A COUNTABLE TEICHMÜLLER MODULAR GROUP

Katsuhiko Matsuzaki

Department of Mathematics, Ochanomizu University

Abstract. We construct an example of a Riemann surface of infinite topological type for which the Teichmüller modular group consists of only a countable number of elements. We also consider distinguished properties which the Teichmüller space of this Riemann surface possesses.

§1. Introduction

In this paper, we construct an example of a Riemann surface of infinite topological type for which the Teichmüller modular group has only a countable number of elements. The Teichmüller space of such a Riemann surface is infinite dimensional. For an analytically finite Riemann surface, its Teichmüller space is finite dimensional and the Teichmüller modular group is finitely generated, in particular countable. Hence our example proves the existence of an infinite dimensional Teichmüller space that has a nature of finite dimension regarding the cardinality of the modular group. This answers a problem raised by Epstein [3].

Actually, the original problem asks whether the reduced Teichmüller modular group, which is the quotient group of the Teichmüller modular group by free homotopy equivalence, can be countable or not. By making our Riemann surface R have no ideal boundary at infinity, we give a stronger example where even the full Teichmüller modular group Mod(R) is countable. Then we consider certain properties which the Teichmüller space T (R) of this R has. For instance, we prove discontinuity of the action of Mod(R) on T (R) and triviality of the action of Mod(R) at the origin of the asymptotic Teichmüller space AT (R). The latter space has been studied by Earle, Gardiner and Lakic [4].

Throughout this paper, we assume that a Riemann surface R is hyperbolic, that is, it is represented as a quotient space ℍ^{2}/Γ of the hyperbolic plane ℍ^{2} by a torsion free Fuchsian group Γ. The Teichmüller space T (R) of R is the set of all equivalence classes of the pair (f, σ), where f : R → R_{σ} is a quasiconformal homeomorphism of R onto another Riemann surface R_{σ} of a complex structure σ. Two pairs (f_{1}, σ_{1}) and (f_{2}, σ_{2}) are defined to be equivalent if σ_{1} = σ_{2} and f_{2} ∘f_{1}^{−1} is homotopic (rel. ∂) to a conformal automorphism of R_{σ_{1}} = R_{σ_{2}}. Here the homotopy is considered to be relative to the boundary at infinity (=rel. ∂) when the corresponding Fuchsian group is of the second kind. A distance between equivalence classes p_{1} = [f_{1}, σ_{1}]

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 30F60; Secondary 32G15.

Key words and phrases. Teichmüller space, Quasiconformal mapping class group, Hyperbolic Riemann surface, Pants decomposition.

and p_{2} = [f_{2}, σ_{2}] in T (R) is defined by d(p_{1}, p_{2}) = log K(h), where h is an extremal quasiconformal homeomorphism in the sense that its maximal dilatation K(h) is minimal in the homotopy class (rel. ∂) of f_{2} ∘ f_{1}^{−1}. Then d is a complete distance on T (R), which is called the Teichmüller distance.

The Teichmüller modular group Mod(R) (or the quasiconformal mapping class group) of R is a group of the homotopy classes (rel. ∂) of quasiconformal automorphisms of R. An element g of Mod(R) acts on T (R) in such a way that [f, σ] ↦ [f ∘ g^{−1}, σ], where g also denotes a representative of the homotopy class. It is evident from definition that Mod(R) acts on T (R) isometrically with respect to the Teichmüller distance. One can consult a monograph [6] for basic facts on Teichmüller spaces.

We consider a Riemann surface R such that the Teichmüller modular group Mod(R) consists only of countably many elements. As necessary conditions for R to have this property, we easily see the following.

Proposition 1. If Mod(R) is countable, then the Riemann surface R = ℍ^{2}/Γ satisfies the following conditions:

(1) The number of simple closed geodesics on R whose lengths are uniformly bounded is finite.

(2) R has no ideal boundary at infinity, that is, the corresponding Fuchsian group Γ is of the first kind.

Proof. (1) First we note that the number of closed geodesics having uniformly bounded lengths and having non-empty intersection with a compact subset B of R is finite. Indeed, if infinite, then by assigning a unit directional vector based on a point in B to each closed geodesic, we have infinitely many elements of a compact subset in the unit tangent bundle over R. This implies that there is a convergent sequence of axes in ℍ^{2} for the corresponding elements γ of the Fuchsian group Γ representing R. Moreover, uniform boundedness of the lengths of the closed geodesics enable us to choose a subsequence satisfying the translation lengths of γ are also convergent. However, this contradicts the discreteness of Γ.

