A NEW CHARACTERIZATION OF GROMOV HYPERBOLICITY FOR NEGATIVELY CURVED SURFACES

In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or, Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces. Running headline: A NEW CHARACTERIZATION OF GROMOV HYPERBOLICITY

A geodesic metric space is called hyperbolic (in the Gromov sense) if it satisfies the "Rips condition": there is an upper bound of the distance of every point in a side of any geodesic triangle to the union of the two other sides (see Definition 2.3).
But, it is not easy to determine if a given space is Gromov hyperbolic or not. One interesting instance is that of a Riemann surface endowed with the Poincaré metric. With that metric structure a Riemann surface is negatively curved, but not all Riemann surfaces are Gromov hyperbolic, since topological obstacles can impede it: for instance, the two-dimensional jungle-gym (a Z 2 -covering of a torus with genus two) is not hyperbolic.
We are interested in studying when Riemann surfaces equipped with their Poincaré metric are Gromov hyperbolic. The following theorem is the main result of this paper, which is a new characterization of Gromov hyperbolicity for Riemann surfaces (see Theorem 5.1): A Riemann surface S is hyperbolic if and only if the c 0 -triangles contained in simple closed geodesics of S satisfy the Rips condition. By a c 0 -triangle we mean a triangle with continuous injective (1, c 0 )quasigeodesic sides, and we require that the vertices and the edges of such triangles are contained in simple closed geodesics of S.
In general, one has to verify the Rips condition for all triangles. Our result is that for Riemann surfaces you only have to verify it for a smaller class of triangles.
Furthermore this theorem provides a bound for the hyperbolicity constant: if the triangles contained in simple closed geodesics satisfy the Rips condition with constant δ 0 , then every geodesic triangle satisfy it with constant δ = max{11, δ 0 + 6}.
A connected question with our main theorem is when a Euclidean bounded domain with its quasihyperbolic metric is Gromov hyperbolic. (Let us recall that in the case of modulated plane domains, quasihyperbolic and Poincaré metrics are equivalent.) Recently, Balogh and Buckley [BB] have made significant progress in this question (see also [BHK] and the references therein).
Theorem 5.1 provides good bounds for the hyperbolicity constants of some classical surfaces such as the punctured disk, the annuli, the Y -pieces and plane domains of finite type (see Lemma 5.4 and corollaries 5.1, 5.2 and 5.3).
It can also be successfully used as a powerful tool to study hyperbolicity of a class of Riemann surfaces by means of its decomposition in Y -pieces and funnels (see Theorem 5.3).
As a consequence of these results, we have obtained interesting examples of hyperbolic Riemann surfaces (see Theorem 5.3 and corollaries 5.1, 5.2 and 5.3), and a result that allows us a better understanding of the role that funnels and half-disks (see Definition 5.4) play in the study of hyperbolicity (see Theorem 5.2). Theorem 5.2 is a useful result which has several applications in [RT2] and [PRT2].
One can think of the following as a natural first result in order to study hyperbolicity: if a Riemann surface has a sequence of funnels {F n } n with lim n→∞ L(∂F n ) = ∞, then it is not hyperbolic. In [RT2] we prove that this reasonable result is false indeed, and an important tool in the proof is Theorem

5.2.
Notations. We denote by X or X n geodesic metric spaces. By d X , L X and B X we shall denote, respectively, the distance, the length and the balls in the metric of X. From now on, when there is no possible confusion, we will not write the subindex X.
We denote by R, S or S 0 Riemann surfaces. We assume that the metric defined on these surfaces is the Poincaré metric, unless the contrary is specified.
If Ω is a plane domain, we shall denote by λ Ω the conformal density of the Poincaré metric in Ω , i.e. the function such that ds = λ Ω (z)|dz| is the Poincaré metric in Ω .
We denote by z and z the real and imaginary part of z, respectively. Acknowledgements. We would like to thank Professor J. L. Fernández for some useful discussions. Also, we would like to thank the referee for his/her careful reading of the manuscript and for some helpful suggestions. §2. Background in Gromov spaces In our study of hyperbolic Gromov spaces we use the notations of [GH]. We give now the basic Theorem A. ( [GH,p.41]) Let us consider a geodesic metric space X.
