Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 315–326

ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE

Hiroshige Shiga

Tokyo Institute of Technology, Department of Mathematics Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan; shiga@math.titech.ac.jp

Dedicated to the memory of Professor Nobuyuki Suita

Abstract. We consider a distance d_{L} on the Teichmüller space T (S_{0}) of a hyperbolic Riemann surface S_{0} . The distance is defined by the length spectrum of Riemann surfaces in T (S_{0}) and we call it the length spectrum metric on T (S_{0}). It is known that the distance d_{L} determines the same topology as that of the Teichmüller metric if S_{0} is a topologically finite Riemann surface.

In this paper we show that there exists a Riemann surface S_{0} of infinite type such that the length spectrum distance d_{L} on T (S_{0}) does not define the same topology as that of the Teichmüller distance. Also, we give a sufficient condition for these distances to have the same topology on T (S_{0}).

1. Introduction and results

On the Teichmüller space T (S_{0}) of a hyperbolic Riemann surface S_{0} , we have the Teichmüller distance d_{T} ( · , ·), which is a complete distance on T (S_{0}). In this paper, we study another distance d_{L}( · , ·) which is defined by the length spectrum on Riemann surfaces in T (S_{0}). Li [4] discussed the distance d_{L}( · , ·) on the Teichmüller space of a compact Riemann surface of genus g ≥ 2 and showed that the distance d_{L} defines the same topology as that of the Teichmüller distance. Recently, Liu [5] showed that the same statement is true even if S_{0} is a Riemann surface of topologically finite type, and he asked whether the statement holds for Riemann surface of infinite type. Our first result gives a negative answer to this question.

Theorem 1.1. There exist a Riemann surface S_{0} of infinite type and a sequence {p_{n}}_{n=0}^{∞} in T (S_{0}) such that

d_{L}(p_{n}, p_{0}) → 0, n → ∞,

while

d_{T} (p_{n}, p_{0}) → ∞, n → ∞.

From the proof of this theorem, we show the incompleteness of the length spectrum distance.

2000 Mathematics Subject Classification: Primary 30F60; Secondary 32G15.

Corollary 1.1. There exists a Riemann surface of infinite type such that the length spectrum distance d_{L} is incomplete in the Teichmüller space.

Next, we give a sufficient condition for the length distance to define the same topology as that of the Teichmüller distance as follows.

Theorem 1.2. Let S_{0} be a Riemann surface. Assume that there exists a pants decomposition S_{0} = ⋃_{k=1}^{∞} P_{k} of S_{0} satisfying the following conditions.

(1) Each connected component of ∂P_{k} is either a puncture or a simple closed geodesic of S_{0} , k = 1, 2, ….

(2) There exists a constant M > 0 such that if α is a boundary curve of some

P_{k} then

0 < M^{−1} < l_{S_{0}}(α) < M

holds. Then d_{L} defines the same topology as that of d_{T} on the Teichmüller space T (S_{0}) of S_{0} .

2. Preliminaries

Let S_{0} be a hyperbolic Riemann surface. We consider a pair (S, f) of a Riemann surface S and a quasiconformal homeomorphism f of S_{0} onto S. Two such pairs (S_{j}, f_{j}), j = 1, 2, are called equivalent if there exists a conformal mapping h: S_{1} → S_{2} which is homotopic to f_{2}∘f_{1}^{−1} , and we denote the equivalence class of (S, f) by [S, f]. The set of all equivalence classes [S, f] is called the Teichmüller space of S_{0} : we denote it by T (S_{0}).

The Teichmüller space T (S_{0}) has a complete distance d_{T} called the Teichmüller distance which is defined by

d_{T} ([S_{1}, f_{1}], [S_{2}, f_{2}]) = inf_{f} log K[f],

where the infimum is taken over all quasiconformal mappings f: S_{1} → S_{2} homotopic to f_{2} ∘ f_{1}^{−1} and K[f] is the maximal dilatation of f.

