THE ORDER OF CONFORMAL AUTOMORPHISMS OF RIEMANN SURFACES OF INFINITE TYPE

EGE FUJIKAWA

Abstract. Let R be a Riemann surface of infinite type such that the injectivity radius at any point in R is less than a positive constant M, and f a conformal automorphism of R fixing a compact subset in R. We show that the order of f is less than a certain constant depending on M.

1. Introduction

On a compact Riemann surface R of genus g ≥ 2, it is known that the order of a conformal automorphism of R is not greater than 2(2g + 1) (Wiman cf. [4, p.96]). Since the hyperbolic area of R is 4π(g − 1), the injectivity radius at any point in R is not greater than a constant depending only on g. This means that the order of a conformal automorphism of R is estimated by the supremum of the injectivity radii which is taken over all points in R. We extend this result to the case of Riemann surfaces which are not necessarily of finite type. That is, for any hyperbolic Riemann surface R such that the injectivity radius at any point in R is less than a positive constant M, if a conformal automorphism f of R fixes a compact subset in R, then the order of f is estimated by M. Note that, in the case that R has the non-abelian fundamental group, a conformal automorphism f of R fixes a compact subset on R if and only if f has the finite order.

2. Main theorems

Let H be the upper-half plane equipped with the hyperbolic metric dλ = |dz|/Imz. We say that a Riemann surface R is hyperbolic if it is represented by H/Γ for a torsion-free Fuchsian group Γ acting on H. The hyperbolic distance on H or on R is denoted by d(,). The injectivity radius at p ∈ R is the supremum of radii r of embedded hyperbolic discs centered at p. The supremum of the injectivity radii is uniformly bounded from below.

Date: July 13, 2002.

Proposition 1. ([7]) For any hyperbolic Riemann surface R, the supremum of the injectivity radii which is taken over all points in R is not less than arcsinh(2/\sqrt{3}).

Before we state the main theorems, we note the following fact.

Proposition 2. Let R = H/Γ, where Γ is a Fuchsian group which is not necessarily torsion-free, and f a conformal automorphism of R with finite order n > 1. Then f fixes either a non-trivial simple closed geodesic, a puncture, a point or a cone point on R.

Proof. Let f̃ be a lift of f to H which is an element of PSL_{2}(R). Since f has the finite order n, we see that f̃^{n} belongs to Γ and that f̃^{m} (1 ≤ m < n) does not belong to Γ. If f̃^{n} is parabolic, then f̃ is parabolic. Hence f fixes a puncture on R. If f̃^{n} is the identity, then f̃ is elliptic with the fixed point p̃ ∈ H. Hence f fixes the point on R which is the projection of p̃. If f̃^{n} is elliptic, then f̃ is elliptic with the fixed point p̃ ∈ H. Hence f fixes the cone point on R which is the projection of p̃. Further, if f̃^{n} is hyperbolic, then f̃ is hyperbolic. Hence f fixes a non-trivial closed geodesic c_{∗} on R. In this case, we prove that f fixes either a non-trivial simple closed geodesic, a puncture, a point or a cone point on R. We consider the quotient R̂ = R/⟨f⟩ by a cyclic group ⟨f⟩ and its Fuchsian model Γ̂ = ⟨Γ, f̃⟩. Then ĉ_{∗} = c_{∗}/⟨f⟩ is a non-trivial closed geodesic on R̂. There exists a subset ĉ′ of ĉ_{∗} such that ĉ′ is a non-trivial simple closed curve and it corresponds to a conjugacy class of an element γ in Γ̂ − Γ. Indeed, suppose that there is no such curves. That is, suppose that every non-trivial simple closed curve ĉ_{i} ⊂ ĉ_{∗} corresponds to a conjugacy class of an element γ_{i} in Γ. Since Γ is a normal subgroup of Γ̂, the curve ĉ_{∗} corresponds to a conjugacy class of the composition of some elements in {γ_{i}, γ_{i}^{−1}}_{i}, which is in Γ. However, this is a contradiction. Let c′ ⊂ c_{∗} be a connected component of the preimage of ĉ′. Then c′ is a simple closed curve fixed by f. If γ is hyperbolic, then there exists a simple closed geodesic c′_{∗} that is homotopic to c′, and it is fixed by f. If γ is parabolic, then ĉ′ surround a puncture p̂ on R̂. Then c′ surround a puncture p which is a lift of p̂, and f fixes p. If γ is elliptic, then ĉ′ surround a cone point p̂ on R̂. In case c′ is trivial, then it surrounds a point p which is a lift of p̂, and f fixes p. In case c′ is non-trivial, then it surrounds a cone point p which is a lift of p̂, and f fixes p.

