4} & if ζ_{2} ∈ Π_{1} ∪ Π_{2}, \\ −v_{5} + (n + 1)v_{4} & if ζ_{2} ∈ Π_{3}, \end{matrix}
v′_{6} = \{ \begin{matrix}v_{5} + nv_{4} & if ζ_{2} ∈ Π_{1} ∪ Π_{2}, \\ −v_{5} + nv_{4} & if ζ_{2} ∈ Π_{3}, \end{matrix}

and set w = ∑_{j=1}^{6} v′_{j}. Then w > 0 and all v′_{j} ≥ 0 are even integers, |v′_{2} − (m + 1)v_{1}| = v_{2}, |v′_{5} − (n + 1)v_{4}| = v_{5}, and

|v′_{2} − (m + 2)v_{1}| = \{ \begin{matrix}|v_{2} − v_{1}| = v_{3} & if ζ_{1} ∈ Π_{1} ∪ Π_{2}, \\ | − v_{2} − v_{1}| = v_{3} & if ζ_{1} ∈ Π_{3}, \end{matrix}
|v′_{5} − (n + 2)v_{4}| = \{ \begin{matrix}|v_{5} − v_{4}| = v_{6} & if ζ_{2} ∈ Π_{1} ∪ Π_{2}, \\ | − v_{5} − v_{4}| = v_{6} & if ζ_{2} ∈ Π_{3}. \end{matrix}

As v′_{2} = v′_{1} + v′_{3} and v′_{5} = v′_{4} + v′_{6} , the point (v′_{1}/w, v′_{2}/w, v′_{3}/w, v′_{4}/w, v′_{5}/w, v′_{6}/w) is a rational point in Π ∩ (Π_{2} × Π_{2}).

From the proof Lemma 4.2 we know that there is an α ∈ 𝒢 ℒ with N_{T} (α) ≥ I_{X}(α) and N_{S}(α) ≥ I_{Y} (α) such that

\begin{aligned}2I_{X}(α) & = v′_{1}, \\ 2N_{T} (α) & = v′_{2}, \\ 2\{N_{T} (α) − I_{X}(α)\} & = v′_{3}, \\ 2I_{Y} (α) & = v′_{4}, \\ 2N_{S}(α) & = v′_{5}, \\ 2\{N_{S}(α) − I_{Y} (α)\} & = v′_{6}. \end{aligned}

Let α′ = 𝒯_{2}^{m+1}𝒯_{1}^{−n−1}(α). From Lemma 3.5,

\begin{aligned}2I_{X}(α′) & = 2I_{X}(α) = v_{1}, \\ 2I_{Y} (α′) & = 2I_{Y} (α) = v_{4}, \\ 2|N_{T} (α′)| & = |2\{N_{T} (α) − (m + 1)I_{X}(α)\}| = |v′_{2} − (m + 1)v_{1}| = v_{2}, \\ 2|N_{T} (α′) − I_{X}(α′)| & = |2\{N_{T} (α) − (m + 2)I_{X}(α)\}| = |v′_{2} − (m + 2)v_{1}| = v_{3}, \\ 2|N_{S}(α′)| & = |2\{N_{S}(α) − (n + 1)I_{Y} (α)\}| = |v′_{5} − (n + 1)v_{4}| = v_{5}, \\ 2|N_{S}(α′) − I_{Y} (α′)| & = |2\{N_{S}(α) − (n + 2)I_{Y} (α)\}| = |v′_{5} − (n + 2)v_{4}| = v_{6}. \end{aligned}

Thus Ψ(α′) = ζ.

4.4. The injectivity of Ψ. So far, we have proved that Ψ maps \overline{πℐ (𝒢)} onto the 3-sphere Δ. Next, we shall prove that Ψ is injective on \overline{πℐ (𝒢)} . This proves the following theorem.

Theorem 4.3. The map Ψ is a homeomorphism of \overline{πℐ (𝒢)} onto Δ, and then \overline{πℐ (𝒢)} is homeomorphic to a 3-sphere.

Since ψ_{2} is a homeomorphism of ℰ onto Π, it remains to show that ψ_{1} is injective on \overline{πℐ (𝒢)} .

Let ℒ_{1}, ℒ_{2} ∈ π^{−1}\overline{πℐ (𝒢)} with ψ_{1}(π(ℒ_{1})) = ψ_{1}(π(ℒ_{2})). There exist sequences {t_{n}} and {s_{n}} of positive numbers, and there exist sequences {α_{n}} and {β_{n}} of elements in 𝒢 such that

lim_{n→∞} t_{n} I_{α_{n}} = ℒ_{1} and lim_{n→∞} s_{n} I_{β_{n}} = ℒ_{2}.

Geometric intersection numbers on a five-punctured sphere 111

Set p = λ(ℒ_{1}) /λ(ℒ_{2}) . By assumption, for i = 1, 2 and for j = 1, 2, 3, we have

ℒ_{1}(γ_{ij}) = pℒ_{2}(γ_{ij}), or, equivalently, lim_{n→∞} t_{n} I_{α_{n}}(γ_{ij}) = lim_{n→∞} ps_{n} I_{β_{n}}(γ_{ij}).

We shall complete the proof by showing that

lim_{n→∞} t_{n} I_{α_{n}}(γ) = lim_{n→∞} ps_{n} I_{β_{n}}(γ) for all γ ∈ 𝒢 .

