THE ORDER OF PERIODIC ELEMENTS OF TEICHMÜLLER MODULAR GROUPS

EGE FUJIKAWA

Abstract. We consider a quasiconformal automorphism of a Riemann surface, which fixes the homotopy class of a simple closed geodesic. Under certain conditions on the injectivity radius of the surface and bounds on the dilatation of the map, the automorphism induces a periodic element of the Teichmüller modular group. We may also estimate the order of the period.

1. Introduction

Let R be an arbitrary Riemann surface with possibly infinitely generated fundamental group. An element χ of the Teichmüller modular group Mod(R) is induced by a quasiconformal automorphism f of R. We would like to determine when the order of χ is finite. When f is a conformal automorphism of R, then the element χ of Mod(R) induced by f fixes the base point of the Teichmüller space T (R). In [3], we proved that, for a Riemann surface R with non-abelian fundamental group, a conformal automorphism f of R has finite order if and only if f fixes either a simple closed geodesic, a puncture or a point on R. In each case, we obtained a concrete estimate for the order of f in terms of the injectivity radius on R. One of our results is the following. For the definition of the upper bound condition, see the next section.

Theorem 1.1 ([3], [4]). Let R be a hyperbolic Riemann surface with nonabelian fundamental group. Suppose that R satisfies the upper bound condition for a constant M > 0 and a connected component R_{M}^{∗} of R_{M}. Let f be a conformal automorphism of R such that f(c) = c for a simple closed geodesic c on R with c ⊂ R_{M}^{∗} and l(c) = l > 0. Then the order n of f satisfies

n < (e^{M} − 1) cosh(l/2).

The purpose of this paper is to extend Theorem 1.1 to a quasiconformal automorphism f. One of the difficulties that arise is that the element χ ∈ Mod(R) induced by f need not have a fixed point on T (R). However, we will show that if the maximal dilatation of f is smaller than some constant, then χ is periodic.

2000 Mathematics Subject Classification. Primary 30F60; Secondary 30C62.

The author would like to express her gratitude to Professor Katsuhiko Matsuzaki for his valuable suggestions.

2. Statement of Theorem

Let ℍ be the upper half-plane equipped with the hyperbolic metric |dz|/Im z. Throughout this paper, we assume that a Riemann surface R is hyperbolic. Namely, it is represented as ℍ/Γ for some torsion-free Fuchsian group Γ acting on ℍ. Furthermore, we also assume that R has a non-abelian fundamental group. The hyperbolic distance on ℍ is denoted by d, and the hyperbolic length of a curve c on R by l(c). For the axis L of a hyperbolic element of the Fuchsian group Γ, we denote by π_{Γ}(L) the projection of L to ℍ/Γ. When there is no fear of confusion, we denote this simply by π(L). Also, for a quasiconformal automorphism f̃ of ℍ, we denote by f̃(L)_{∗} the geodesic having the same end points as those of f̃(L).

We recall the definition of Teichmüller spaces and Teichmüller modular groups. Fix a Riemann surface R. We say that two quasiconformal maps f_{1} and f_{2} on R are equivalent if f_{2} ∘ f_{1}^{−1} is homotopic to a conformal map of f_{1}(R) onto f_{2}(R). The reduced Teichmüller space T (R) with the base Riemann surface R is the set of all equivalence classes [f] of quasiconformal maps f on R. The Teichmüller distance d_{T} on T (R) is defined by d_{T} ([f_{1}], [f_{2}]) = log K(g), where g is an extremal quasiconformal map in the sense that its maximal dilatation K(g) is minimal in the homotopy class of f_{2} ∘ f_{1}^{−1}. This is a complete metric on T (R). The reduced Teichmüller modular group Mod(R) of R is a group of the homotopy classes [h] of quasiconformal automorphisms h of R. Each element [h] of Mod(R) induces an automorphism of T (R) by [f] ↦ [f ∘ h^{−1}], which is an isometry with respect to d_{T}.

We now make a couple of definitions given in terms of the hyperbolic geometry of Riemann surfaces.

Definition. For a constant M > 0, we define R_{M} to be the set of points p ∈ R for which there exists a non-trivial simple closed curve c_{p} passing through p with l(c_{p}) < M. The set R_{ϵ} is called the ϵ-thin part of R if ϵ > 0 is smaller than the Margulis constant. Furthermore, a connected component of the ϵ-thin part corresponding to a puncture is called the cusp neighborhood.