We present now the class of maps which play the main role in the theory.
Definition 2.4. A function between two metric spaces f : Such a function is called an (a, b)-quasi-isometry. An (a, b)-quasigeodesic in X is an (a, b)-quasiisometry between an interval of R and X.
Let us observe that a quasi-isometry does not have to be continuous (for instance, the map f : R −→ Z such that f ([n, n + 1)) = n for every integer n is a (1, 1)-quasi-isometry).
Quasi-isometries are important since they are maps which preserve hyperbolicity: GH,p.88]) Let us consider an (a, b)-quasi-isometry between two geodesic metric only on δ, a and b.
Definition 2.5. Let us consider H > 0, a metric space X, and subsets Y, Z ⊆ X. The set The following is a beautiful and useful result: GH,p.87]) For each δ, b ≥ 0 and a ≥ 1, there exists a constant H = H(δ, a, b) with the following property: Let us consider a δ-hyperbolic geodesic metric space X and an (a, b)-quasigeodesic g joining x and y. If γ is a geodesic joining x and y, then H(g, γ) ≤ H.
This property is known as geodesic stability. Mario Bonk has proved that, in fact, geodesic stability is equivalent to hyperbolicity [Bo].
Along this paper we will work with topological subspaces of a geodesic metric space X. There is a natural way to define a distance in these spaces: Definition 2.6. If X 0 is a subset connected by rectifiable paths of a metric space (X, d), then we associate to it the inner or intrinsic distance We want to remark that almost every constant appearing in the results of this paper depends just on a small number of parameters (in fact, we give explicit expressions for them). This is a common place in the theory of hyperbolic spaces (see e.g. theorems A, B and C) and is also typical of surfaces with curvature −1 (see the Collar Lemma in [R] and [S], and Theorem 3.1 in [PRT2]).
We need some technical results which we collect in the following lemmas.
Lemma 3.1. Let us consider a geodesic metric space X and ε > 0. If γ is a continuous curve Proof. Let us consider γ with its arc-length parametrization γ : We assume that there are 0 ≤ t, s ≤ l with |t − s| > d X (γ(t), γ(s)) + ε. Without loss of generality we can assume t < s. We define a curve γ 0 as a concatenation of three curves: γ([0, t]), a geodesic η connecting γ(t) with γ(s), and γ ([s, l]). Since γ 0 is a continuous curve connecting x with y, we have which is a contradiction.
Lemma 3.2. Let us consider a metric space X with a closed geodesic g of length l. If γ is a continuous injective (1, c)-quasigeodesic in X with its arc-length parametrization, and it is contained Remarks. 1. It is clear that every closed geodesic is only a local geodesic, but not a geodesic (see Definition 2.2); however, since there is no possible confusion, we call it closed geodesic instead of closed local geodesic.
Proof. Let us consider γ with its arc-length parametrization γ : [0, l 0 ] −→ X. Assume that l 0 > (l + c)/2; then l − l 0 < l 0 − c. Observe that d(γ(0), γ(l 0 )) ≤ l − l 0 , since g \ γ is a continuous curve of length l − l 0 joining γ(0) and γ(l 0 ) (γ is continuous and injective). Hence, we have that Proof. Given an (a, b)-quasigeodesic triangle in X of sides q 1 , q 2 , q 3 , Theorem C gives that there exist geodesics g 1 , g 2 , g 3 , such that g i has the same end points as q i and H( Since X is 4δ-thin, we can find y ∈ g j ∪ g k with d(y , x ) ≤ 4δ. We also have a point y ∈ q j ∪ q k with The following result is a modification of Theorem 2.4 in [RT1] (using a completely different line of argument). Furthermore, this proof gives an explicit expression for the constants involved. It can be applied to the study of hyperbolicity of Riemann surfaces (see Theorem 5.3). In order to state it, we need one definition.