We define another distance on T (S_{0}) by length spectrum of Riemann surfaces. Let Σ(S) be the set of closed geodesics on a hyperbolic Riemann surface S. For any two points [S_{j}, f_{j}], j = 1, 2, in T (S_{0}), we set

ϱ([S_{1}, f_{1}], [S_{2}, f_{2}]) = sup_{c∈Σ(S_{1})} max\{\frac{l_{S_{1}}(c)}{l_{S_{2}}(f_{2} ∘ f_{1}^{−1}(c))}, \frac{l_{S_{2}}(f_{2} ∘ f_{1}^{−1}(c))}{l_{S_{1}}(c)}\},

where l_{S}(α) is the hyperbolic length of a closed geodesic on S freely homotopic to a closed curve α. For two points [S_{j}, f_{j}] ∈ T (S_{0}), j = 1, 2, we define a distance d_{L} called the length spectrum distance by

d_{L}([S_{1}, f_{1}], [S_{2}, f_{2}]) = log ϱ([S_{1}, f_{1}], [S_{2}, f_{2}]).

Wolpert ([7]) shows that l_{S_{2}}(f(c)) ≤ K[f]l_{S_{1}}(c) holds for every quasiconformal mapping f: S_{1} → S_{2} and for every c ∈ Σ(S_{1}). Thus, we have immediately:

Lemma 2.1. An inequality

d_{L}(p, q) ≤ d_{T} (p, q)

holds for every p, q ∈ T (S_{0}).

3. Proofs of Theorem 1.1 and Corollary 1.1

3.1. Proof of Theorem 1.1. First, we take monotone divergent sequences {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞} of positive numbers so that a_{n+1} = b_{n} and b_{n}/a_{n} > n, n = 1, 2, …. For each n, we take a right-angled hexagon H_{n} so that the lengths of three edges are a_{n}, b_{n}, b_{n} as Figure 1.

Figure 1.

We take b_{n} so large that

(1)  t_{n} > na_{n}

holds, where t_{n} is the height of H_{n}, the distance from the bottom edge of length a_{n} to the top edge. Gluing two copies of H_{n}, we obtain a pair of pants P_{n} as in Figure 2.

The hyperbolic lengths of the three boundary curves of P_{n} are 2a_{n}, 2b_{n}, 2b_{n}. Note that the lengths of boundaries of P_{n} are long but the distances between any two boundaries are short. From these pairs of pants {P_{n}}_{n=1}^{∞}, we construct a Riemann surface S_{0} as follows.

Step 1. For each k ∈ N, glue two copies of P_{k} along the boundaries of length 2a_{k}. Then we obtain a Riemann surface of type (0, 4), say S_{k,1} . Let γ_{k} denote the “core” curve in S_{k,1} with length 2a_{k} (see Figure 3).

Figure 2.

Figure 3.

Step 2. Make four copies of P_{k+1} and glue each copy with S_{k,1} along their boundary curves of length 2b_{k} = 2a_{k+1} . The resulting Riemann surface S_{k,2} is of type (0, 8).

Step 3. Continue the above construction inductively. Namely, make 2^{n+1} copies of P_{k+n} and glue each copy with S_{k,n−1} along their boundary curves of length 2b_{k+n−1} = 2a_{k+n}. Then we obtain a Riemann surface S_{k,n} of type (0, 2^{n+1}).

Step 4. Take n(k) ∈ N so large that the distance between γ_{k} and ∂S_{k,n(k)} is greater

than ka_{k}. Put S^{(k)} = S_{k,n(k)} . From the construction, we see the following:

Observation. For any points p, q ∈ γ_{k}, let γ be a geodesic arc from p to q. Then the hyperbolic length l_{S_{0}}(γ) of γ satisfies

(2)  l_{S_{0}}(γ) > ka_{k},

if γ ⊄ γ_{k}. Step 5. Make another copy S^{(−k)} of S^{(k)} for each k ∈ N and construct a Riemann

surface S_{0} of infinite type from {S^{(k)}}_{k=−∞}^{−1} ∪ {S^{(k)}}_{k=1}^{∞} and another pair of pants with geodesic boundaries such that each S^{(k)} is isometrically embedded in S_{0} . Let f_{n} be the positive Dehn twist for γ_{n}, n ∈ N. Here, the “positive” Dehn

twist means the Dehn twist with left turning. Set p_{n} = [S_{0}, f_{n}] and p_{0} = [S_{0}, id]. The following lemma shows that d_{T} (p_{n}, p_{0}) → ∞.

Lemma 3.1. Let K_{n} be the maximal dilatation of the extremal quasiconformal mapping which is homotopic to f_{n}. Then, lim_{n→∞} K_{n} = ∞. Thus, lim_{n→∞} d_{T} (p_{n}, p_{0}) = ∞ as n → ∞.