This proposition immediately gives the following well known result.

Corollary 1. Let R be a compact Riemann surface, and f a conformal automorphism of R. If f is irreducible, then f has a fixed point on R.

Assume that R has the non-abelian fundamental group. Then the action of Aut(R) is properly discontinuous (cf. [6, Theorem X.48]). Thus, if a conformal automorphism f of R fixes either a non-trivial simple closed geodesic, a puncture, a point or a cone point on R, then f has the finite order. Hence, by Proposition 2, f has the finite order if and only if f fixes either a non-trivial simple closed geodesic, a puncture, a point or a cone point on R. In each case, we estimate the order of f concretely in terms of the injectivity radius on R.

Theorem 1. (hyperbolic case) Let R be a hyperbolic Riemann surface. Suppose that there exists a positive constant M such that the injectivity radius at any point in R is less than M. Let c be a non-trivial simple closed geodesic on R whose length is ℓ. Further, let f be a conformal automorphism of R that satisfies f(c) = c, and n the order of f. Then

n < (e^{2M} − 1) cosh(ℓ/2).

Theorem 2. (parabolic case) Let R be a hyperbolic Riemann surface with a puncture p. Suppose that there exists a positive constant M such that the injectivity radius at any point in R is less than M. Let f be a conformal automorphism of R that satisfies f(p) = p, and n the order of f. Then

n < e^{2M} − 1.

Theorem 3. (elliptic case) (i) Let R be a hyperbolic Riemann surface. For a positive constant M, let p be a point in R at which the injectivity radius is less than M. Let f be a conformal automorphism of R that satisfies f(p) = p, and n the order of f. Then

n < 2π cosh M.

(ii) Let Γ be a Fuchsian group which contains an elliptic element γ with order m > 1. Let p̃ be the fixed point of γ, and p its projection to the orbifold R = H/Γ. Suppose that there exists a positive constant M such that the injectivity radius at any point in R is less than M. Let f be a conformal automorphism of R that satisfies f(p) = p, and n the order of f. Then

n < (e^{2M} − 1)\frac{π}{m} (\frac{1}{sin^{2} \frac{π}{m}} + \frac{1}{sinh^{2} M} )^{\frac{1}{2}} .

Remark 1. In the assumptions of Theorems 1, 2 and 3 (ii), the injectivity radius at any point in R is uniformly bounded from above. Then R must have the non-abelian fundamental group. Further, in the assumption of Theorem 3 (i), the conformal automorphism f fixes a point on R at which the injectivity radius is bounded. Thus, if the fundamental group of R is abelian, then the order of f is not greater than 2. Hence we may assume that R has the non-abelian fundamental group.

Remark 2. By Proposition 1, the constant M in Theorems 1 and 2 satisfies M ≥ arcsinh(2/\sqrt{3}).

Remark 3. The upper bound of the order of f obtained in Theorem 2 is the limiting case of that in Theorem 1 as ℓ → 0. It is also the limiting case of that in Theorem 3 (ii) as m → ∞.

3. The collar, cusp and cone lemmas

The proofs of the theorems are based on the collar, cusp and cone lemmas (cf. [3] and [5]).

Definition 1. A subset S ⊂ H is said to be precisely invariant under a subgroup Γ_{S} of a Fuchsian group Γ, if γ(S) = S for all γ ∈ Γ_{S} and γ(S) ∩ S = ∅ for all γ ∈ Γ − Γ_{S}.

Collar Lemma. Let Γ be a Fuchsian group (which is not necessarily torsion-free) acting on H, and L an axis of a hyperbolic element γ ∈ Γ whose translation length is less than ℓ. Assume that there is no fixed points of elements in Γ on L. Then a collar

C(L) = {z ∈ H | d(z, L) ≤ ω(ℓ)}

is precisely invariant under the cyclic subgroup ⟨γ⟩ generated by γ, where sinh ω(ℓ) = (2 sinh(ℓ/2))^{−1}. Equivalently, the boundaries ∂C(L) of C(L) and the real axis make an angle θ, where tan θ = 2 sinh(ℓ/2).