Remark. The injectivity radius at a point p ∈ R is the supremum of radii of embedded hyperbolic discs centered at p. Note that R_{M} coincides with the set of those points having the injectivity radius less than M/2.

Definition. We say that R satisfies the lower bound condition if there exists a constant ϵ > 0 such that ϵ-thin part of R consists of only cusp neighborhoods or neighborhoods of geodesics which are homotopic to boundary components. We also say that R satisfies the upper bound condition if there exist a constant

PERIODIC ELEMENTS OF TEICHMÜLLER MODULAR GROUPS 3

M > 0 and a connected component R_{M}^{∗} of R_{M} such that the homomorphism of π_{1}(R_{M}^{∗}) to π_{1}(R) induced by the inclusion map of R_{M}^{∗} into R is surjective.

Remark. The lower and upper bound conditions are quasiconformally invariant notions (see [5, Lemma 8]).

We shall obtain a range of maximal dilatations of quasiconformal automorphisms f inducing periodic elements χ ∈ Mod(R). Moreover, we get a concrete estimate for the order of χ.

Theorem 2.1. Let R be a Riemann surface satisfying the lower bound condition for a constant ϵ > 0 as well as the upper bound condition for a constant M > 0 and a connected component R_{M}^{∗} of R_{M}. For a given constant l > 0, there exists a constant K_{0} = K_{0}(ϵ, M, l) > 1 depending only on ϵ, M and l that satisfies the following: Let f be a quasiconformal automorphism of R such that f(c) is homotopic to c for a simple closed geodesic c on R with c ⊂ R_{M}^{∗} and l(c) = l. Suppose K(f) < K_{0}. Then there exists a positive integer n ≤ N_{0} such that f^{n} is homotopic to the identity. Here

N_{0} = N_{0}(M, l) = −\frac{l}{log(tanh(D + 13.5))},

D = D(M, l) = \{ \begin{matrix}2 arccosh (\frac{sinh(M/2)}{sinh(l/2)}) + M & if l ≤ M, \\ M & if l ≥ M. \end{matrix}

In particular, when K(f) = 1, we have the following:

Theorem 2.2. Let R be a Riemann surface satisfying the upper bound condition for a constant M > 0 and a connected component R_{M}^{∗} of R_{M} as well as the lower bound condition. Let f be a conformal automorphism of R such that f(c) = c for a simple closed geodesic c on R with c ⊂ R_{M}^{∗} and l(c) = l > 0. Then the order n of f satisfies

n ≤ −\frac{l}{log(tanh(D/2))},

where D = D(M, l) is the same constant as in Theorem 2.1.

Note that for M ≥ arcsinh(2/\sqrt{3}) = 0.98 ⋯ and every l > 0, we have

−\frac{l}{log(tanh(M/2))} < (e^{M} − 1) cosh(l/2).

Here the constant arcsinh(2/\sqrt{3}) is the smallest possible value of M for which R satisfies the upper bound condition (see [6]). Hence when l ≥ M, the upper bound of the order of f obtained in Theorem 2.2 is smaller than that in Theorem 1.1. However, when l < M, the estimate in Theorem 1.1 is still better than that in Theorem 2.2 for all sufficiently small l. In fact, (e^{M} −1) cosh(l/2) converges to e^{M} − 1 as l → 0, while −l/(log(tanh(D/2))) diverges to +∞.

In connection with Theorems 2.1 and 2.2, we would like to mention the result about the discreteness of the orbit of a certain subgroup of the Teichmüller modular group.

Proposition 2.3 ([5]). Let R be a Riemann surface satisfying the lower and upper bound conditions. For a simple closed geodesic c on R, let G be a subgroup of Mod(R) such that g(c) is homotopic to c for every [g] ∈ G. Then for every point p ∈ T (R), the orbit G(p) of p is a discrete subset in T (R). Furthermore, for any point p ∈ T (R), there exist only finitely many elements [g] in G that fix p.

3. Proof of Theorems

For a proof of these theorems, we first prove some properties on the hyperbolic geometry of Riemann surfaces.