Proof. We consider a neighbourhood U_{n} of γ_{n} which is defined by

U_{n} = {z ∈ S_{0} | d_{S_{0}}(z, γ_{n}) < ε}

for some ε > 0, where d_{S_{0}}( · , ·) is the hyperbolic distance on S_{0} . We take ε > 0 small enough that U_{n} is conformally equivalent to an annulus. We define the Dehn twist f_{n} such that f_{n} | U_{n} is the standard Dehn twist on the annulus and the identity on S_{0} \ U_{n}. That is, f_{n} | U_{n} is defined in terms of the polar coordinates in the annulus by

r exp (iθ) ↦ r exp \{i(θ + 2π \frac{r − 1}{R − 1})\},

if U_{n} is equivalent to {1 < |z| < R}.

Let α_{n} be a simple closed geodesic in S_{n,1} ⊂ S_{0} perpendicular to γ_{n}. We consider connected components of π^{−1}(U_{n}), π^{−1}(γ_{n}) and π^{−1}(α_{n}) on H, where π: H → S_{0} is a universal covering map. We may assume that the connected component of π^{−1}(γ_{n}) is the imaginary axis and that of π^{−1}(α_{n}) is δ := {|z| = 1} ∩H, the unit circle in H. Let Ũ_{n} denote the connected component of π^{−1}(U_{n}) containing the imaginary axis.

Let F_{n}: H → H be a lift of an extremal quasiconformal mapping which is homotopic to f_{n}. We may take F_{n} so that F_{n}(0) = 0, F_{n}(i) = i and F_{n}(∞) = ∞. It is well known that F_{n} can be extended to a homeomorphism of \overline{H} and the boundary mapping F_{n} | R depends only on the homotopy class of f_{n} up to Aut(H).

Let z_{1}, z_{2} , Re z_{1} < 0 < Re z_{2} , be the points of δ∩∂Ũ_{n}. Since f_{n} is the positive Dehn twist, we see that F_{n}(z_{1}) = z_{1} and F_{n}(z_{2}) = e^{2a_{n}}z_{2} . Hence, F_{n}(δ ∩ Ũ_{n}) is an arc connecting z_{1} and e^{2a_{n}}z_{2} in Ũ_{n}. Applying the similar argument to a subarc of δ in each component of π^{−1}(U_{n}), we see that

−1 < F_{n}(−1) < e^{2a_{n}} < F_{n}(1).

In particular, lim_{n→∞} F_{n}(1) = ∞. Therefore, for the cross ratio

[a, b, c, d] = (a − b)(c − d)(a − d)^{−1}(c − b)^{−1}

we have

[−1, 0, 1, ∞] = −1

and

λ_{n} = [F_{n}(−1), F_{n}(0), F_{n}(1), F_{n}(∞)] = [F_{n}(−1), 0, F_{n}(1), ∞] = \frac{F_{n}(−1)}{F_{n}(1)}.

Thus, we have lim_{n→∞} λ_{n} = 0. Therefore, the conformal modulus of a quadrilateral H with vertices F_{n}(−1), F_{n}(0), F_{n}(1) and F_{n}(∞) degenerates as n → ∞. Since the quasiconformal mapping F_{n} maps a quadrilateral H with vertices −1, 0, 1 and ∞ onto the quadrilateral H with vertices F_{n}(−1), F_{n}(0), F_{n}(1) and F_{n}(∞), we have K_{n} = K(F_{n}) → +∞.

Next, we shall show that d_{L}(p_{n}, p_{0}) → 0.

Let α be a closed geodesic on S_{0} . If α ∩ γ_{n} = ∅, then it is obvious that l_{S_{0}}(α) = l_{S_{0}}(f_{n}(α)).

Suppose that ^{#}(α ∩ γ_{n}) = m > 0. Since each point of α ∩ γ_{n} makes a Dehn twist, we have

(3)  l_{S_{0}}(f_{n}(α)) ≤ l_{S_{0}}(α) + ml_{S_{0}}(γ_{n}) = l_{S_{0}}(α) + 2ma_{n}.

On the other hand, from (2) we have

(4)  l_{S_{0}}(α) > mna_{n}.