Cusp Lemma. Let Γ be a Fuchsian group (which is not necessarily torsion-free) acting on H. Suppose that Γ contains a parabolic element γ with the fixed point ζ. Then there exists a horoball C(ζ) tangent at ζ such that C(ζ) is precisely invariant under the cyclic subgroup ⟨γ⟩ generated by γ, and that the area of cusp neighborhood C(ζ)/⟨γ⟩ is 1.

Cone Lemma. Let Γ be a Fuchsian group acting on H. Suppose that a point p ∈ H is fixed by an elliptic element γ ∈ Γ whose order is n > 2. Then a hyperbolic disc

C(p) = {z ∈ H | d(z, p) < ρ(n)}

is precisely invariant under the cyclic subgroup ⟨γ⟩ generated by γ, where for 2 < n < 7, ρ(n) is a constant μ ≈ .075, and for n ≥ 7, cosh ρ(n) = (2 sin(π/n))^{−1}.

4. Proofs of the theorems

In this section, we prove the theorems. First we give a proof of Theorem 1 which is based on Collar Lemma. The proof follows from the fact that there exists a wider collar of the non-trivial simple closed geodesic c, in proportion to the order of a conformal automorphism f that fixes c.

Proof of Theorem 1: Let Γ be a Fuchsian model of R, and f̃ a lift of f which is a hyperbolic element in PSL_{2}(R). Note that f̃^{n} is a hyperbolic element in Γ which is corresponding to c. We consider the quotient R̂ = R/⟨f⟩ by the cyclic group ⟨f⟩ and its Fuchsian model Γ̂ = ⟨Γ, f̃⟩. Then ĉ = c/⟨f⟩ is a non-trivial simple closed geodesic on R̂ whose length is ℓ/n. Since f̃ is corresponding to ĉ, we may assume that f̃(z) = exp(ℓ/n)z with the axis L = {iy | y > 0}. Applying Collar Lemma for Γ̂ and γ = f̃, we can take a collar

C̃(L) = {re^{iθ} ∈ H | 0 < r, θ_{0} < θ < π − θ_{0}}

so that it is precisely invariant under ⟨f̃⟩ ⊂ Γ̂, where

tan θ_{0} = 2 sinh(ℓ/2n).

In particular, γ(C̃(L)) ∩ C̃(L) = ∅ for any γ ∈ Γ − ⟨f̃^{n}⟩. Then we can take a tubular neighborhood C(c) = C̃(L)/⟨f̃^{n}⟩ of c on R whose fundamental region is

A = {re^{iθ} ∈ H | 1 < r < e^{ℓ}, θ_{0} < θ < π − θ_{0}}.

We may assume that d(c, ∂C(c)) = ω(ℓ/n) > M. Indeed, if

ω(ℓ/n) = arcsinh (2 sinh \frac{ℓ}{2n})^{−1} ≤ M,

then we have

n ≤ ℓ{2 arcsinh(2 sinh M)^{−1}}^{−1} = N(M, ℓ).

By Remark 2, we have M ≥ arcsinh(2/\sqrt{3}) > 0.987. Then

{arcsinh(2 sinh M)^{−1}}^{−1} < e^{2M} − 1.

Indeed, this inequality is equivalent to

2 sinh(e^{2M} − 1)^{−1} sinh M < 1.

For x < 1, we have sinh x < 3x/2. Since M > 0.987, we have (e^{2M} − 1)^{−1} < 1. Thus,

sinh(e^{2M} − 1)^{−1} < 3{2(e^{2M} − 1)}^{−1}.

Further, sinh M < e^{M}/2. Hence, for M > 0.987,

2 sinh(e^{2M} − 1)^{−1} sinh M < 3e^{M}{2(e^{2M} − 1)}^{−1}
= 3/(4 sinh M)
< 1.

Furthermore, ℓ/2 < cosh(ℓ/2) for ℓ > 0. Hence

N(M, ℓ) < (e^{2M} − 1) cosh(ℓ/2),

and we have nothing to prove.

We take a point p in C(c) which satisfies d(p, ∂C(c)) = M. Here ∂C(c) is a boundary curve of C(c). From the assumption, the injectivity radius at p is less than M. That is, the length r_{p} of the shortest non-trivial simple closed curve γ passing through p is less than 2M. Since d(p, ∂C(c)) = M, the curve γ is in C(c). Let p̃ = re^{iθ} ∈ A (θ_{0} < θ < π/2) be a lift of p. Setting z_{1}(t) = re^{it} for t ≥ 0, we have

M = d(p̃, ∂C̃(L)) = ∫_{θ_{0}}^{θ} \frac{|z′_{1}(t)|}{Imz_{1}(t)}dt = ∫_{θ_{0}}^{θ} \frac{1}{sin t}dt ≥ ∫_{θ_{0}}^{θ} \frac{1}{t}dt = log \frac{θ}{θ_{0}} .