Proposition 3.1. Let R = ℍ/Γ be a Riemann surface satisfying the upper bound condition for a constant M > 0 and a connected component R_{M}^{∗} of R_{M}. Suppose that L is the axis of a hyperbolic element of Γ such that the projection π(L) is a simple closed geodesic c on R with c ⊂ R_{M}^{∗} and l(c) = l > 0. Then there exists an axis L′ of a hyperbolic element of Γ such that L ∩ L′ = ∅, d(L, L′) ≤ D and π(L′) = π(L). Here D = D(M, l) is the same constant as in Theorem 2.1.

Proof. First we assume that l > M. Since c ⊂ R_{M}^{∗}, there exists a non-trivial simple closed curve α passing through p ∈ c with l(α) < M. It follows from the assumption l > M that α is not homotopic to c, which implies that there exists an axis L′ (≠ L) such that π(L′) = c and d(L, L′) < M.

Next we assume that l ≤ M. We further assume that there exists an annular neighborhood A(c) of c with width ω(c), where

ω(c) = arccosh (\frac{sinh(M/2)}{sinh(l/2)}) .

Then, for any q ∈ ∂A(c), the boundary of A(c), the shortest simple closed curve γ passing through q and homotopic to c has length M.

Indeed, we may assume that L = {iy | y > 0}, and q̃ = e^{iθ} and q̃′ = e^{l+iθ} are lifts of q to ℍ. Then, by the equality (7.20.3) in [2], we have

\frac{1}{sin θ} = \frac{1}{cos(π/2 − θ)} = cosh d(q̃, L) = cosh ω(c) = \frac{sinh(M/2)}{sinh(l/2)}.

Thus, by Theorem 7.2.1 in [2], we see that

sinh \frac{1}{2}d(q̃, q̃′) = \frac{|q̃ − q̃′|}{2 (Imq̃Imq̃′)^{1/2}} = \frac{e^{l} − 1}{2 e^{l/2} sin θ} = \frac{sinh(l/2)}{sin θ} = sinh \frac{M}{2} ,

which implies that l(γ) = d(q̃, q̃′) = M.

We can take a point q_{0} ∈ ∂A(c) such that q_{0} ∈ R_{M}^{∗}. Indeed, otherwise, ∂A(c) ∩ R_{M}^{∗} = ∅. Since c ⊂ R_{M}^{∗}, this means that R_{M}^{∗} is an annular neighborhood of c, contradicting the upper bound condition.

By the definition of R_{M}, there exists a non-trivial simple closed curve β passing through q_{0} with l(β) < M. By the consideration above, we see that the curve β is not homotopic to c. Hence there exists an axis L′ (≠ L) such that π(L′) = c and d(L, L′) < 2 ω(c) + M.

Finally, we assume that l ≤ M and that the width of the maximal annular neighborhood A(c) of c is less than ω(c). Then there exists an axis L′ (≠ L) such that π(L′) = c and d(L, L′) < 2 ω(c).

We now estimate the number of axes satisfying Proposition 3.1.

Definition. For an element γ of a Fuchsian group, we say that two axes L_{1} and L_{2} are γ-equivalent if γ^{n}(L_{1}) = L_{2} for some n ∈ ℤ.

Proposition 3.2. Let R = ℍ/Γ be a Riemann surface and D_{0} > 0 a constant. Furthermore, let L be the axis of a hyperbolic element γ ∈ Γ such that the projection π(L) is a simple closed geodesic c on R with l(c) = l > 0. Let S be the set of axes L′ of hyperbolic elements of Γ satisfying the following: (i) L ∩ L′ = ∅, (ii) d(L, L′) ≤ D_{0}, (iii) π(L′) = c and (iv) there exists an arc α connecting L and L′ whose projection to R has no intersection with c except at the end points. Then the number of γ-equivalence classes of axes in S is dominated by

−\frac{l}{log(tanh(D_{0}/2))}.

Proof. We may assume that L = {iy | y > 0}. We take θ_{0} (0 < θ_{0} < π/2) so that cosh D_{0} = (cos θ_{0})^{−1} and set θ = π/2 − θ_{0}. Furthermore, we set

T_{+} = {re^{iθ} | 1 ≤ r < e^{l}} and T_{−} = {re^{i(π−θ)} | 1 ≤ r < e^{l}}.