Assume that there exist infinitely many simple closed geodesics whose lengths are uniformly bounded. We can choose mutually disjoint {c_{n}}_{n=1}^{∞} from them, for otherwise, infinitely many such geodesics must intersect a single one, contradicting the claim in the above paragraph. Then the Dehn twist χ_{n} along each c_{n} is an element of Mod(R) with uniformly bounded maximal dilatation. Also any infinite composition chosen from {χ_{n}}_{n=1}^{∞} gives an element of Mod(R). See [7]. Since such choices are uncountably many, so are the elements of Mod(R).

(2) Assume that there exists ideal boundary ∂R at infinity. Then there are uncountable many (actually a continuous family of) quasiconformal automorphisms of R that have different boundary maps on ∂R but freely homotopic to the identity. They define uncountably many distinct elements of Mod(R). □

In Section 3, we construct a Riemann surface R that satisfies these conditions (1) and (2). A proof that R satisfies (2) is a crucial point of our argument, which is given in Section 4. Then R becomes naturally the required surface. A proof that Mod(R) is countable is given in Section 5.

Condition (1) and the following proposition due to Wolpert [9] impose strong restriction on the elements of Mod(R).

Proposition 2. Let c be a simple closed geodesic on a Riemann surface R with the length ℓ(c) and f : R → R′ a quasiconformal homeomorphism onto another Riemann surface R′ with the maximal dilatation K ≥ 1. Then the geodesic length ℓ′(f(c)) of the free homotopy class of f(c) on R′ satisfies

\frac{1}{K} ℓ(c) ≤ ℓ′(f(c)) ≤ Kℓ(c).

§2. Discontinuity

In this section, which is not directly related to the construction of our example though, we show that any Teichmüller space T (R) with countable Mod(R) has a similar property to finite dimensional Teichmüller spaces. The main result of this section is Theorem 1, where we show that if Mod(R) is countable it necessarily acts discontinuously on T (R). We say that the action of Mod(R) is discontinuous if, for every point p ∈ T (R), there exists a neighborhood U of p such that the number of elements g ∈ Mod(R) satisfying g(U) ∩ U ≠ ∅ is finite. Since Teichmüller modular groups of analytically finite Riemann surfaces are necessarily countable, Theorem 1 generalizes the well-known fact that such groups act discontinuously (cf. [6]).

Lemma 1. Let p = [f, σ] be a point of the Teichmüller space T (R) and R_{σ} the corresponding Riemann surface. If the number of simple closed geodesics on R_{σ} whose lengths are less than some positive constant L is positive finite, then the orbit G(p) under the Teichmüller modular group G = Mod(R) is a closed set. Moreover, the isotropy subgroup Stab_{G}(p), which is isomorphic to the group Aut(R_{σ}) of all biholomorphic automorphisms of R_{σ}, is finite.

Proof. Suppose that a sequence of points p_{n} = [f_{n}, σ] in the orbit G(p) converges to a point q = [f_{∞}, τ] ∈ T (R). Then we may choose f_{n} and f_{∞} so that the maximal dilatations K(f_{n} ∘ f_{∞}^{−1}) of f_{n} ∘ f_{∞}^{−1} : R_{τ} → R_{σ} converge to 1. It follows from Proposition 2 that there exists a simple closed geodesic c on R_{τ} whose length is less than L. Also, for any sufficiently large n, f_{n} ∘ f_{∞}^{−1} sends c to one of the finitely many simple closed geodesics on R_{σ} with length less than L. Hence a subsequence of f_{n} ∘ f_{∞}^{−1} converges locally uniformly to a quasiconformal homeomorphism h : R_{τ} → R_{σ}. Eventually K(h) = 1, that is, h is conformal and σ = τ. This implies that q belongs to the orbit G(p).

For the latter assertion, we have only to assume all p_{n} and q are coincident with p in the above proof. Then we may regard f_{n} ∘ f_{∞}^{−1} as conformal automorphisms of R_{σ}. They have a convergent subsequence as we have seen above, while Aut(R_{σ}) is discrete. This means that Stab_{G}(p) consists only of finitely many elements. □

Remark. For the first statement in Lemma 1, the finiteness of simple closed geodesics with bounded length is necessary. In fact, there is an example of a Teichmüller space T (R) in which the orbit G(p) under G = Mod(R) is not closed for some p ∈ T (R).

Theorem 1. If Mod(R) is countable, then it acts on T (R) discontinuously.