Hence θ ≤ e^{M}θ_{0}. We put a = e^{iθ} and b = e^{ℓ+iθ}. Then r_{p} = d(a, b). From Theorem 7.2.1 in [1], we have

sinh \frac{1}{2}d(a, b) = \frac{|a − b|}{2(Ima Imb)^{\frac{1}{2}}} = \frac{e^{ℓ} − 1}{2e^{\frac{ℓ}{2}} sin θ} = \frac{sinh \frac{ℓ}{2}}{sin θ} ≥ \frac{sinh \frac{ℓ}{2}}{θ}

≥ \frac{sinh \frac{ℓ}{2}}{e^{M}θ_{0}} = \frac{sinh \frac{ℓ}{2}}{e^{M}arctan(2 sinh \frac{ℓ}{2n})} ≥ \frac{sinh \frac{ℓ}{2}}{2e^{M} sinh \frac{ℓ}{2n}}

= \frac{n sinh \frac{ℓ}{2}}{e^{M}ℓ} \frac{\frac{ℓ}{2n}}{sinh \frac{ℓ}{2n}} ≥ \frac{n sinh \frac{ℓ}{2}}{e^{M}ℓ} \frac{ℓ}{sinh ℓ} = \frac{n sinh \frac{ℓ}{2}}{e^{M} sinh ℓ}
= \frac{n}{2e^{M} cosh \frac{ℓ}{2}} .

For the last inequality, we have used the fact that x(sinh x)^{−1} is a monotone decreasing function for x > 0. Since r_{p} < 2M, this implies that

n < 2e^{M} sinh M cosh(ℓ/2)
= (e^{2M} − 1) cosh(ℓ/2).

Next, we give a proof of Theorem 2 which is based on Cusp Lemma. The idea is similar to that in Theorem 1.

Proof of Theorem 2: Let Γ be a Fuchsian model of R, and f̃ a lift of f to H which is a parabolic element in PSL_{2}(R). We may assume that f̃(z) = z + 1. Note that f̃^{n} belongs to Γ. We set Γ̂ = ⟨Γ, f̃⟩, which is a Fuchsian model of R̂ = R/⟨f⟩. Applying Cusp Lemma for Γ̂ and γ = f̃, we can take a horoball

C̃(∞) = {z ∈ H | 1 < Imz}

so that it is precisely invariant under ⟨f̃⟩ ⊂ Γ̂. In particular, we have γ(C̃(∞))∩C̃(∞) = ∅ for any γ ∈ Γ−⟨f̃^{n}⟩. We set C(p) = C̃(∞)/⟨f̃^{n}⟩, whose fundamental region is

{z ∈ H | 0 < Rez < n, 1 < Imz}.

We take a point q in C(p) so that d(q, ∂C(p)) = M. Here ∂C(p) is the boundary curve of C(p). From the assumption, the injectivity radius at q is less than M. That is, the length r_{q} of the shortest non-trivial simple closed curve γ passing through q is less than 2M. Since d(q, ∂C(p)) = M, the curve γ is in C(p). We put a = e^{M}i and b = n + e^{M}i. Then r_{q} = d(a, b). From Theorem 7.2.1 in [1], we have

sinh \frac{1}{2}d(a, b) = \frac{|a − b|}{2(Ima Imb)^{\frac{1}{2}}} = \frac{n}{2e^{M}} .

Since r_{q} < 2M, this implies that

n < 2e^{M} sinh M
= e^{2M} − 1.

Finally, we prove Theorem 3. The proof is based on Cusp Lemma.

Proof of Theorem 3 (i): Since 2π cosh M > 6 for M > 0, we may assume that n ≥ 7. Let p̃ be a lift of p to H, and f̃ a lift of f to H fixing the point p̃, which is an elliptic element in PSL_{2}(R). Note that f̃^{n} is the identity. We set Γ̂ = ⟨Γ, f̃⟩, which is a discrete group. Applying Cone Lemma for Γ̂ and γ = f̃, we can take a hyperbolic disc

C̃(p̃) = {z ∈ H | d(z, p̃) < ρ(n)}

so that it is precisely invariant under ⟨f̃⟩ ⊂ Γ̂, where

cosh ρ(n) = (2 sin(π/n))^{−1}.