Combining (3) and (4), we have

\frac{l_{S_{0}}(f_{n}(α))}{l_{S_{0}}(α)} < 1 + \frac{1}{2n} → 0,

as n → ∞. Since f_{n}^{−1} is also a Dehn twist, from the same argument as above, we have

\frac{l_{S_{0}}(α)}{l_{S_{0}}(f_{n}(α))} < 1 + \frac{1}{2n}.

Therefore, we note that lim_{n→∞} d_{L}(p_{n}, p_{0}) = 0 and complete the proof of Theorem 1.1.

3.2. Proof of Corollary 1.1. We use the same Riemann surface S_{0} and the same quasiconformal mappings f_{n} as in the proof of Theorem 1.1. Set F_{n} = f_{1} ∘ f_{2} ∘ ⋯ ∘ f_{n} and q_{n} = [S_{0}, F_{n}]. Then, we see that lim_{m,n→∞} d_{L}(q_{m}, q_{n}) = 0 and {q_{n}}_{n∈N} is a Cauchy sequence in T (S_{0}) with respect to the length spectrum distance d_{L}. However, it does not converge to any point in T (S_{0}) because F_{∞} = ∏_{n=1}^{∞} f_{n} is not homotopic to a quasiconformal mapping. Indeed, by using the same argument as in the proof of Lemma 3.1, we see that the maximal dilatation of any homeomorphism homotopic to F_{∞} is not finite. Hence, the distance d_{L} is not complete in T (S_{0}).

4. Proof of Theorem 1.2

It follows from Lemma 2.1 that d_{L}(p_{n}, p_{0}) → 0 when d_{T} (p_{n}, p_{0}) → 0. Thus, it suffices to show that d_{T} (p_{n}, p_{0}) → 0 as d_{L}(p_{n}, p_{0}) → 0.

By using Lemma 2.1 again, we see that the condition of Theorem 1.2 is quasiconformal invariant, that is, any Riemann surface which is quasiconformally equivalent to S_{0} satisfies the condition of Theorem 1.2 for some constant. Hence we may assume that p_{0} = [S_{0}, id].

Put p_{n} = [S_{n}, f_{n}] ∈ T (S_{0}) and assume that lim_{n→∞} d_{L}(p_{n}, p_{0}) = 0. Let 𝒞_{P} be the set of closed geodesics which are boundaries of some P_{k} in S_{0} . For each α ∈ 𝒞_{P} , there exist a closed geodesic in S_{n} homotopic to f_{n}(α). We denote the closed geodesic by [f_{n}(α)]. The set {[f_{n}(α)]}_{α∈𝒞_{P}} together with punctures of S_{n} gives a pants decomposition of S_{n}. Let P_{k}^{(n)} denote a pair of pants in the pants decomposition of S_{n} such that each boundary component is a closed geodesic homotopic to a component of f_{n}(P_{k}) or a puncture of ∂f_{n}(P_{k}).

From the definition of d_{L}, we have

(d_{L}(p_{n}, p_{0}))^{−1}l_{S_{0}}(α) ≤ l_{S_{n}}([f_{n}(α)]) ≤ d_{L}(p_{n}, p_{0})l_{S_{0}}(α)

for any α ∈ 𝒞_{P} . As M^{−1} ≤ l_{S_{0}}(α) ≤ M for α ∈ 𝒞_{P} , we see that {l_{S_{n}}([f_{n}(α)])}_{n=1}^{∞} converges to l_{S_{0}}(α) uniformly on 𝒞_{P} , that is, for any ε > 0, there exists n_{0} ∈ N such that if n ≥ n_{0} , then

(5)  |l_{S_{n}}([f_{n}(α)]) − l_{S_{0}}(α)| < ε

holds for any α ∈ 𝒞_{P} .

It is known that the lengths of boundaries determine the moduli of the pair of pants. Hence, if n is sufficiently large, then from (5) we verify that there exists a quasiconformal mapping g_{k,n} of P_{k} onto P_{k}^{(n)} with small dilatation. However, we need to find quasiconformal mappings g_{n} on the whole surface S_{0} such that lim_{n→∞} K[g_{n}] = 1. To obtain such mappings, we consider the “Fenchel–Nielsen coordinates” of the infinite-dimensional Teichmüller space T (S_{0}). The classical Fenchel–Nielsen coordinates are defined in the Teichmüller space of Riemann surfaces of finite type. We define the coordinates in T (S_{0}) by using an exhaustion of S_{0} .