Since

lim_{n→∞} t_{n}I_{X}(α_{n}) = lim_{n→∞} t_{n} I_{α_{n}}(γ_{11}) = lim_{n→∞} ps_{n} I_{β_{n}}(γ_{11}) = lim_{n→∞} ps_{n}I_{X}(β_{n}), and
lim_{n→∞} t_{n}I_{Y} (α_{n}) = lim_{n→∞} t_{n} I_{α_{n}}(γ_{21}) = lim_{n→∞} ps_{n} I_{β_{n}}(γ_{21}) = lim_{n→∞} ps_{n}I_{Y} (β_{n}),

then, by using the geometric intersection formula, we only have to show that

lim_{n→∞} t_{n}|I_{X}(α_{n})N_{T} (γ) − I_{X}(γ)N_{T} (α_{n})|= lim_{n→∞} ps_{n}|I_{X}(β_{n})N_{T} (γ) − I_{X}(γ)N_{T} (β_{n})|

and

lim_{n→∞} t_{n}|I_{Y} (α_{n})N_{S}(γ) − I_{Y} (γ)N_{S}(α_{n})|= lim_{n→∞} ps_{n}|I_{Y} (β_{n})N_{S}(γ) − I_{Y} (γ)N_{S}(β_{n})|.

To simplify notation, set A_{n} = t_{n}I_{X}(α_{n}), B_{n} = ps_{n}I_{X}(β_{n}), C_{n} = t_{n}N_{T} (α_{n}), D_{n} = ps_{n}N_{T} (β_{n}), I = I_{X}(γ) and N = N_{T} (γ). Thus

lim_{n→∞} A_{n} = lim_{n→∞} B_{n} and lim_{n→∞} |C_{n}| = lim_{n→∞} |D_{n}|.

It is clear that

lim_{n→∞} C_{n} = lim_{n→∞} D_{n} if lim_{n→∞} |C_{n}| = lim_{n→∞} |D_{n}| = 0.

If

lim_{n→∞} |C_{n}| = lim_{n→∞} |D_{n}| ≠ 0,

by the continuity of Ψ we may choose α_{n} and β_{n} so that C_{n}D_{n} > 0, and then we also have

lim_{n→∞} C_{n} = lim_{n→∞} D_{n}.

The inequality

||A_{n}N − C_{n}I| − |B_{n}N − D_{n}I|| ≤ |A_{n} − B_{n}| · |N| + |C_{n} − D_{n}| · I proves that

lim_{n→∞}{|A_{n}N − C_{n}I| − |B_{n}N − D_{n}I|} = 0,

or equivalently,

lim_{n→∞} t_{n}|I_{X}(α_{n})N_{T} (γ) − I_{X}(γ)N_{T} (α_{n})| = lim_{n→∞} ps_{n}|I_{X}(β_{n})N_{T} (γ) − I_{X}(γ)N_{T} (β_{n})|.

By the same reasoning, one shows that

lim_{n→∞} t_{n}|I_{Y} (α_{n})N_{S}(γ) − I_{Y} (γ)N_{S}(α_{n})| = lim_{n→∞} ps_{n}|I_{Y} (β_{n})N_{S}(γ) − I_{Y} (γ)N_{S}(β_{n})|.

The proof is complete.

4.5. An embedding of h \overline{πℐ (𝒢)} into R^{4} . Let 𝒞 = Π_{1} ∪ Π_{2} ∪ Π_{3} be the set given at the beginning of this section, and let φ: 𝒞 ⟶ R^{2} be defined by

φ(r_{1}, r_{2}, r_{3}) = \{ \begin{matrix}(r_{1}, r_{2}) & if (r_{1}, r_{2}, r_{3}) ∈ Π_{1} ∪ Π_{2}, \\ (r_{1}, −r_{2}) & if (r_{1}, r_{2}, r_{3}) ∈ Π_{3}. \end{matrix}

It is easy to see that (r_{1}, r_{2}, r_{3}) ∈ (Π_{1} ∪ Π_{2}) ∩ Π_{3} if and only if r_{2} = 0. This implies that φ is continuous on 𝒞 . Moreover, φ is injective as proved below.

Let (r_{1}, r_{2}, r_{3}) and (t_{1}, t_{2}, t_{3}) be two points of 𝒞 , φ(r_{1}, r_{2}, r_{3}) = φ(t_{1}, t_{2}, t_{3}). By the definition, we have r_{1} = t_{1} . Also, we see easily that r_{2} = 0 if and only if t_{2} = 0. If r_{2} = 0, then (r_{1}, r_{2}, r_{3}), (t_{1}, t_{2}, t_{3}) ∈ Π_{3} , and thus (r_{1}, r_{2}, r_{3}) = (t_{1}, t_{2}, t_{3}). Assume that r_{2}t_{2} ≠ 0, i.e. r_{2} > 0 and t_{2} > 0. Then either

\begin{aligned}(r_{1}, r_{2}) & = φ(r_{1}, r_{2}, r_{3}) = φ(t_{1}, t_{2}, t_{3}) = (t_{1}, t_{2}), or \\ (r_{1}, −r_{2}) & = φ(r_{1}, r_{2}, r_{3}) = φ(t_{1}, t_{2}, t_{3}) = (t_{1}, −t_{2}), \end{aligned}

and thus (r_{1}, r_{2}, r_{3}) = (t_{1}, t_{2}, t_{3}). Therefore, φ is injective.

By the definition of Π_{1} , Π_{2} and Π_{3} , we obtain the inverse of φ immediately given by φ^{−1}(t_{1}, t_{2}) = (t_{1}, |t_{2}|, |t_{1} − t_{2}|) for all (t_{1}, t_{2}) ∈ φ(𝒞).

Since r_{1} + r_{2} + r_{4} + r_{5} > 0 whenever (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) ∈ Δ, then the function ψ_{3}: Δ ⟶ R^{4} defined by

ψ_{3}(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = (\frac{φ(r_{1}, r_{2}, r_{3})}{r_{1} + r_{2} + r_{4} + r_{5}} , \frac{φ(r_{4}, r_{5}, r_{6})}{r_{1} + r_{2} + r_{4} + r_{5}} )

is continuous on Δ. We shall prove that ψ_{3} is injective.

Let (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) and (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) be any two points of Δ with

ψ_{3}(r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = ψ_{3}(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}).