Proof. Suppose that the action of G = Mod(R) is not discontinuous. Then either (a) there exists a point p ∈ T (R) such that the isotropy subgroup Stab_{G}(p) is infinite or (b) there exists a point p ∈ T (R) and a sequence of elements {g_{n}} of G such that p_{n} = g_{n}(p) are different from p and converge to p as n → ∞. See Fujikawa [5].

By Propositions 1 and 2, for p = [f, σ], there exists a positive constant L such that the number of simple closed geodesics on R_{σ} whose lengths are less than L is positive finite. Then condition (a) is impossible by Lemma 1. If condition (b) holds, then p is a point of accumulation of the orbit G(p). By the group invariance, this implies that every point of G(p) is a point of accumulation. Since G(p) is closed by Lemma 1, G(p) is a closed perfect set. Since a closed perfect set in a complete metric space is uncountable, G(p) is uncountable and so is G. This contradicts the assumption. □

§3. Construction

The fundamental piece of our construction is a pair of pants P, which is a hyperbolic surface with geodesic boundaries c and is homeomorphic to a threepunctured sphere. Every pair of pants admits the canonical orientation reversing isometric involution. The fixed point set of this involution consists of three geodesic segments σ, which we call the symmetry axes. Cutting along the symmetry axes, we have two congruent right-angled hexagons D.

Let P_{0} be a pair of pants the lengths of whose geodesic boundary components are 0! and 1! and 1!. Let P_{1} be a pair of pants with the lengths 1! and 2! and 2!. In the same way, for every non-negative integer n, let P_{n} be a pair of pants with the lengths n! and (n + 1)! and (n + 1)!. The three symmetry axes divide P_{n} into two congruent right-angled hexagons D_{n}. The geodesic boundary components of length n! and (n + 1)! in P_{n} are denoted by c_{n} and c_{n+1} respectively. The two symmetry axes of P_{n} connecting c_{n} and c_{n+1} are denoted by σ_{n}. See Figure 1.

Figure 1. A pair of pants and the right-angled hexagon

We prepare 2^{n+1} copies of P_{n} and glue the geodesic boundary components as follows: Take 2 copies of P_{0} and glue the geodesic boundary component c_{0} of each P_{0} together so that the symmetry axes σ_{0} of both P_{0} meet. The resulting hyperbolic surface with 4 geodesic boundary components c_{1} is denoted by R_{1}. Next take 4 copies of P_{1} and glue the geodesic boundary component c_{1} of each P_{1} with the 4 boundary components of R_{1} so that the symmetry axes σ_{1} of P_{1} meet the symmetry axes of P_{0}. The resulting hyperbolic surface with 8 geodesic boundary components c_{2} is denoted by R_{2}.

Continuing this process, we obtain, for every positive integer n, a hyperbolic surface R_{n} with 2^{n+1} geodesic boundary components c_{n} made of R_{n−1} and 2^{n} copies of P_{n−1}. Then take the exhaustion of these surfaces R_{n}, which is R′ = ∪_{n=1}^{∞}R_{n}. In other words, R′ admits a pants decomposition whose dual graph is the trivalent regular tree. However, this is not yet our required Riemann surface. The reason why not is that R′ is not complete, which is equivalent to saying that a complete hyperbolic surface containing R′ as a deformation retract has ideal boundary at infinity.

At each step, the symmetry axes of P_{n} and P_{n+1} meet and hence they all together constitute geodesic lines {σ} in R′. We measure the length of each σ. In each pair of pants P_{n} (n ≥ 0), the lengths of the three symmetry axes are calculated by trigonometry on the right-angled hexagon D_{n}. See Buser [2, Chap. 2] and Figure 1.

Proposition 3. Let ℓ(σ_{n}) be the length of the symmetry axes σ_{n} of P_{n} connecting the boundary components c_{n} and c_{n+1}. Then

arcsinh \{\frac{1}{sinh(n!/4)}\} < ℓ(σ_{n}) < 2 arcsinh \{\frac{1}{2 sinh(n!/4)}\} .

Proof. By trigonometry on a right-angled hexagon, we have

ℓ(σ_{n}) = arccosh \{\frac{cosh((n + 1)!/2) + cosh(n!/2) cosh((n + 1)!/2)}{sinh(n!/2) sinh((n + 1)!/2)}\} .