Definition 3.1. We say that the closed geodesic metric spaces {X n } n∈Λ are a (c 1 , c 2 )-regular decomposition of the geodesic metric space X if they verify the following conditions: (a) X = ∪ n∈Λ X n and X n ∩ X m = η nm , where for each n ∈ Λ, {η nm } m∈Λ\{n} are pairwise disjoint closed subsets of X n (η nm = ∅ is allowed); furthermore any geodesic in X with finite length meets at most a finite number of η nm 's.
(b) For any n, m ∈ Λ, diam Xn (η nm ) ≤ c 1 and if η nm = ∅, then X \ η nm is not connected and a, b are in different connected components of X \ η nm for any a ∈ X n \ η nm , b ∈ X m \ η nm .
(c) For each n ∈ Λ there exist disjoint sets A n , B n ⊆ Λ, verifying the following properties: if m / ∈ A n ∪ B n , then η nm = ∅; diam X n (∪ m∈A n η nm ) ≤ c 2 ; and every geodesic joining two points in X n cannot escape from X n across a η nm with m ∈ B n , Remarks. 1. The sets Λ, A n and B n do not need to be countable.
2. The hypothesis on X \ η nm guarantees that the graph R = (V, E) constructed in the following way is a tree: 3. We can think that the hypothesis "a geodesic joining two points in X n cannot escape from X n across a η nm with m ∈ B n ", is very restrictive; however, Lemma 5.5 below gives a very simple condition which allows one to assure this hypothesis.
4. If X is a Riemann surface, {X n } n∈Λ are bordered Riemann surfaces and η nm ⊂ ∂X n ∩ ∂X m , condition "a, b are in different components of X \ η nm for any a ∈ X n \ η nm , b ∈ X m \ η nm " in (b), is a consequence of "X \ η nm is not connected".

5.
We wish to emphasize that condition diam X n (η nm ) ≤ c 1 is not very restrictive: if the space is "wide" at every point (in the sense of long injectivity radius, as in the case of simply connected spaces) or "narrow" at every point (as in the case of trees), it is easier to study its hyperbolicity; if we can found narrow parts (as η nm ) and wide parts, the problem is more difficult and interesting.
Proof. Let us consider a geodesic triangle T = {a, b, c} in X. If T ⊆ X n for some n, then T is δ 0 -thin, by hypothesis. We assume now that T intersects several X n 's. We intend to study T in each of those X n 's separately.
Let us take z ∈ T . If z belongs to two sides of T , there is nothing to prove; if z only belongs to one side of T , we denote by A the union of the sides of T which do not intersect z.
Let us fix n ∈ Λ. We assume first that the three sides of T intersect X n .
We construct a geodesic polygon P n in X n modifying T ∩ X n in the following way: Let us consider a side γ i (i = 1, 2, 3) of T . If γ i ⊆ X n , we define g i := γ i . If γ i is not contained in X n , then we In other case, we define ] in X n . This minimum and this maximum exist since γ i is a continuous function in a compact interval and γ i ∩ (∪ m∈A n η nm ) is a compact set: each η nm is a closed set and γ i meets at most a finite number of η nm 's.
It is possible that g 1 ∪ g 2 ∪ g 3 is not a polygon, since there can exist gaps between two g i 's. Since diam X n (η nm ) ≤ c 1 and X \ η nm is not connected for any m ∈ Λ, we can find three geodesics h 1 , h 2 , h 3 in X n of length less or equal than c 1 such that g 1 ∪ h 1 ∪ g 2 ∪ h 2 ∪ g 3 ∪ h 3 is a geodesic polygon P n in X n (some h i can be a point). It is clear that P n has at most 12 sides, and then it is 10δ 0 -thin.
Without loss of generality we can assume that z ∈ g 1 . In order to simplify the notation, we define Let us assume now that We consider now the next four possibilities: Let us assume now that only two sides of T intersect X n . As in the previous case, we can replace each γ i which intersect X n by g i . Then we can construct in a similar way a geodesic polygon P n in X n with at most 8 sides, which is 6δ 0 -thin. Hence the previous argument gives the same result with even sharper constant.