Write

\begin{aligned}φ(r_{1}, r_{2}, r_{3}) & = (r′_{1}, r′_{2}), \\ φ(r_{4}, r_{5}, r_{6}) & = (r′_{4}, r′_{5}), \\ φ(t_{1}, t_{2}, t_{3}) & = (t′_{1}, t′_{2}) and \\ φ(t_{4}, t_{5}, t_{6}) & = (t′_{4}, t′_{5}). \end{aligned}

Then r_{j} = r′_{j} and t_{j} = t′_{j} for j = 1, 4; r_{j} = |r′_{j}| and t_{j} = |t′_{j}| for j = 2, 5;

r_{3} = |r′_{1} − r′_{2}|, r_{6} = |r′_{4} − r′_{5}|, t_{3} = |t′_{1} − t′_{2}|, t_{6} = |t′_{4} − t′_{5}|.

Let

p = \frac{r_{1} + r_{2} + r_{4} + r_{5}}{t_{1} + t_{2} + t_{4} + t_{5}} = \frac{r′_{1} + |r′_{2}| + r′_{4} + |r′_{5}|}{t′_{1} + |t′_{2}| + t′_{4} + |t′_{5}|}.

By assumption, r′_{j} = pt′_{j} for j = 1, 2, 4, 5. Since ∑_{j=1}^{6} r_{j} = ∑_{j=1}^{6} t_{j} = 1, then

\begin{aligned}1 & = r′_{1} + |r′_{2}| + |r′_{1} − r′_{2}| + r′_{4} + |r′_{5}| + |r′_{4} − r′_{5}| \\ & = p\{t′_{1} + |t′_{2}| + |t′_{1} − t′_{2}| + t′_{4} + |t′_{5}| + |t′_{4} − t′_{5}|\} = p. \end{aligned}

Therefore, (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}) = (t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}).

From Theorem 4.3 together with the above discussion, we have shown the following theorem.

Theorem 4.4. The composition Φ of Ψ followed by ψ_{3} is a homeomorphism of \overline{πℐ (𝒢)} onto a 3-sphere lying in R^{4} . Moreover,

Φ(α) = (\frac{I_{X}(α)}{σ(α)}, \frac{N_{T} (α)}{σ(α)}, \frac{I_{Y} (α)}{σ(α)}, \frac{N_{S}(α)}{σ(α)}) for all α ∈ 𝒢 ℒ ,

where σ(α) = I_{X}(α) + |N_{T} (α)| + I_{Y} (α) + |N_{S}(α)|.

5. Words for geodesics in 𝒢̂ and their traces

In this section, we consider the Maskit embedding ℳ_{5} of the Teichmüller space of Σ_{5} , which is a family of regular B -groups G(μ, ν) parametrized by complex numbers μ and ν. Each G(μ, ν) representing a five-punctured sphere and three thrice-punctured spheres. The regular set Ω(μ, ν) of G(μ, ν) has a unique simply connected component Ω_{0}(μ, ν) invariant under G(μ, ν) such that Ω_{0}(μ, ν)/G(μ, ν) is a five-punctured sphere. Every geodesic γ ∈ 𝒢̂ corresponds to a cyclic semi-reduced Γ-word W (γ; μ, ν) in G(μ, ν). The trace tr W (γ; μ, ν) is a polynomial in μ and ν. The main work of this section is to compute the high order terms of the trace polynomials tr W (γ; μ, ν). This section is a part of the author’s Ph.D. thesis [3].

5.1. Cyclic semi-reduced Γ-words for geodesics in 𝒢̂ . In this subsection, we shall give a complete description of cyclic semi-reduced Γ-words representing geodesics in 𝒢̂ . Furthermore, we shall write them in exactly two canonical forms. This reduces the difficulty of computing the high-order terms of the trace polynomials tr W (γ; μ, ν).

From Proposition 2.7 and [4, Theorem 3.2], we have

Theorem 5.1. Let γ ∈ 𝒢̂ . If I_{Y} (γ) = 0, then γ is represented by a cyclic semi-reduced Γ-word of the form

∏_{i=1}^{m} T^{r_{i}}X^{ω_{i}}T^{t_{i}}S^{δ_{i}},

where δ_{i}, ω_{i} ∈ {1, −1}, m = I_{X}(γ) = I_{S}(γ), and r_{i} and t_{i} are integers satisfying the following conditions:

(i) −1 ≤ (r_{i} + t_{i})ω_{i} ≤ 0 and −1 ≤ (r_{i+1} + t_{i})δ_{i} ≤ 0, where r_{m+1} = r_{1} .

(ii) |r_{i}|, |t_{i}| ∈ {r, r + 1}, where r = min{|r_{i}|, |t_{i}| : i = 1, …, m}.

(iii) r_{i} ≥ 0, t_{i} ≤ 0 whenever γ ∈ 𝒢T^{+} , and r_{i} ≤ 0, t_{i} ≥ 0 whenever γ ∈ 𝒢T^{−}. (iv) ∑_{i=1}^{m}(r_{i} − t_{i}) = N_{T} (γ).

By considering the function Θ_{2} , we have

Corollary 5.2. Let γ ∈ 𝒢̂ . If I_{X}(γ) = 0, then γ is represented by a cyclic semi-reduced Γ-word of the form

∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}T^{δ_{i}},

where δ_{i}, ε_{i} ∈ {1, −1}, n = I_{Y} (γ) = I_{T} (γ), and p_{i} and q_{i} are integers satisfying the following conditions:

(i) −1 ≤ (p_{i} + q_{i})ε_{i} ≤ 0 and −1 ≤ (p_{i+1} + q_{i})δ_{i} ≤ 0, where p_{n+1} = p_{1} .

(ii) |p_{i}|, |q_{i}| ∈ {p, p + 1}, where p = min{|p_{i}|, |q_{i}| : i = 1, …, n}.