Since

g(ξ) = \frac{cosh(ξ) + cosh(n!/2) cosh(ξ)}{sinh(n!/2) sinh(ξ)}

is a monotone decreasing function of ξ, we have

arcsinh \{\frac{1}{sinh(n!/4)}\}

= arccosh \{\frac{1 + cosh(n!/2)}{sinh(n!/2)}\} < ℓ(σ_{n}) < arccosh \{\frac{cosh(n!/2) + cosh^{2}(n!/2)}{sinh^{2}(n!/2)}\}

= 2 arcsinh \{\frac{cosh(n!/4)}{sinh(n!/2)}\}

= 2 arcsinh \{\frac{1}{2 sinh(n!/4)}\} .

This is the desired estimate. □

The right hand side of this inequality is further estimated from above by

2 arcsinh \frac{1}{2 sinh(n!/4)} < \frac{1}{sinh(n!/4)}.

Hence the sum of ℓ(σ_{n}) taken over all n converges, which implies that the length of every σ is finite. Therefore R′ = ∪_{n=1}^{∞}R_{n} is not complete.

In order to obtain a complete hyperbolic surface we twist R′ along each simple closed geodesic c_{n}. By twisting the appropriate amount we can make the distance between c_{0} and P_{n} large. See Basmajian [1]. Note that since the trivalent regular tree corresponding to R′ has uncountably many topological ends, so does R′. In order that the surface obtained by twisting R′ be complete, the twisting along each component of every c_{n} should be in some sense given equal weight so that ideal boundary components at infinity, which are at most countable, do not appear. However, this intuition does not tell explicitly how much to twist, and it is a subtle problem to determine the right amount. For example if we apply a 1/4-twist along each c_{n}, we do not know whether the resulting surface is complete.

We now proceed to the construction, and define the twists as follows. See Figure 2. For every P_{n}, we take a point x_{n} on the boundary component c_{n} so that it is the midpoint between the symmetry axes and call it the center of P_{n}. Starting from x_{n}, we draw the shortest geodesic segment α_{n} in P_{n} to the boundary component c_{n+1} right; α_{n} and c_{n+1} right intersect in the right angle. Along the c_{n+1} right, we twist P_{n+1} so that the center x_{n+1} of P_{n+1} comes to the endpoint of α_{n}. Next, along the c_{n} left, we also twist P_{n+1} left in the same direction and by the same amount as P_{n+1} right. Then the center x_{n+1} of P_{n+1} left comes to a certain point on the boundary component c_{n+1} left. The geodesic segment in P_{n} connecting x_{n} and this new place for x_{n+1} left is denoted by β_{n}. Along c_{0} between the two P_{0}, we give no twist. Namely, the two centers of the two P_{0} sit on the same point.

Figure 2. Twist and the trirectangle

¿From the following proposition, we can see that the length of the geodesic segment α_{n} in P_{n} is uniformly bounded away from zero for every n ≥ 0. Also the intersection of α_{n} and c_{n+1} (the new place for x_{n+1} right) is always on the right side of the midpoint of c_{n+1}. The latter implies that all the directions of the twists are the same; P_{n+1} turns to the right if we observe it from P_{n}.

Proposition 4. In each pair of pants P_{n} (n ≥ 0), consider a trirectangle made of the geodesic segments α_{n}, σ_{n}, c′_{n} and c′′_{n+1}, where c′_{n} and c′′_{n+1} are the portions of c_{n} and c_{n+1} (Figure 2). Let ℓ(·) denote the length of each segment, and ϕ_{n} the angle at x_{n} between c′_{n} and α_{n}. Then ℓ(α_{n}) > arcsinh 1 and ℓ(c′′_{n+1}) < n!/4 for every n ≥ 0. Moreover, ϕ_{n} > π/4.

Proof. A trirectangle is determined by the lengths of two sides and we know that ℓ(c′_{n}) = n!/4 and sinh ℓ(σ_{n}) > 1/ sinh(n!/4) by Proposition 3. The other lengths are obtained from the following formulas ([2]):

sinh ℓ(α_{n}) = sinh ℓ(σ_{n}) cosh ℓ(c′_{n})
tanh ℓ(c′′_{n+1}) = \frac{1}{cosh ℓ(σ_{n})} tanh ℓ(c′_{n}) < tanh ℓ(c′_{n}).

From the first formula, we have

sinh ℓ(α_{n}) > \frac{1}{sinh(n!/4)} cosh(n!/4) > 1.

From the second formula, we have ℓ(c′′_{n+1}) < n!/4.

The angle ϕ_{n} is also obtained from the following ([2]):

tan ϕ_{n} = \frac{cosh ℓ(σ_{n})}{sinh(n!/4) sinh ℓ(σ_{n})}.

Then, by simple calculation as in Proposition 3, this is estimated from below by

\frac{1 + cosh(n!/2)}{sinh(n!/2)\sqrt{1 + (2 sinh(n!/4))^{−2}}} > 1.