Then d(L, T_{+}) = D_{0} and d(L, T_{−}) = D_{0}. To estimate the number of γequivalence classes of elements in S, we have only to consider the maximal number n of disjoint axes L′ that are tangent to T_{+} or T_{−}.

Let C be the Euclidean circle on ℂ that is tangent to the segment T_{+} and has center a > 0 with radius r. Then r = a sin θ, and the circle C passes through two points,

x_{1} = (1 − sin θ)a and x_{2} = (1 + sin θ)a.

The ratio of these points is given by

s = \frac{x_{2}}{x_{1}} = \frac{1 + sin θ}{1 − sin θ} = \frac{1 + cos θ_{0}}{1 − cos θ_{0}} = \frac{cosh D_{0} + 1}{cosh D_{0} − 1} = \frac{1}{(tanh(D_{0}/2))^{2}} .

Hence it is easy to see that

n ≤ 2 · \frac{l}{log s} = −\frac{l}{log(tanh(D_{0}/2))}.

The following proposition gives a relationship between the hyperbolic distance of two axes and that of their images under a quasiconformal map.

Proposition 3.3 ([1]). Let f be a K-quasiconformal automorphism of ℍ. Then there exists a constant C = C(K) > 0 depending only on K such that, for any two geodesics L_{1} and L_{2} in ℍ, the inequality

K^{−1} · d(L_{1}, L_{2}) − C ≤ d(f(L_{1})_{∗}, f(L_{2})_{∗}) ≤ K · d(L_{1}, L_{2}) + C

holds. The constant C(K) satisfies C(K) → 0 as K → 1, and may be taken to be

(1/2)arccosh (2^{−(K−1)^{2}}e^{6(K+1)^{2}\sqrt{K−1}}) .

The following proposition gives a sufficient condition for the maximal dilatations of quasiconformal maps to be bounded away from one.

Proposition 3.4 ([4]). Let R = ℍ/Γ be a Riemann surface. Suppose that R satisfies the lower bound condition for a constant ϵ > 0 as well as the upper bound condition for a constant M > 0 and a connected component R_{M}^{∗} of R_{M}. Let B > 0 and l > 0 be constants. Then there exists a constant A_{0} = A_{0}(ϵ, M, B, l) > 1 depending only on ϵ, M, B, l and satisfying the following conditions: Given a quasiconformal automorphism f of R, suppose that there exist three disjoint axes L_{i} (i = 1, 2, 3) of hyperbolic elements of Γ such that

1. their projections π(L_{i}) on R are simple closed geodesics c_{i} (i = 1, 2, 3) with c_{i} ⊂ R_{M}^{∗} and l(c_{i}) ≤ l,

2. d(L_{1}, L_{2}) ≤ B,

3. f̃(L_{1})_{∗} = L_{1}, f̃(L_{2})_{∗} = L_{2}, f̃(L_{3})_{∗} ≠ L_{3} for a lift f̃ of f to ℍ. Then K(f) ≥ A_{0}.

We now prove our theorems.

Proof of Theorem 2.1. We set B := D = D(M, l) in Proposition 3.4 and let A_{0} = A_{0}(ϵ, M, l) > 1 be a constant depending only on ϵ, M and l obtained in Proposition 3.4. Setting A = min{A_{0}, 2}, we prove the statement for K_{0} = A^{1/(N_{0}+1)}. Namely, we show that, if K(f) < K_{0}, then there exists an integer n ≤ N_{0} such that f^{n} is homotopic to the identity.

Let Γ be a Fuchsian model of R. Furthermore let L_{1} be an axis such that π(L_{1}) = c and γ_{1} the primitive hyperbolic element of Γ with axis L_{1}. By applying Proposition 3.1 to L_{1}, we see that there exists an axis L_{2} of a hyperbolic element γ_{2} of Γ such that L_{1} ∩ L_{2} = ∅, d(L_{1}, L_{2}) ≤ D and π(L_{1}) = π(L_{2}).

Let f̃ be a lift of f to ℍ satisfying f̃(L_{1})_{∗} = L_{1}. Since K(f) < K_{0} = A^{1/(N_{0}+1)}, we have K(f^{k}) < A for k ≤ N_{0} + 1. Then, by Proposition 3.3,

(1) d(L_{1}, f̃^{k}(L_{2})_{∗}) = d(f̃^{k}(L_{1})_{∗}, f̃^{k}(L_{2})_{∗}) ≤ A · d(L_{1}, L_{2}) + C(A)
≤ 2D + C(2) = 2D + (1/2)arccosh(e^{54}/2)
≤ 2D + 27

for all k ≤ N_{0} + 1.