(iii) p_{i} ≤ 0, q_{i} ≥ 0 whenever γ ∈ 𝒢S^{+} , and p_{i} ≥ 0, q_{i} ≤ 0 whenever γ ∈ 𝒢S^{−}. (iv) ∑_{i=1}^{n}(q_{i} − p_{i}) = N_{S}(γ).

In the following, we assume that γ ∈ 𝒢̂ with I_{X}(γ)I_{Y} (γ) > 0. From Proposition 2.1, we may assume that γ ∈ 𝒢S^{+} with I_{X}(γ) ≥ I_{Y} (γ). Let I_{Y} (γ) = n. Then γ is represented by a cyclic semi-reduced Γ-word W of the form

W = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}W_{i},

where ε_{i} = ±1, where p_{i} ≥ 0 and q_{i} ≥ 0 are integers, and where each W_{i} is a semi-reduced Γ-word as given in equation (5). Since

𝒯_{1}^{2}(W) = ∏_{i=1}^{n} S^{−p_{i}−1}Y^{ε_{i}}S^{q_{i}+1}W_{i},

by considering the geodesic 𝒯_{1}^{2}(γ) we may assume that p_{i} > 0 and q_{i} > 0 for all i.

Now, we shall determine the subwords W_{i}. Note that each W_{i} is always followed by S^{−1} since p_{i+1} > 0 for each i, where p_{n+1} = p_{1} . Consider the admissible subarc γ_{i} represented by the reduced word W̃_{i} = S⃗W_{i}S^{−1} . Note that

I_{X}(γ_{i}) = I_{X^{−1}}(γ_{i}) > 0, I_{Y} (γ_{i}) = I_{Y^{−1}}(γ_{i}) = 0 and I_{S^{−1}}(γ_{i}) = 2 + I_{S}(γ_{i}),

for every i, and that

I_{X}(γ) = ∑_{i=1}^{n} I_{X}(γ_{i}).

To simplify notation, for every fixed i we write a = m_{i} and write

W̃_{i} = S⃗E_{1} ⋯ E_{a}S^{−1}.

Let l be the strand of γ_{i} joining the S^{−1} -side to the E_{1} -side, and let l′ be the strand of γ_{i} joining the E_{a}^{−1} -side to the S^{−1} -side. Let P_{0} and P′_{0} be the endpoints of l and l′ on the S^{−1} -side respectively, and let Q_{0} be the point on the S -side such that Q_{0} = S(P_{0}).

Claim. If P is the endpoint of a strand of γ_{i} on the S^{−1} -side, and if P ≠ P_{0} and P ≠ P′_{0} , then P ≺ P_{0} and P ≺ P′_{0} .

Proof of the claim. Note that such a point P exists only when I_{S^{−1}}(γ_{i}) > 2. Let Q = S(P). Then Q is an endpoint of a strand L of γ_{i} connecting the S -side to the E -side for some E ∈ {X^{±1}, T^{±1}}.

If P_{0} ≺ P, then Q_{0} ≺ Q. By the definition of W_{i} and that of Q_{0} , the point Q_{0} is an endpoint of a strand L_{0} of γ connecting the S -side and the E′ -side with E′ ∈ {S^{−1}, Y^{±1}}. This implies that L_{0} intersects L. This is impossible since γ is simple. Hence, P ≺ P_{0} . Similarly, P ≺ P′_{0} . The proof of the claim is complete.

Figure 8.

Let P_{k} ≺ ⋯ ≺ P_{1} be all the points where the lift of γ_{i} to 𝒟 meets the S^{−1} - side, where k = I_{S^{−1}}(γ_{i}) ≥ 2. From the above claim, we have {P_{1}, P_{2}} = {P_{0}, P′_{0}}.

Let l_{1} be the strand of γ_{i} with P_{1} an endpoint, and let A_{1} be the other endpoint of l_{1} . Note that A_{1} lies on the E -side for some E ∈ {X^{±1}, T^{±1}}. Let Q_{2} = S(P_{2}). Since I_{Y} (γ_{i}) = I_{Y^{−1}}(γ_{i}) = 0, there is a simple arc l̂⊂ 𝒟 joining Q_{2} to A_{1} which is disjoint from all strands of γ_{i} except possibly l_{1} (see Figure 8).

Let γ̂_{i} be the curve on Σ_{5} obtained from γ_{i} by replacing l_{1} by l̂. Clearly, γ̂_{i} is a simple loop in 𝒢̂ with I_{Y} (γ̂_{i}) = 0 and I_{X}(γ̂_{i}) = I_{X}(γ_{i}).

By Theorem 5.1, the free homotopy class [γ̂_{i}] is represented by a cyclic semireduced Γ-word Ŵ_{i} of the form

Ŵ_{i} = ∏_{j=1}^{m′_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}},

where m′_{i} = I_{X}([γ̂_{i}]) = I_{X}(γ_{i}), and r_{ij}, t_{ij}, ω_{ij} and δ_{ij} are integers satisfying the conditions given in Theorem 5.1.

Let γ̂_{i} be oriented so that the initial point of the projection of l̂ to Σ_{5} is the projection of A_{1} , and the terminal point is the projection of Q_{2} . We write Ŵ_{i} so that Ŵ_{i} represents the oriented closed curve γ̂_{i}. Then δ_{im′_{i}} = 1, and

W̃_{i} = S⃗(∏_{j=1}^{m′_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ′_{ij}})S^{−1},

where δ′_{im′_{i}} = 0 and δ′_{ij} ∈ {1, −1} for 1 ≤ j < m′_{i}, and thus

(7)  W = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}(∏_{j=1}^{m′_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ′_{ij}}).

Theorem 5.3. Let γ ∈ 𝒢̂ with m = I_{X}(γ) > 0 and n = I_{Y} (γ) > 0.

(A) If m ≥ n, then γ is represented by a cyclic semi-reduced Γ-word W (γ) of the form

W (γ) = ∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}(∏_{j=1}^{m_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}}),

where ε_{i}, ω_{ij} ∈ {1, −1}, m_{i} > 0, and p_{i}, q_{i}, r_{ij}, t_{ij} and δ_{ij} are integers satisfying the following conditions:

(i) ∑_{i=1}^{n} m_{i} = m.