Hence we have ϕ_{n} > π/4. □

Remark. (1) By estimating from the opposite side and taking the limit, we see that the constants obtained in the above proposition are sharp: lim_{n→∞} ℓ(α_{n}) = arcsinh 1 and lim_{n→∞} ϕ_{n} = π/4. (2) Since ℓ(c′′_{n+1}) < n!/4, the rotation number of the twist is at least n/4(n + 1), which converges to 1/4 as n → ∞.

We denote the exhaustion ∪_{n=1}^{∞}R_{n} with the twists along c_{n} by R. In the next section, we prove that R is a complete hyperbolic surface.

§4. Completeness

In order to prove that R is complete, we suppose to the contrary that R is not complete. Then, since R = ∪_{n=1}^{∞}R_{n} has the exhaustion by the hyperbolic surfaces R_{n} with geodesic boundaries, any boundary component of R is geodesic. We develop R to the hyperbolic plane ℍ^{2} to obtain a simply connected domain R̃.

Then R̃ is bounded by countably many geodesic lines L in ℍ^{2} together with ideal boundary at infinity. For each geodesic line L, a sequence {c̃_{n}} of geodesic lines in ℍ^{2} converges to L, where c̃_{n} is a developed image of a simple closed geodesic c_{n} in R.

We choose a fundamental domain of R as follows. In each pair of pants P_{n}, take the right-angled hexagon D_{n} containing the center x_{n} among the congruent two. Since the center x_{n+1} of D_{n+1} is on the boundary of D_{n}, the union of all the D_{n} constitutes a simply connected domain D such that D and the other congruent half D′ together make R. We develop D and D′ into ℍ^{2} as simply connected domains, which are denoted by D̃ and D̃′. The union of D̃ and D̃′ makes a fundamental domain of R. Since they are congruent, it is enough to consider only D̃.

We estimate the least distance necessary to cross over D̃, which is the infimum of the distances between any two distinct components of the complement of D̃ in R̃. It is easy to see that we have only to measure a path along c̃_{n}. By Proposition 4, its length is bounded from below by n!/4 for n ≥ 0. Hence the distance for crossing over D̃ is at least 1/4.

Take a geodesic line L that is a boundary component of R̃. We draw an arc η̃ of finite length from a point inside R̃ to an interior point of L. (The role of η̃ is not important; it is merely for a convenience of explanation.) There are two cases to be considered: η̃ crosses over either infinitely many orbits of D̃ or D̃′ under the holonomy group, or only finitely many of them. In the former case, η̃ would contain infinitely many subarcs of length greater than 1/4, which contradicts the fact that the length of η̃ is finite. In the latter case, without loss of generality, we may assume that η̃ starts from x̃_{0} and lies entirely in D̃, where x̃_{0} ∈ D̃ corresponds to x_{0} ∈ R. This means that the closure of D̃ has the intersection with L in ℍ^{2} and η̃ lands on L after passing through the geodesic lines c̃_{n} for all n. At each level n, η̃ goes choosing one of the left and the right c̃_{n+1} towards L.

We consider an infinite tree T based on x_{0} that consists of the geodesic segments α_{n} and β_{n} (n ≥ 0). Any path in T towards infinity is determined by choosing one of α_{n} and β_{n} (equivalently choosing one of the next vertices x_{n+1}) at each vertex x_{n}. We take a path γ in T that has the same itinerary as the arc η̃. Let γ̃ ⊂ D̃ be the developed image of γ. This is a piecewise geodesic ray starting from x̃_{0} and connecting x̃_{1}, x̃_{2}, … consecutively. For each n, η̃ and γ̃ intersect the same geodesic line c̃_{n} where the x̃_{n} lies.

We consider possibility of the existence of the path γ constructed above. The essential case in our arguments is when γ is the α-path, which is the piecewise geodesic ray α_{0} ∪ α_{1} ∪ α_{2} · · ·. We can see that the length of the α-path is infinite by Proposition 4. However, the following Lemma 2 with Figure 3 shows that if γ is the α-path then its length must be finite. Thus the possibility that γ is the α-path is eliminated.