In particular, γ(C̃(p̃)) ∩ C̃(p̃) = ∅ for any γ ∈ Γ − {id}. Then there exists a hyperbolic disc C(p) centered at p with radius ρ(n). Thus the length of any non-trivial simple closed curve passing through p is greater than 2ρ(n). On the other hand, the injectivity radius at p is less than M by the assumption. That is, there exists a non-trivial simple closed curve passing through p whose length is less than 2M. Hence we have ρ(n) < M, and this implies that

n < π{arcsin(2 cosh M)^{−1}}^{−1}
< 2π cosh M.

Proof of Theorem 3 (ii): Let f̃ be a lift of f to H, which is an elliptic element in PSL_{2}(R). Note that the order of f̃ is mn, and that f̃^{n} ∈ Γ. Applying Cone Lemma for Γ̂ = ⟨Γ, f̃⟩, we can take a hyperbolic disc

C̃(p̃) = {z ∈ H | d(z, p̃) < ρ(mn)}

so that it is precisely invariant under ⟨f̃⟩ ⊂ Γ̂, where

cosh ρ(mn) = (2 sin(π/mn))^{−1}.

In particular, g(C̃(p̃)) ∩ C̃(p̃) = ∅ for any g ∈ Γ − ⟨f̃^{n}⟩. Then the fundamental region of C(p) = C̃(p̃)/Γ is conformally equivalent to

{re^{iθ} ∈ C | 0 ≤ r < ρ(mn), 0 < θ < 2π/m}.

We may assume that ρ(mn) > M. Indeed, if

ρ(mn) = arccosh(2 sin(π/mn))^{−1} ≤ M,

then

n ≤ \frac{π}{m} (arcsin(2 cosh M)^{−1})^{−1} < \frac{2π}{m} cosh M < \frac{2πe^{M}}{m}

< \frac{2πe^{M}}{m} \{(\frac{sinh M}{sin \frac{π}{m}} )^{2} + 1\}^{\frac{1}{2}}

= (e^{2M} − 1)\frac{π}{m} (\frac{1}{sin^{2} \frac{π}{m}} + \frac{1}{sinh^{2} M} )^{\frac{1}{2}} .

and we have nothing to prove.

We take a point q in C(p) that satisfies d(q, ∂C(p)) = M. Here ∂C(p) is the boundary curve of C(p). From the assumption, the injectivity radius at q is less than M. That is, the length r_{q} of the shortest nontrivial simple closed curve γ passing through q is less than 2M. Since

CONFORMAL AUTOMORPHISMS OF RIEMANN SURFACES 9

d(q, ∂C(p)) = M, the curve γ is in C(p). From the cosine rule of triangles (cf. [1, p.148]), we have

cosh r_{q} = cosh^{2}(ρ(mn) − M) − sinh^{2}(ρ(mn) − M) cos(2π/m)
= sinh^{2}(ρ(mn) − M)(1 − cos(2π/m)) + 1.

Then r_{q} < 2M implies that

(1)  ρ(mn) < arcsinh (\frac{cosh 2M − 1}{1 − cos \frac{2π}{m}})^{\frac{1}{2}} + M
= arcsinh (\frac{sinh M}{sin \frac{π}{m}} ) + M
=: M_{1}.

To obtain (1), we have used the fact that the inverse function of y = sinh^{2} x for x > 0 is arcsinh\sqrt{y}. Since

ρ(mn) = arccosh(2 sin(π/mn))^{−1},

the inequality (1) implies that

n < (π/m) {arcsin (2 cosh M_{1})^{−1}}^{−1}
< (2π/m) cosh M_{1}
= (2π/m) cosh \{arcsinh (\frac{sinh M}{sin \frac{π}{m}} ) + M\} .

We set

X = \frac{sinh M}{sin \frac{π}{m}} .

Using the hyperbolic cosine formula and the fact that cosh(arcsinh x) = \sqrt{x^{2} + 1} for x > 0, we have

n < (2π/m) cosh {arcsinhX + M}
= (2π/m) {cosh (arcsinhX) cosh M + sinh (arcsinhX) sinh M}
< (2π/m) {cosh (arcsinhX) cosh M + cosh (arcsinhX) sinh M}
= (2π/m) e^{M} cosh (arcsinhX)
= (2π/m) e^{M}(X^{2} + 1)^{\frac{1}{2}}
= \frac{2πe^{M}}{m} \{(\frac{sinh M}{sin \frac{π}{m}} )^{2} + 1\}^{\frac{1}{2}}
= (e^{2M} − 1)\frac{π}{m} (\frac{1}{sin^{2} \frac{π}{m}} + \frac{1}{sinh^{2} M} )^{\frac{1}{2}} .