(ii) For 1 ≤ i ≤ n, δ_{im_{i}} = 0, and if m_{i} > 1, then |δ_{ij}| = 1 for 1 ≤ j < m_{i}.

(iii) For 1 ≤ i ≤ n,
−1 ≤ (p_{i} + q_{i})ε_{i} ≤ 0 and |p_{i}|, |q_{i}| ∈ {p, p + 1},

where p = min{|p_{i}|, |q_{i}| : 1 ≤ i ≤ n}. Moreover, p_{i} ≤ 0, q_{i} ≥ 0 for all i when γ ∈ 𝒢S^{+} , and p_{i} ≥ 0, q_{i} ≤ 0 for all i when γ ∈ 𝒢S^{−}.

(iv) For 1 ≤ i ≤ n and 1 ≤ j ≤ m_{i},

−1 ≤ (r_{ij} + t_{ij})ω_{ij} ≤ 0 and |r_{ij}|, |t_{ij}| ∈ {r, r + 1},

where r = min{|r_{ij}|, |t_{ij}| : 1 ≤ i ≤ n, 1 ≤ j ≤ m_{i}}. Moreover, r_{ij} ≤ 0, t_{ij} ≥ 0

when γ ∈ 𝒢T^{−}, and r_{ij} ≥ 0, t_{ij} ≤ 0 when γ ∈ 𝒢T^{+} .

(v) N_{S}(γ) = ∑_{i=1}^{n}(q_{i} − p_{i}) and N_{T} (γ) = ∑_{i=1}^{n} ∑_{j=1}^{m_{i}}(r_{ij} − t_{ij}).

(B) If n ≥ m, then γ is represented by a cyclic semi-reduced Γ-word W (γ) of the form

W (γ) = ∏_{i=1}^{m} T^{r_{i}}X^{ω_{i}}T^{t_{i}}(∏_{j=1}^{n_{i}} S^{p_{ij}}Y^{ε_{ij}}S^{q_{ij}}T^{δ_{ij}}),

where ε_{ij}, ω_{i} ∈ {1, −1}, n_{i} > 0, and r_{i}, t_{i}, p_{ij}, q_{ij} and δ_{ij} are integers satisfying the following conditions:

(i) ∑_{i=1}^{m} n_{i} = n.

(ii) For 1 ≤ i ≤ m, δ_{in_{i}} = 0, and if n_{i} > 1, then δ_{ij} = ±1 for 1 ≤ j < n_{i}.

(iii) For 1 ≤ i ≤ m,
−1 ≤ (r_{i} + t_{i})ω_{i} ≤ 0 and |r_{i}|, |t_{i}| ∈ {r, r + 1},

where r = min{|r_{i}|, |t_{i}| : 1 ≤ i ≤ m}. Moreover, r_{i} ≤ 0, t_{i} ≥ 0 for all i when γ ∈ 𝒢T^{−}, and r_{i} ≥ 0, t_{i} ≤ 0 for all i when γ ∈ 𝒢T^{+} .

(iv) For 1 ≤ i ≤ m and 1 ≤ j ≤ n_{i},
−1 ≤ (p_{ij} + q_{ij})ε_{ij} ≤ 0 and |p_{ij}|, |q_{ij}| ∈ {p, p + 1},

where p = min{|p_{ij}|, |q_{ij}| : 1 ≤ i ≤ m, 1 ≤ j ≤ n_{i}}. Moreover, p_{ij} ≤ 0, q_{ij} ≥ 0

when γ ∈ 𝒢S^{+} , and p_{ij} ≥ 0, q_{ij} ≤ 0 when γ ∈ 𝒢S^{−}.

(v) N_{T} (γ) = ∑_{i=1}^{m}(r_{i} − t_{i}) and N_{S}(γ) = ∑_{i=1}^{m} ∑_{j=1}^{n_{i}}(q_{ij} − p_{ij}).

Remark 5.1. If I_{X}(γ) = I_{Y} (γ) = n, then

W (γ) = ∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}T^{r_{i}}X^{ω_{i}}T^{t_{i}}.

Proof of Theorem 5.3. From Propositions 2.1 and 2.3, the assertion (B) will follow from (A) by considering the geodesic Θ_{2}(γ). Thus, we shall assume that m ≥ n. On the other hand, since I_{E}(Θ_{1}(γ)) = I_{E}(γ) for E ∈ {X, Y }, we may assume that γ ∈ 𝒢S^{+} .

Let W be a cyclic semi-reduced Γ-word representing γ. Then 𝒯_{1}^{2}(W) is of the form as given in equation (7):

𝒯_{1}^{2}(W) = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}(∏_{j=1}^{m_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}})

with p_{i} > 0 and q_{i} > 0 for all i, and thus

W = ∏_{i=1}^{n} S^{−p′_{i}}Y^{ε_{i}}S^{q′_{i}}(∏_{j=1}^{m_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}}),

where p′_{i} = p_{i} − 1 ≥ 0 and q′_{i} = q_{i} − 1 ≥ 0 for i = 1, …, n.

It follows from Proposition 2.7 that

N_{S}(γ) = ∑_{i=1}^{n}(q′_{i} − p′_{i}) and N_{T} (γ) = ∑_{i=1}^{n} ∑_{j=1}^{m_{i}}(r_{ij} − t_{ij}).

This proves condition (v).

It remains to prove that if γ is represented by the word W given in (A), then (iii)′ |p_{i}|, |q_{i}| ∈ {p, p + 1} for 1 ≤ i ≤ n, and (iv)′ |r_{ij}|, |t_{ij}| ∈ {r, r + 1} for 1 ≤ i ≤ n and 1 ≤ j ≤ m_{i}, where

p = min{|p_{i}|, |q_{i}| : 1 ≤ i ≤ n} and r = min{|r_{ij}|, |t_{ij}| : 1 ≤ i ≤ n, 1 ≤ j ≤ m_{i}}.