We take a subregion S_{0}^{(m)} , m = 1, 2, …, of S_{0} satisfying the following conditions (after rearrangement of the numbers of {P_{k}}_{k=1}^{∞}).

(1) S_{0}^{(m)} = Int(⋃_{k=1}^{k(m)} \overline{P}_{k}) for some k(m) ∈ N,

(2) S_{0}^{(1)} ⊂ S_{0}^{(2)} ⊂ ⋯ ⊂ S_{0}^{(m)} ⊂ S_{0}^{(m+1)} ⊂ ⋯, and

(3) S_{0} = ⋃_{m=1}^{∞} S_{0}^{(m)} .

Similarly, we take an exhaustion {S_{n}^{(m)}}_{m=1}^{∞} of S_{n} from {P_{k}^{(n)}}_{k=1}^{∞}.

Let Ŝ_{0}^{(m)} , respectively Ŝ_{n}^{(m)} be the Nielsen extension of S_{0}^{(m)} , respectively S_{n}^{(m)} . Since (f_{n})_{∗}(π_{1}(S_{0}^{(m)})) = π_{1}(S_{n}^{(m)}), we see that there exists a quasiconformal mapping f_{n,m}: Ŝ_{0}^{(m)} → Ŝ_{n}^{(m)} such that

(f_{n,m})_{∗} ∘ (ι_{0}^{(m)})_{∗} = (ι_{n}^{(m)})_{∗} ∘ (f_{n})_{∗} | π_{1}(S_{0}^{(m)})

on π_{1}(S_{0}^{(m)}), where ι_{0}^{(m)} and ι_{n}^{(m)} are the natural inclusion maps from S_{0}^{(m)} to Ŝ_{0}^{(m)} and from S_{n}^{(m)} to Ŝ_{n}^{(m)} , respectively. A pair (Ŝ_{n}^{(m)}, f_{n,m}) gives a point in T (Ŝ_{0}^{(m)}). Obviously, if m′ > m, then

(6)  (f_{n,m′})_{∗} | π_{1}(S_{0}^{(m)}) = (f_{n,m})_{∗}.

Now, we consider the Fenchel–Nielsen coordinates (cf. [3]) on T (Ŝ_{0}^{(m)}) with respect to the pants decomposition given by {P_{k}}_{k=1}^{k(m)} . The Fenchel–Nielsen coordinates consist of length coordinates and twist coordinates. The length coordinates are the collection of lengths of boundaries of the pants decomposition and the twist coordinates are the collection of twist angles along the boundaries of the pants decomposition. From (6) or the construction of Ŝ_{n}^{(m)} , we see that if α ∈ S_{0}^{(m)} and m′ > m, then

lŜ_{n}^{(m)}([f_{n,m}(α)]) = lŜ_{n}^{(m′)}([f_{n,m′}(α)]) = l_{S_{n}}([f_{n}(α)]).

Thus, it follows from (5) that the length coordinates of p̂_{n}^{(m)} = [Ŝ_{n}^{(m)}, f_{n,m}] converge to those of p̂_{0}^{(m)} = [Ŝ_{0}^{(m)}, id] as n → ∞.

Similarly, it is also seen that the twist parameter along [f_{m,n}(α)] does not depend on m if [f_{n}(α)] ⊂ S_{n}^{(m)} . Thus, we may denote the twist parameter along [f_{n}(α)], α ∈ 𝒞_{P} , by θ_{n}(α).

Lemma 4.1. The sequence {θ_{n}(α)}_{n∈N} converges to θ_{0}(α), the twist parameter along α of p_{0} ∈ T (S_{0}), uniformly on 𝒞_{P} . Namely, for any ε > 0 there exists n_{0} ∈ N such that if n ≥ n_{0} , then

|θ_{n}(α) − θ_{0}(α)| < ε

holds for any α ∈ 𝒞_{P} .

Proof. For any closed geodesic α ∈ 𝒞_{P} , the following two cases occur (Figure 4): (A) α is a non-dividing curve and is contained in a subregion T (α) of S_{0} of genus

one with one geodesic boundary curve in 𝒞_{P} . (B) α is a dividing curve and is contained in a planar subregion of S_{0} bounded

by four geodesic curves in 𝒞_{P} .