We consider the set S_{0} of all axes L′ of hyperbolic elements of Γ satisfying the following conditions: (i) L_{1}∩L′ = ∅, (ii) d(L_{1}, L′) ≤ 2D+27, (iii) π(L′) = c and (iv) there exists an arc α connecting L_{1} and L′ such that the projection of α to R has no intersection with c except at the end points. We see that the set S′ = {f̃^{k}(L_{2})_{∗}}_{k=1}^{N_{0}+1} is contained in S_{0}. Indeed, by the proof of Proposition 3.1, the axis L_{2} satisfies the property (iv), and since f̃^{k} is a homeomorphism, the axes f̃^{k}(L_{2})_{∗} satisfy the same property. The other properties (i), (ii), (iii) are also satisfied.

By Proposition 3.2, the number of γ_{1}-equivalence classes of elements in S_{0} is dominated by N_{0}. Hence there exist at least two elements in S′, say f̃^{m_{1}}(L_{2})_{∗} and f̃^{m_{2}}(L_{2})_{∗} (1 ≤ m_{1} < m_{2} ≤ N_{0} + 1), that are γ_{1}-equivalent to each other. Thus there exists j ∈ ℤ such that γ_{1}^{j} ∘ f̃^{n}(L_{2})_{∗} = L_{2}, where n = m_{2} − m_{1} (≤ N_{0}). With this n, we will prove that f^{n} is homotopic to the identity. We set F = γ_{1}^{j} ∘ f̃^{n}, which is a lift of f^{n} to ℍ.

Suppose to the contrary that f^{n} is not homotopic to the identity. We set χ(γ) = F ∘ γ ∘ F^{−1} for γ ∈ Γ. Then there exists γ_{3} ∈ Γ such that χ(γ_{3}) ≠ γ_{3}. Setting γ′_{i} = γ_{3} ∘ γ_{i} ∘ γ_{3}^{−1} for i = 1, 2, we claim that either χ(γ′_{1}) ≠ γ′_{1} or χ(γ′_{2}) ≠ γ′_{2} is satisfied. Suppose that both χ(γ′_{1}) = γ′_{1} and χ(γ′_{2}) = γ′_{2} are satisfied. Since χ(γ_{i}) = γ_{i}, we have β ∘ γ_{i} ∘ β^{−1} = γ_{i} (i = 1, 2), where β = γ_{3}^{−1} ∘ χ(γ_{3}). Thus, β fixes all fixed points of γ_{1} and γ_{2}. Since γ_{1} and γ_{2} are non-commutative, the Möbius transformation β fixes four points and must be the identity. This contradicts that χ(γ_{3}) ≠ γ_{3}.

Hence either F (γ_{3}(L_{1}))_{∗} ≠ γ_{3}(L_{1}) or F (γ_{3}(L_{2}))_{∗} ≠ γ_{3}(L_{2}) is satisfied, and we may assume without loss of generality that F (γ_{3}(L_{1}))_{∗} ≠ γ_{3}(L_{1}). Since π(γ_{3}(L_{1})) = π(L_{1}) = c, we can apply Proposition 3.4 to the lift F of f^{n} and to the three axes L_{1}, L_{2} and γ_{3}(L_{1}). Then we have K(f^{n}) ≥ A_{0}, a contradiction, since we assumed K(f^{n}) < A ≤ A_{0}. Hence if K(f) < A^{1/(N_{0}+1)}, then f^{n} is homotopic to the identity.

Proof of Theorem 2.2. In the proof of Theorem 2.1, we can replace the inequality (1) with

d(L_{1}, f̃^{k}(L_{2})_{∗}) = d(f̃^{k}(L_{1})_{∗}, f̃^{k}(L_{2})_{∗}) = d(L_{1}, L_{2}) = D.

Hence we have only to replace the constant 2D+27 with D in Theorem 2.1.

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Department of mathematical and Computing Sciences Tokyo Institute of Technology Oh-okayama Meguro-ku Tokyo, 152-8552 Japan

 E-mail address: ege.fujikawa@is.titech.ac.jp