Note that the other conditions follow from Lemma 2.6.

We shall prove condition (iii)′. Condition (iv)′ will follow by a similar argument. By applying a cyclic permutation to the word W, we may assume that p = min{|p_{1}|, |q_{1}|}. By considering W^{−1} , we may assume that ε_{1} = 1.

Without loss of generality, we assume that γ ∈ 𝒢S^{+} , and write

W = ∏_{i=1}^{n} S^{−p_{i}}Y^{ε_{i}}S^{q_{i}}(∏_{j=1}^{m_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}}), p_{i}, q_{i} ≥ 0 for all i.

Since q_{1} − p_{1} = (q_{1} − p_{1})ε_{1} ≤ 0, then p = q_{1} .

There is nothing to prove if n = 1. Assume that n > 1. Suppose that there is an i_{0} > 1 such that max{p_{i_{0}}, q_{i_{0}}} > p + 1.

𝒯_{1}^{−2p}(W) = ∏_{i=1}^{n} S^{−p′_{i}}Y^{ε_{i}}S^{q′_{i}}(∏_{j=1}^{m_{i}} T^{r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}}),

where p′_{i} = p_{i} − p and q′_{i} = q_{i} − p for all i.

Let γ′ = 𝒯_{1}^{−2p}(γ). Since q′_{1} = q_{1} − p = 0, then γ′ has a strand joining the Y^{−1} -side to the E -side for some E ∈ {X^{±}, T^{±}}. On the other hand, max{p′_{i_{0}}, q′_{i_{0}}} > 1, then γ′ has a strand joining the S -side to the S^{−1} -side. This is impossible! The proof is complete.

5.2. Trace polynomials. In what follows, let G be the subgroup of PSL(2, C) generated by the following four parabolic transformations:

S = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) , T = ( \begin{matrix}1 & 4 \\ 0 & 1 \end{matrix} ) ,

X = ( \begin{matrix}1 + 4i & 16 \\ 1 & 1 − 4i \end{matrix} ) and Y = ( \begin{matrix}1 + 4i & 4 \\ 4 & 1 − 4i \end{matrix} ) .

By using Maskit’s first combination theorem ([8, Theorem VII.C.2]), one can prove that G is a regular B -group representing a five-punctured sphere and three thrice punctured spheres. The regular set of G has a simply connected component Ω_{0} invariant under G such that Ω_{0}/G = Σ_{5} . Such a Kleinian group G will be called a Maskit five-punctured group.

There is a connected and simply connected fundamental domain 𝒟 for G acting on Ω_{0} (see Figure 9) with Γ = {S^{±1}, T^{±1}, X^{±1}, Y^{±1}} the set of side pairings. The domain 𝒟 may be schematically drawed as in Figure 1 with sides labelled as before. Thus every geodesic in 𝒢 is represented by a cyclic semi-reduced Γ-word given in Theorem 5.1, Corollary 5.2 or Theorem 5.3.

Now, we consider the quasiconformal conjugates of G. Let f be a quasiconformal automorphism of Ĉ such that fGf^{−1} is a Kleinian group. If f is normalized

Figure 9. The fundamental domain 𝒟.

to fix 0, 1 and ∞, then fGf^{−1} is the subgroup of PSL(2, C) generated by S, T, X_{μ} and Y_{ν}, where

X_{μ} = ( \begin{matrix}1 + μ & −μ^{2} \\ 1 & 1 − μ \end{matrix} ) and Y_{ν} = ( \begin{matrix}1 + 2ν & 4 \\ −ν^{2} & 1 − 2ν \end{matrix} )

with complex numbers μ and ν satisfying |μ| ≥ 1, |ν| ≥ \frac{1}{2} and |μν + 2| ≥ 1.

For any two non-zero complex numbers μ and ν, let G(μ, ν) be the subgroup of PSL(2, C) generated by S, T, X_{μ} and Y_{ν}. We refer to the set ℳ_{5} of all (μ, ν) ∈ C^{2} with Im μ > 0 and Im ν > 0 such that G(μ, ν) is a Maskit fivepunctured group as the Maskit embedding of the Teichmüller space of Σ_{5} .

For every (μ, ν) ∈ ℳ_{5} , let ρ_{(μ,ν)}: G ⟶ G(μ, ν) be the isomorphism defined by

ρ_{(μ,ν)}(S) = S, ρ_{(μ,ν)}(T) = T, ρ_{(μ,ν)}(X) = X_{μ} and ρ_{(μ,ν)}(Y) = Y_{ν}.

For every γ ∈ 𝒢̂, let W (γ) ∈ G be a cyclic semi-reduced Γ-word representing γ, and let W (γ; μ, ν) = ρ_{(μ,ν)}(W (γ)). Write the trace polynomial tr W (γ; μ, ν) as

F (γ; μ, ν) = tr W (γ; μ, ν) = a_{1}μ^{r}ν^{s} + a_{2}μ^{r−1}ν^{s} + a_{3}μ^{r}ν^{s−1} + O(r + s − 2),

where a_{1} ≠ 0, a_{2} and a_{3} are integers, and where O(r + s − 2) is a polynomial in μ and ν of degree ≤ r + s − 2. We call a_{1}μ^{r}ν^{s} + a_{2}μ^{r−1}ν^{s} + a_{3}μ^{r}ν^{s−1} the high order terms of F (γ; μ, ν).

If I_{Y} (γ) = 0 and I_{X}(γ) = m > 0, then from [4, Theorem 3.4] we have
(8)  F (γ; μ, ν) = ±{μ^{2m} + 4N_{T} (γ)μ^{2m−1}} + O(μ^{2m−2}),

where O(μ^{2m−2}) is a polynomial in μ of degree ≤ 2m − 2.