5. Application

In this section, we apply our main theorem to investigating a certain property on the hyperbolic geometry on Riemann surfaces. The property we observe is as follows.

Definition 2. We say that a Riemann surface R satisfies the lower bound condition if there exists a positive constant ϵ such that the ϵthin part of R consists only of cusp neighborhoods. Further, we say that R satisfies the upper bound condition if there exists a positive constant M such that the injectivity radius at any point in R is less than M.

In [2], we have defined the (generalized) upper bound condition, and have shown the following.

Proposition 3. ([2]) Let R be a hyperbolic Riemann surface of analytically finite type, and R̃ a normal covering surface of R which is not a universal cover. Then R̃ satisfies the lower and (generalized) upper bound conditions.

In connection with this result, we show that Riemann surfaces inherit the lower and upper bound conditions from its normal covering surface.

Proposition 4. Let R be a hyperbolic Riemann surface, and R̃ a normal covering surface of R. If R̃ satisfies the lower and upper bound conditions, then R also satisfies these conditions.

Proof. It is clear that R satisfies the upper bound condition. Suppose that R does not satisfy the lower bound condition. Then R has a sequence {c_{n}} of disjoint simple closed geodesics with ℓ_{n} = ℓ(c_{n}) → 0 (n → ∞). Here ℓ(·) means the hyperbolic length of a curve. Let c̃_{n} ⊂ R̃ be a connected component of the preimage of c_{n}. Since R̃ satisfies the lower bound conditions, there exists a positive constant ϵ such that ℓ(c̃_{n}) > ϵ for all n. We take a positive constant M so that R̃ satisfies the upper bound condition for M. Assume that ℓ(c̃_{n}) ≤ 2M for infinitely many n. Then, by Theorem 1, the order of a conformal automorphism f̃_{n} of R̃ that fixes c̃_{n} is less than N = (e^{2M} − 1) cosh M. Then we have ℓ(c_{n}) > ϵ/N. However, this contradicts ℓ(c_{n}) → 0 (n → ∞). Next, we assume that ℓ(c̃_{n}) > 2M (including the case that c̃_{n} is not closed) for infinitely many n. By Collar Lemma, there exists a tubular neighborhood C(c_{n}) of c_{n} with width ω(ℓ_{n}), where sinh ω(ℓ_{n}) = (2 sinh(ℓ_{n}/2))^{−1}. From the proof of Theorem 1, there exists a tubular neighborhood of c̃_{n} with width ω(ℓ_{n}). Since R̃ satisfies the upper bound condition for the constant M, there exists a non-trivial simple closed curve passing through p̃_{n} ∈ c̃_{n} whose length is less than 2M. However, since ℓ(c̃_{n}) > 2M and since ω(ℓ_{n}) → ∞ as n → ∞, we have a contradiction.

The following example shows that, in Proposition 4, if the normal covering surface R̃ of R satisfies only one of the two conditions, then R does not necessarily satisfy the two conditions.

Example 1. Let

R̃ = C − ⋃_{n=1}^{∞} ⋃_{m∈Z} \{\frac{m}{n} ± n^{2}\sqrt{−1}\} ,

and set R = R̃/⟨f⟩, where f(z) = z + 1. This Riemann surface R satisfies the upper bound condition but does not satisfy the lower bound condition. On the other hand, the normal covering surface R̃ of R satisfies the lower bound condition but does not satisfy the upper bound condition.

References

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[2] E. Fujikawa, Limit sets and regions of discontinuity of Teichmüller modular groups, preprint, 2001.

[3] N. Halpern, A proof of the collar lemma, Bull. London Math Soc., 13, 1981, 141-144.

[4] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. 17, 1966, 86-97.

[5] J. P. Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43, 1976, 829-840.

[6] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1959.

[7] A. Yamada, On Marden’s universal constant of Fuchsian groups II, J. d’Analyse Math., 41, 1982, 234-248.

Department of Mathematics, Tokyo Institute of Technology, Ohokayama Meguro-ku Tokyo 152-8551, Japan

 E-mail address: fujikawa@math.titech.ac.jp