Finally, let us assume that only one side of T intersects X n . Then z belongs to some [b 1 , b 2 ], and the same inequality holds.
The same proof of Theorem 3.1 gives sharper constants in some particular cases.
An open non-exceptional Riemann surface (or a non-exceptional Riemann surface without boundary) S is a Riemann surface whose universal covering space is the unit disk D = {z ∈ C : |z| < 1}, endowed with its Poincaré metric, i.e. the metric obtained by projecting the Poincaré metric of the unit disk or, equivalently, the upper half plane U = {z ∈ C : Im z > 0}, with the metric ds = λ U (z)|dz| = |dz|/ Im z. Observe that, with this definition, every compact non-exceptional Riemann surface without boundary is open. With this metric, S is a complete Riemannian manifold with constant curvature −1; therefore S is a geodesic metric space. The only Riemann surfaces which are left out are the sphere, the plane, the punctured plane and the tori. It is easy to study hyperbolicity of these particular cases.
It is well-known (see e.g. [An,p.100], [B, p.131], [JS,p.227], [N, p.18]) that A collar in S about a simple closed geodesic γ is a doubly connected domain in S "bounded" by two Jordan curves (called the boundary curves of the collar) orthogonal to the pencil of geodesics emanating from γ; such a collar is equal to {p ∈ S : d S (p, γ) < d}, for some positive constant d. The constant d is called the width of the collar. The Collar Lemma [R] says that there exists a collar of γ of As we remarked after Lemma 3.2, every closed geodesic is a local geodesic, but not a geodesic; however, since there is no possible confusion, we call it closed geodesic instead of closed local geodesic.
A puncture in a non-exceptional Riemann surface is a doubly connected end in which we can find homotopically non-trivial curves with arbitrarily small length. A puncture is an isolated point in ∂S in the case that S ⊂ C. We can think of a puncture as a boundary geodesic of zero length. It is not difficult to see that if any ball in R intersects at most a finite number of connected components of V , and the boundary of S is locally Lipschitz, then S is a geodesic metric space.
A funnel is a bordered non-exceptional Riemann surface which is topologically a cylinder and whose boundary is a simple closed geodesic. Given a positive number a, there is a unique (up to conformal mapping) funnel such that its boundary curve has length a. Every funnel is conformally equivalent, for some β > 1, to the subset {z ∈ C : 1 ≤ |z| < β} of the annulus {z ∈ C : 1/β < |z| < β}.
Every doubly connected end of an open non-exceptional Riemann surface is a puncture (if there are homotopically non-trivial curves with arbitrary small length) or a funnel (in other case).
A Y -piece is a bordered non-exceptional Riemann surface which is conformally equivalent to a sphere minus three open disks and whose boundary curves are simple closed geodesics (and then it is triply connected). Given three positive numbers a, b, c, there is a unique (up to conformal mapping) Y -piece such that their boundary curves have lengths a, b, c (see e.g. [Ra,p.410] A generalized Y -piece is a non-exceptional Riemann surface (with or without boundary) which is conformally equivalent to a sphere without n open disks and m points, with integers n, m ≥ 0 such that n + m = 3, the n boundary curves are simple closed geodesics and the m deleted points are punctures.
Observe that a generalized Y -piece is topologically the union of a Y -piece and m cylinders. §5.

Results in Riemann surfaces
Although one should expect Gromov hyperbolicity in non-exceptional Riemann surfaces due to its constant curvature −1, this turns out to be untrue in general, since topological obstacles can impede it: for instance, the two-dimensional jungle-gym (a Z 2 -covering of a torus with genus two) is not hyperbolic.
In [RT2] we prove that there is no inclusion relationship between hyperbolic Riemann surfaces and the usual classes of Riemann surfaces, such as O G , O HP , O HB , O HD , surfaces with linear isoperimetric inequality, or the complements of these classes (even in the case of plane domains). This fact shows that the study of hyperbolic Riemann surfaces is more complicated and interesting than one might think at first sight. One can find other results on hyperbolicity of Riemann surfaces in [RT1], [RT2], [PRT1] and [PRT2].