If I_{X}(γ) = 0 and I_{Y} (γ) = n > 0, then from Lemma 5.4(ii) given below we have

(9)  F (γ; μ, ν) = ±4^{n}{ν^{2n} + 2N_{S}(γ)ν^{2n−1}} + O(ν^{2n−2}),

where O(ν^{2n−2}) is a polynomial in ν of degree ≤ 2n − 2.

Lemma 5.4. If γ ∈ 𝒢̂ with I_{X}(γ) = m and I_{Y} (γ) = n, then

(i) F (Θ_{1}(γ); μ, ν) = F (γ; −μ, −ν), (ii) F (Θ_{2}(γ); μ, ν) = F (γ; −2ν, −\frac{1}{2}μ), (iii) F (𝒯_{1}(γ); μ, ν) = (−1)^{n}F (γ; μ, ν + 1), (iv) F (𝒯_{1}^{−1}(γ); μ, ν) = (−1)^{n}F (γ; μ, ν − 1),

(v) F (𝒯_{2}(γ); μ, ν) = (−1)^{m}F (γ; μ − 2, ν), and (vi) F (𝒯_{2}^{−1}(γ); μ, ν) = (−1)^{m}F (γ; μ + 2, ν).

Proof. Let

C_{1} = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) and C_{2} = ( \begin{matrix}0 & −2i \\ 1/2i & 0 \end{matrix} ) ,

and let χ_{j}(A) = C_{j}AC_{j}^{−1} for all A ∈ PSL(2, C). Set ρ_{j} = χ_{j}Θ_{j}. A direct computation gives

\begin{aligned}ρ_{j}(S) & = S, \\ ρ_{1}(Y_{ν}) & = Y_{−ν}, \end{aligned} \begin{aligned}ρ_{j}(T) & = T, \\ ρ_{2}(X_{μ}) & = X_{−2ν}, \end{aligned} \begin{aligned}ρ_{1}(X_{μ}) & = X_{−μ}, \\ ρ_{2}(Y_{ν}) & = Y_{−μ/2}. \end{aligned}

By a similar argument as that in the proof of Lemma 3.3 of [4], the assertions (i) and (ii) will follow.

Since the transformations S, T and X_{μ} are invariant under 𝒯_{1} , and since

𝒯_{1}(Y_{ν}) = Y_{ν}^{−1}S = −Y_{ν+1} and 𝒯_{1}^{−1}(Y_{ν}) = SY_{ν}^{−1} = −Y_{ν−1},

then (iii) and (iv) are valid. From (ii) and (iii), we have

\begin{aligned}F (𝒯_{2}(γ); μ, ν) & = F (Θ_{2}𝒯_{1}Θ_{2}(γ); ν, μ) = F (𝒯_{1}Θ_{2}(γ); −2ν, −\frac{1}{2}μ) \\ & = (−1)^{I_{Y} (Θ_{2}(γ))}F (Θ_{2}(γ); −2ν, −\frac{1}{2}μ + 1) = (−1)^{m}F (γ; μ − 2, ν). \end{aligned}

This proves (v). Similarly, the equation given in (vi) will follow from (ii) and (iv).

Geometric intersection numbers on a five-punctured sphere 121 In the rest of this section, we shall compute the high-order terms of F (γ; μ, ν)

for γ ∈ 𝒢̂ with I_{X}(γ)I_{Y} (γ) > 0.

Let I_{X}(γ) = m and I_{Y} (γ) = n. Assume that m ≥ n, and that γ ∈ 𝒢T^{−}. Then γ is represented by a cyclic semi-reduced Γ-word given below:

W = ∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}(∏_{j=1}^{m_{i}} T^{−r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}}),

where r_{ij}, t_{ij} ≥ 0. Note that

N_{T} (γ) = − ∑_{i=1}^{n} ∑_{j=1}^{m_{i}}(r_{ij} + t_{ij}) and N_{S}(γ) = ∑_{i=1}^{n}(q_{i} − p_{i}).

For integers r ≥ 0, t ≥ 0, p and q, and for ω, δ, ε ∈ {1, −1}, we have:

T^{−r}X^{ω}T^{t} = ( \begin{matrix}ωμ + 1 − 4rω & −ωμ^{2} + 4(r + t)ωμ + const. \\ ω & −ωμ + 1 + 4tω \end{matrix} ) ,
S^{p}Y^{ε}S^{q} = ( \begin{matrix}2εν + 1 + 4εq & 4ε \\ −εν^{2} + 2ε(p − q)ν + const. & −2εν + 1 + 4εp \end{matrix} ) ,
T^{−r}X^{ω}T^{t}S^{δ} =

( \begin{matrix}−ωδμ^{2} + (1 + 4(r + t)δ)ωμ + const. & −ωμ^{2} + 4(r + t)ωμ + const. \\ −ωδμ + const. & −ωμ + 1 + 4tω \end{matrix} ) .

For i = 1, …, n, let ξ_{i} = ω_{i1} when m_{i} = 1, let

ξ_{i} = (∏_{j=1}^{m_{i}} ω_{ij})( ∏_{j=1}^{m_{i}−1} δ_{ij}) when m_{i} > 1, λ_{i} = 4 ∑_{j=1}^{m_{i}}(r_{ij} + t_{ij}),

and let

W_{i} = ∏_{j=1}^{m_{i}} T^{−r_{ij}}X^{ω_{ij}}T^{t_{ij}}S^{δ_{ij}} = ( \begin{matrix}a_{i}(μ) & b_{i}(μ) \\ c_{i}(μ) & d_{i}(μ) \end{matrix} ) .

If m_{i} = 1, then

a_{i}(μ) = ξ_{i}(μ + const.) = ξ_{i}(μ^{2m_{i}−1} + ⋯),
b_{i}(μ) = −ξ_{i}(μ^{2} − λ_{i}μ + const.) = −ξ_{i}(μ^{2m_{i}} − λ_{i}μ^{2m_{i}−1} + ⋯),
c_{i}(μ) = ξ_{i} = ξ_{i}(μ^{2m_{i}−2} + ⋯), and
d_{i}(μ) = −ξ_{i}(μ + const.) = −ξ_{i}(μ^{2m_{i}−1} + ⋯).