Lemma 2. Let L be a geodesic line and x̃_{0} ∉ L a point in the hyperbolic plane ℍ^{2}. Suppose that a sequence of mutually disjoint geodesic lines {c̃_{n}}_{n=1}^{∞} separates L from x̃_{0} consecutively: c̃_{1} separates L from x̃_{0}, and inductively c̃_{n+1} separates L from c̃_{n}. Let x̃_{1} ∈ c̃_{1} be the nearest point from x̃_{0}, and inductively let x̃_{n+1} ∈ c̃_{n+1} be the nearest point from x̃_{n}. Then the sum of the distances ∑_{n=0}^{∞} d(x̃_{n}, x̃_{n+1}) is bounded from above by twice the length from x̃_{0} to L along a horocircle tangent at the endpoint of L. In particular, the sum converges.

Figure 3. Bounded by horocircles

Proof. Let H and H′ be the horocircles tangent at the endpoints of L and passing through x̃_{0}. For a point z ∈ c̃_{n} within the region bounded by H and H′, consider the distance f(z) = d(z, c̃_{n+1}) between z and c̃_{n+1} as a continuous function of z. It takes its maximum at one of the endpoints of the segment c̃_{n} bounded by H and H′. Since the maximum is further bounded by the arclength along H or H′ between c̃_{n} and c̃_{n+1}, so is d(x̃_{n}, x̃_{n+1}). Taking the sum over all n, we have the assertion. □

Next we consider the case when γ is the β-path, which is the piecewise geodesic ray β_{0} ∪β_{1} ∪β_{2} · · ·. We can also eliminate this possibility as follows. Along each c̃_{n} (n ≥ 1), the adjacent hexagon shifts to the right at least n!/4−(n−1)!/4 length by Proposition 4. Since c̃_{n} sit nearly in parallel to L for sufficiently large n, this forces the arc η̃ in D̃ to land at ∂ℍ^{2} on the right side of L, not at an interior point of L. See Figure 4. (To give another reason why this case cannot occur, we have only to notice that the other α-path in the backside D′ exists in parallel to the β-path in D and they together travel passing through the same c_{n}. Since the α-path cannot behave like this as is seen above, γ cannot be the β-path.)

The case when γ contains finitely many α- or β-segments is excluded in the same reason as above. The remaining possibility is that γ contains both α- and β-segments infinitely many. By Proposition 4, each α_{n} makes at least angle π/4 measured from c_{n}. Since β-segments are given by turning the x_{n+1} left to the right, the angle made by β_{n} and c_{n} is greater than the one made by α_{n}, namely, it is also greater than π/4. An elementary geometric observation shows that, for all sufficiently large n, the base point x̃_{n} of an α_{n} in γ̃ must be close to the left endpoint of L and the base point of a β_{n} in γ̃ must be close to the right endpoint of L. Then both the endpoints of L are accumulation points of γ̃. However, this is impossible for the reason that outgoing angles of γ̃ at the intersection with c̃_{n} are uniformly bounded below by π/4. (We may alternatively use the fact lim_{n→∞} ℓ(α_{n}) = arcsinh 1 for this reasoning.) See Figure 5.

¿From the arguments in this section, we conclude:

Figure 4. The β-path goes right

Figure 5. Alternate α- and β-segments

Theorem 2. The hyperbolic surface R defined in Section 3 is complete.

§5. Countability

In this section, we prove that the Teichmüller modular group Mod(R) for our Riemann surface R consists of countably many elements.

Proposition 5. In the pair of pants P_{n} of R, consider the shortest geodesic arc δ_{n}(⊄ c_{n}) that connects the boundary component c_{n} to itself. Then its hyperbolic length ℓ(δ_{n}) satisfies

ℓ(δ_{n}) = 2 arcsinh \{\frac{cosh((n + 1)!/2)}{sinh(n!/4)}\} > n! × n.

Proof. By symmetry, it is easy to see that δ_{n} is an arc starting from the center x_{0} perpendicularly to the backside center on c_{n}. Using trigonometry on the rightangled hexagon D_{n} or trigonometry on the right-angled pentagon, we can calculate the half length of δ_{n}. See Figure 1. □

Based on this proposition, we have the following.

Lemma 3. (1) The hyperbolic length of a closed geodesic contained in R − \overline{R_{n}} is greater than (n + 1)! > n! × n. (2) The hyperbolic length of a closed geodesic (≠ c_{n}) having the intersection with ∂R_{n} is greater than n! × n.

Proof. Every closed geodesic contained in R − \overline{R_{n}} is either coincident with c_{n+i} for some i ≥ 1 or containing a subarc in P_{n+i} for some i ≥ 1 with the both endpoints on c_{n+i}. In the former case, the length is (n + i)!, and in the latter case the length is greater than (n + i)! × (n + i) by Proposition 5. In both cases, it is greater than (n + 1)!. Similarly, every closed geodesic (≠ c_{n}) having the intersection with ∂R_{n} contains a subarc in P_{n+i} for some i ≥ 0 with the both endpoints on c_{n+i}. Hence its length is greater than n! × n. □

We consider quasiconformal automorphisms of R. Proposition 2 gives strong restriction to the possibility of the images of R_{n}.