The main result in this paper is The results in this section give many examples of hyperbolic Riemann surfaces, and provide criteria in order to decide whether a Riemann surface is hyperbolic or not.
Definition 5.1. By a simply connected polygon in a non-exceptional Riemann surface we mean a polygon isometric to a polygon in the Poincaré disk. We say that two sides in a polygon are disjoint if their interiors are disjoint.
We collect in the following lemmas some technical results which we need in order to clarify the proof of Theorem 5.1. Let us assume also that C meets orthogonally the sides A and B. We have that: (1) cosh l 0 = cosh a cosh b cosh c − sinh a sinh b.
Remark. It is clear by the triangle inequality that l 0 ≤ a + b + c.
Proof. Since the quadrilateral is simply connected, we can assume that it is contained in the unit disk D. Part (1) can be found in [F, p.88].
We show part (2). Let us observe that the function f (t) := 2(cosh t−1) Let us assume also that C meets orthogonally the sides A and B. If η and C are not disjoint, then we have that: (1) cosh l 0 = cosh a cosh b cosh c + sinh a sinh b.
Proof. Since the quadrilateral is simply connected, we can assume that it is contained in the unit disk D. Part (1) can be found in [F, p.89].
We show part (2). The inequality is a consequence of Proof. Since Q is simply connected, we can assume that it is contained in the upper half plane U.
Without loss of generality we can assume that Q is the quadrilateral with vertices i, it, ie −iθ , ie −iφ t, with 0 < θ, φ < π/2 and t = e c ≥ e c0 .

It is clear that
Without loss of generality we can assume that is not the case, we can change the roles of θ and φ).
It is obvious that d(i, η) is less than the distance of i to the geodesic η 0 joining 1 and t.
The Möbius transformation T z := (z − t)/(z − 1) maps η 0 onto the imaginary half-axis I, and T i = (t + 1 + i(t − 1))/2. A computation gives (see e.g. [B, p.162]) Proof. Let us consider a geodesic triangle T = {a, b, c} in A l . If T is homotopic to a point, then it is the boundary of a simply connected closed set E, and consequently E, with its intrinsic distance, is isometric to some subset of D; this implies that T is δ 0 -thin, with δ 0 := log(1 + √ 2 ), since D is δ 0 -thin (see [An,p.130]). Then we can assume that T is freely homotopic to the simple closed geodesic g, if l > 0, or to the puncture, if l = 0.
Let us assume first that l > 0 and T ∩ g = ∅. We denote by F 1 and F 2 the two funnels whose union is A l (the closures of the two connected components of A l \ g).
Let us observe that the funnels are geodesically convex (every geodesic connecting two points of the funnel is contained in the funnel). Hence, without loss of generality we can assume that a is in the interior of F 1 and b, c are in the interior of F 2 (the case in which there is some vertex in g is easier).
We define B := [a, b] ∩ g and C := [a, c] ∩ g. There are two local geodesics g 1 , g 2 ⊂ g joining B and C; let us observe that L A l (g i ) ≤ l.
Let us consider the triangle T 1 = {a, B, C}, where we choose as [B, C] the local geodesic g i ⊂ g such that [a, B]∪ [B, C]∪ [C, a] is homotopic to a point; since T 1 is homotopic to a point, the above argument C], we obtain a similar result.
Let us consider the quadrilateral Q 1 = {b, c, C, B}, where we choose as [B, C] the local geodesic [B, b] is homotopic to a point; since Q 1 is homotopic to a point, the above argument implies that Q 1 is 2δ 0 -thin. In a similar way to the case of T 1 , given any Let us assume now that l > 0 and T ∩ g = ∅. Next, we find an upper bound for d A l (T, g). Given Hence, without loss of generality we can assume that . We consider first the case x ∈ (a, b). We can assume that t := d A l (a 0 , x 0 ) ≥ l/6.
If x = a or x = b, a similar argument with t := d A l (a 0 , b 0 ) gives sinh s sinh t < 1, and we obtain sinh s < 1/ sinh(l/3), which also implies (5.1).