Figure 4.

We shall show the statement of the lemma in Case (A) since the proof is similar in Case (B).

Let α^{∗} be the shortest simple closed geodesic in all closed curves intersecting α (Figure 5). From the second condition in Theorem 1.2 we verify that there exists a constant C_{1} = C_{1}(M) > 0 depending only on M such that

(7)  0 < C_{1}^{−1} < l_{S_{0}}(α^{∗}) < C_{1}.

Indeed, since l_{S_{0}}(α) < M, the collar theorem (cf. [2, Chapter 4]) guarantees that the geodesic α ⊂ S_{0} has a collar with some width depending only on M. Hence the existence of a positive lower bound of l_{S_{0}}(α^{∗}) is obvious and the lower bound depends only on M. On the other hand, since the hyperbolic length of any boundary curve of any P_{k} is less than M, the hyperbolic distance between any two boundary curves of P_{k} is less than some constant C_{1} depending only on M. Therefore, it is easy to see l_{S_{0}}(α^{∗}) < C_{1} .

Next, we show that there exists a constant Θ depending only on M such that

(8)  |θ_{n}(α)| < Θ

holds for any α ∈ 𝒞_{P} and for any n ∈ N.

Figure 5.

Indeed, if such a constant does not exist, then there exist sequences {n_{t}}_{t∈N} in N and {α_{n_{t}}}_{t∈N} in 𝒞_{P} such that

lim_{t→∞} |θ_{n_{t}}(α_{n_{t}})| = +∞.

Considering that l_{S_{n_{t}}} (α_{n_{t}}) > M^{−1} > 0, we have that

lim_{t→∞} l_{S_{n_{t}}} (f_{n_{t}}(α_{n_{t}}^{∗})) = +∞.

Therefore, from (7) we see that

d_{L}(p_{n_{t}}, p_{0}) ≥ log \frac{l_{S_{n_{t}}} (f_{n_{t}}(α_{n_{t}}^{∗}))}{l_{S_{0}}(α_{n_{t}}^{∗})} → +∞

as t → ∞, and we have a contradiction.

Now, we prove the lemma. Assume that {θ_{n}(α)}_{n∈N} does not uniformly converge to θ_{0}(α). Then, there exist a constant ε_{0} > 0, a sequence {n_{t}}_{t∈N} in N and {α_{n_{t}}}_{t∈N} in 𝒞_{P} such that

(9)  |θ_{n_{t}}(α_{n_{t}}) − θ_{0}(α_{n_{t}})| ≥ ε_{0}

holds for any n_{t}. From this inequality and (8), we can find a simple closed geodesic β in S_{0} such that

|l_{S_{n_{t}}} (f_{n_{t}}(β)) − l_{S_{0}}(β)| > δ

hold for some δ > 0 and for all n_{t} (cf. [8]). Hence, we have

|\frac{l_{S_{n_{t}}} (f_{n_{t}}(β))}{l_{S_{0}}(β)} − 1| > \frac{δ}{l_{S_{0}}(β)} > 0.

It contradicts lim_{t→∞} d_{L}(p_{n_{t}}, p_{0}) = 0 and thus we complete the proof of the lemma.

Now, the proof of Theorem 1.2 is immediate. From the boundedness of the length and the angle parameters and from the uniform convergence of them, we can construct quasiconformal mappings from Ŝ_{0}^{(m)} onto Ŝ_{n}^{(m)} with small dilatations. More precisely, for any ε > 0 there exist n_{0} ∈ N and quasiconformal mappings g_{n,m} from Ŝ_{0}^{(m)} onto Ŝ_{n}^{(m)} , m = 1, 2, …, such that

(1) g_{n,m} is homotopic to f_{n,m}.

(2) If n ≥ n_{0} , then the maximal dilatation K[g_{n,m}] of g_{n,m} is less than (1 + ε). By taking the limit of {g_{n,m}}_{m∈N} as m → ∞, we have a quasiconformal mapping g_{n} from S_{0} onto S_{n} with K[g_{n}] ≤ 1 + ε. From the construction, [S_{n}, g_{n}] = [S_{n}, f_{n}] = p_{n} and we conclude that lim_{n→∞} d_{T} (p_{n}, p_{0}) = 0 as we desired.

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Received 4 December 2002