Lemma 4. Let g : R → R be a K-quasiconformal automorphism of R. Then, for every n ≥ K, the image g(R_{n}) of the subdomain R_{n} is freely homotopic to R_{n} in R.

Proof. If not, there exists a geodesic boundary component c_{n} of ∂R_{n} such that g(c_{n}) is freely homotopic to a simple closed geodesic either contained in R − \overline{R_{n}} or having the intersection with ∂R_{n}. By Lemma 3, the geodesic length ℓ(g(c_{n})) of the free homotopy class g(c_{n}) is greater than n!×n. On the other hand, since ℓ(c_{n}) = n! and g is K-quasiconformal, Proposition 2 asserts that ℓ(g(c_{n})) ≤ K · n! ≤ n! × n. This contradiction proves the lemma. □

If we apply this lemma to R_{n} for each integer n ≥ K, we see that any Kquasiconformal automorphism of R maps every pair of pants P_{n} homotopically onto a pair of pants of the same size. Our final work is then to eliminate the possibility of Dehn twists along each c_{n} for n ≥ K. We succeed in doing that and conclude the following.

Theorem 3. Let g : R → R be a K-quasiconformal automorphism of R. Then, on each connected component E_{n} of R − \overline{R_{n}} for n ≥ max\{K, 5\}, the g restricted to E_{n} is homotopic to a conformal homeomorphism of E_{n} onto another connected component of R − \overline{R_{n}}.

Proof. We apply Lemma 4 to R_{n+i} for every i ≥ 0. Then we can see that the image of each pair of pants P_{n+i} under g is homotopic to some other pair of pants P_{n+i} with the same size. This implies that g restricted to each P_{n+i} is homotopic to a conformal homeomorphism. Hence, on each connected component E_{n} of R − \overline{R_{n}} for n ≥ K, g is homotopic to a conformal homeomorphism possibly with the composition of half Dehn twists along simple closed geodesics c_{n+i} for i ≥ 0.

We will prove that g does not cause a half Dehn twist χ along a simple closed geodesic c_{n+i}. The self-composition χ^{2} is the full Dehn twist along c_{n+i} and the maximal dilatation of the extremal quasiconformal automorphism in the homotopy class of χ^{2} can be estimated from below as

K(χ^{2}) ≥ \sqrt{\{\frac{(n + i)!}{π}\}^{2} + 1} ≥ \frac{n!}{π}.

See [7]. Since K(χ)^{2} ≥ K(χ^{2}), we have K(χ) ≥ {n!/π}^{1/2} > K. This estimate is also valid for any quasiconformal automorphism that is composed by multiple twists along c_{n+i}. However, since g is K-quasiconformal, no such twist is possible. □

As a consequence, we obtain the required property for the Teichmüller modular group.

Corollary 1. The Teichmüller modular group Mod(R) for the Riemann surface R consists only of countably many elements.

Proof. We have only to see that, for every integer n ≥ 5, a subset Mod(R)_{n} consisting of the elements in Mod(R) that have an n-quasiconformal automorphism g as a representative has at most a countable number of elements. By Theorem 3, Mod(R)_{n} is embedded in the reduced Teichmüller modular group Mod^{#}(R_{n}), which is the free homotopy classes of (quasiconformal) automorphisms of the bordered surface R_{n}. Since R_{n} is topologically finite, Mod^{#}(R_{n}) is finitely generated and in particular countable. Hence so is Mod(R)_{n}. (In fact, it consists of finitely many elements.) □

§6. By-products

The Riemann surface R constructed in the previous sections has distinguished properties other than the countability of Mod(R). We note the following consequences as corollaries to Theorem 3 and Lemma 3.

An asymptotically conformal homeomorphism f : R → R′ is a quasiconformal homeomorphism having a property that, for every ϵ > 0, there exists a compact subset V of R such that the maximal dilatation of f restricted to R −V is less than 1 + ϵ. This concept plays a central role in the theory of asymptotic Teichmüller spaces developed by Earle, Gardiner and Lakic [4], [6, Chap. 14]. The asymptotically conformal mapping class group Mod_{0}(R) is defined to be a subgroup of Mod(R) consisting of all homotopy classes that have an asymptotically conformal automorphism of R as a representative. For an analytically finite Riemann surface R, it is clear that Mod_{0}(R) is coincident with Mod(R). Theorem 3 implies that our Riemann surface R gives an example of infinite topological type that holds this property.