By induction, one can show that for m_{i} ≥ 1

\begin{aligned}a_{i}(μ) & = (−1)^{m_{i}}ξ_{i}(−μ^{2m_{i}−1} + ⋯), \\ b_{i}(μ) & = (−1)^{m_{i}}ξ_{i}(μ^{2m_{i}} − λ_{i}μ^{2m_{i}−1} + ⋯), \\ c_{i}(μ) & = (−1)^{m_{i}}ξ_{i}(−μ^{2m_{i}−2} + ⋯), and \\ d_{i}(μ) & = (−1)^{m_{i}}ξ_{i}(μ^{2m_{i}−1} + ⋯). \end{aligned}

For every i = 1, …, n, let

S^{p_{i}}Y^{ε_{i}}S^{q_{i}}W_{i} = ( \begin{matrix}ã_{i}(μ, ν) & b̃_{i}(μ, ν) \\ c̃_{i}(μ, ν) & d̃_{i}(μ, ν) \end{matrix} ) ,

and for every n let

∏_{i=1}^{n} S^{p_{i}}Y^{ε_{i}}S^{q_{i}}W_{i} = ( \begin{matrix}A_{n}(μ, ν) & B_{n}(μ, ν) \\ C_{n}(μ, ν) & D_{n}(μ, ν) \end{matrix} ) .

A direct computation gives:

deg ã_{i} = 2m_{i}, deg b̃_{i} = 2m_{i} + 1 = deg c̃_{i}, deg d̃_{i} = 2m_{i} + 2

and d̃_{i}(μ, ν) = (−1)^{m_{i}−1}ξ_{i}ε_{i}(ν^{2}μ^{2m_{i}} − λ_{i}ν^{2}μ^{2m_{i}−1} + 2(q_{i} − p_{i})νμ^{2m_{i}} + ⋯). By applying induction to n, we have

deg A_{n}(μ, ν) = 2(n − 1) + 2 ∑_{i=1}^{n} m_{i},
deg B_{n}(μ, ν) = 2n − 1 + 2 ∑_{i=1}^{n} m_{i} = deg C_{n}(μ, ν),
deg D_{n}(μ, ν) = 2n + 2 ∑_{i=1}^{n} m_{i},

and the high-order terms of D_{n}(μ, ν) are determined by

∏_{i=1}^{n} d̃_{i}(μ, ν) = ∏_{i=1}^{n}(−1)^{m_{i}−1}ξ_{i}ε_{i}(ν^{2}μ^{2m_{i}} − λ_{i}ν^{2}μ^{2m_{i}−1} + 2(q_{i} − p_{i})νμ^{2m_{i}} + ⋯).

Since F (γ; μ, ν) = A_{n}(μ, ν) + D_{n}(μ, ν) and deg A_{n}(μ, ν) < deg D_{n}(μ, ν) − 1, then the high-order terms of F (γ; μ, ν) are determined by D_{n}(μ, ν).

For any two polynomials

\begin{aligned}f(μ, ν) & = a_{1}μ^{r}ν^{s} + a_{2}μ^{r−1}ν^{s} + a_{3}μ^{r}ν^{s−1} + ⋯ and \\ g(μ, ν) & = b_{1}μ^{r′}ν^{s′} + b_{2}μ^{r′−1}ν^{s′} + b_{3}μ^{r′}ν^{s′−1} + ⋯, \end{aligned}

the high-order terms of the polynomial f(μ, ν)g(μ, ν) is

a_{1}b_{1}μ^{r+r′}ν^{s+s′} + (a_{1}b_{2} + a_{2}b_{1})μ^{r+r′−1}ν^{s+s′} + (a_{1}b_{3} + a_{3}b_{1})μ^{r+r′}ν^{s+s′−1}.

Thus, we have

\begin{aligned}F (γ; μ, ν) & = ±\{ν^{2n}μ^{2m} − (∑_{i=1}^{n} λ_{i})ν^{2n}μ^{2m−1} + 2(∑_{i=1}^{n}(q_{i} − p_{i}))μ^{2m}ν^{2n−1} + ⋯\} \\ & = ±\{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯\}. \end{aligned}

From Proposition 2.1 and Lemma 5.4, the above equations are also valid for

γ ∈ 𝒢T^{+} with I_{X}(γ) ≥ I_{Y} (γ).

If n = I_{Y} (γ) ≥ I_{X}(γ) = m, then, by Proposition 2.1 and Lemma 5.4 again, we have

\begin{aligned}F (γ; μ, ν) & = F (Θ_{2}(γ); −2ν, −\frac{1}{2}μ) \\ & = ±4^{n−m}\{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯\}. \end{aligned}

Summing up above discussion together with equations (8) and (9), we have proved the following theorem.

Theorem 5.5 (trace formula). Let γ ∈ 𝒢̂ with I_{X}(γ) = m and I_{Y} (γ) = n. If m ≥ n, then

F (γ; μ, ν) = ±{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯}.

If m ≤ n, then

F (γ; μ, ν) = ±4^{n−m}{μ^{2m}ν^{2n} + 4N_{T} (γ)μ^{2m−1}ν^{2n} + 2N_{S}(γ)μ^{2m}ν^{2n−1} + ⋯}.

References

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[3] Chiang, Y.: Pleating Varieties in the Maskit Embeddings of Teichmüller Spaces of Punctured Spheres. - The City University of New York, GSUC, Ph.D. Thesis, 1995.

124 Yungyen Chiang

[4] Chiang, Y.: Geometric intersection numbers on a four-punctured sphere. - Conform. Geom. Dyn. 1, 1997, 87–103.

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Received 11 January 1999