Corollary 2. The asymptotically conformal mapping class group Mod_{0}(R) is coincident with Mod(R) for the Riemann surface R.

The asymptotic Teichmüller space AT (R) is the Teichmüller space T (R) modulo the subspace T_{0}(R) consisting of the asymptotically conformal Teichmüller classes. For an analytically finite Riemann surface R, it is clear that T (R) = T_{0}(R) and hence AT (R) is trivial. When R is not analytically finite, it is proved in [4] that Mod(R) acts on AT (R) and this action is faithful if and only if R is conformally equivalent to the unit disk or a punctured unit disk. The asymptotically conformal mapping class group Mod_{0}(R) is the isotropy subgroup of Mod(R) with respect to the origin of AT (R). Remark that, as the action of Mod(R) on AT (R) is not transitive, the isotropy subgroups are not necessarily conjugate to each other. Corollary 2 says that our Riemann surface R gives an example where every element of Mod(R) fixes the origin of AT (R).

Next, for a Riemann surface R with no ideal boundary at infinity, we consider the group of all homotopy classes g of orientation preserving homeomorphic automorphisms of R and its subgroup Mod_{ls}(R) with the following definition: g belongs to Mod_{ls}(R) if there exists a constant κ ≥ 1 such that

\frac{1}{κ}ℓ(c) ≤ ℓ(g(c)) ≤ κℓ(c)

for every simple closed geodesic c on R. By Proposition 2, Mod(R) is a subgroup of Mod_{ls}(R). For a topologically finite Riemann surface R, they are coincident. If R is not topologically finite, then Mod_{ls}(R) always seems uncountable, and this is true for our Riemann surface R. Indeed, statement (2) in Lemma 3 says that the length of any simple closed geodesic c intersecting c_{n} is relatively long to the length of c_{n}. Hence the Dehn twist along c_{n} does not change the ratio for c so much. Since an arbitrary infinite composition of such Dehn twists belongs to Mod_{ls}(R), it contains uncountably many elements. On the other hand, Mod(R) is countable by Corollary 1. Hence we have the following.

Corollary 3. The Teichmüller modular group Mod(R) is a proper subgroup of Mod_{ls}(R) for the Riemann surface R.

On the Teichmüller space T (R), the length spectrum distance between [f_{1}, σ_{1}] and [f_{2}, σ_{2}] is defined by measuring the ratio of the lengths

sup_{c} |log \frac{ℓ_{σ_{1}}(f_{1}(c))}{ℓ_{σ_{2}}(f_{2}(c))}| ,

where the supremum is taken over all simple closed geodesics c on R. By Proposition 2 again, this is not greater than the usual Teichmüller distance. On the Teichmüller space of an analytically finite Riemann surface, these distances are topologically equivalent. Problems around this fact were studied by T. Sorvali, Z. Li and L. Liu among others.

Recently, Shiga [8] considered a Riemann surface of infinite topological type for which the Teichmüller distance and the length spectrum distance induce different topologies on the Teichmüller space. Our construction of the Riemann surface R is related to his work, however, our requirement for R is much stronger. Actually, again from statement (2) in Lemma 3, we see that our Riemann surface R satisfies a condition given in [8] from which the difference of the two distances is deduced.

Corollary 4. On the Teichmüller space T (R) of the Riemann surface R, the Teichmüller distance and the length spectrum distance induce different topologies.

References

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2. P. Buser, Geometry and spectra of compact Riemann surface, Progress in Mathematics 106, Birkhäuser, 1992.

3. A. Epstein, Effectiveness of Teichmüller modular groups, In the tradition of Ahlfors and Bers, Contemporary Math. 256, AMS, 2000, pp. 69–74.

4. C. Earle, F. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence (preprint).

5. E. Fujikawa, Limit sets and regions of discontinuity of Teichmüller modular groups, Proc. Amer. Math. Soc. 132 (2004), 117–126.

6. F. Gardiner and N. Lakic, Quasiconformal Teichmüller theory, SURV 76, American Mathematical Society, 2000.

7. K. Matsuzaki, The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces, Kodai Math. J. 26 (2003), 279–287.

8. H. Shiga, On a distance defined by the length spectrum on Teichmüller space, Ann. Acad. Sci. Fenn. 28 (2003), 315–326.

9. S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. 109 (1979), 323–351.

Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, JAPAN E-mail address: matsuzak@math.ocha.ac.jp