Without loss of generality we can assume that consider the local geodesic g x starting and finishing in x, which is freely homotopic to g. We consider first the case x ∈ (a, b). We denote by 2d x the length of g x and by y the point in g x at distance d x of x.
Let us assume now that l = 0, i.e. that we deal with the case A 0 = D * ; then T is freely homotopic to the puncture. We consider the universal covering map π : U −→ D * , given by π(z) = exp(2πiz).
We denote by z 0 a point of U 0 in which this maximum is attained.
Recall that d(l) := Arcsinh sinh(l/2) cotanh(l/6) if l > 0 and d(0) := Arcsinh 3. Then there exists a point p ∈ T such that the local geodesic g p in A l starting and finishing in p, which is freely homotopic to g or to the puncture, has length 2d p < 2d(l).
Let us assume first that p is not a vertex of T ; without loss of generality we can assume also that p ∈ [a, c]. Since g p is freely homotopic to T , we have a geodesic pentagon P : It is clear that if x , y , are the corresponding points in P to the points x, y ∈ P , we have d A l (x, y) ≤ If α j is the corresponding point in P to α j , we have that We define now β 1 : Let us denote by β j the corresponding point in P to β j .

Therefore, in the four situations we have d
If p is a vertex of T , the proof is easier since we construct a quadrilateral instead of a pentagon, and we do not need to split a side of T . This finishes the proof of Lemma 5.4.
The following is the main result of this paper; it allows one to check the Rips condition only for triangles contained in simple closed geodesics. We would like to remark the simplification that Theorem 5.1 means in the applications: Let us consider an annulus A with simple closed geodesic γ.
A generic triangle T in A is determined by the coordinates of three points, i.e., by six real coordinates; however, a generic triangle T 0 in the simple closed geodesic γ is determined by three real coordinates.
Therefore Theorem 5.1 is a remarkable improvement of Rips condition in the context of Riemann surfaces.
Definition 5.2. By a c 0 -triangle we mean a triangle with continuous injective (1, c 0 )-quasigeodesic sides, with its arc-length parametrization.
2. Even though this theorem reduces drastically the triangles in which we have to check the Rips condition, we must "pay" for it by working with quasigeodesic triangles; however the situation is advantageous since the class of quasigeodesics that we need is very restrictive: recall that we only consider continuous injective (1, c 0 )-quasigeodesics, and Lemma 3.2 gives a bound of its length which will be good enough in the applications (see Theorem 5.3 and corollaries 5.2 and 5.3).
Proof. The heart of the proof of Theorem 5.1 is to relate any geodesic triangle T in S with a c 0 -triangle contained in a simple closed geodesic γ in S. In some way, we can consider T and γ as "subsets" of the annulus A l (with l := L S (γ)). The geodesic triangles in the simple closed geodesic of A l are (l/4)-thin, and this value is sharp (it is enough to consider a triangle with sides of lengths l/4, l/4 and l/2). However, the problem in a general Riemann surface is more difficult (and recall that we can find simple closed geodesics arbitrarily long). Therefore, if l is big we need a narrow metric relationship between T and γ.
If S is hyperbolic, Lemma 3.3 guarantees that every c 0 -triangle in S is δ 0 -thin.
Let us assume that every c 0 -triangle contained in a simple closed geodesic in S is δ 0 -thin. First, we want to remark that if S has boundary, the hypothesis on ∂S gives that it is the union of pairwise disjoint simple local geodesics (closed or non-closed).
In this case, we can construct an open non-exceptional Riemann surface R by pasting to S a funnel in each simple closed geodesic, and a half-disk in each non-closed simple geodesic.
Since S is geodesically convex in R (every geodesic connecting two points of S is contained in S), then d R (z, w) = d S (z, w) for every z, w ∈ S, and any simple closed geodesic in R is contained in S.
Let us consider a geodesic triangle T in S. By Lemma 2.1 in [RT1], we can assume that T is a simple closed curve.
We have three possibilities: T is homotopic to a point, T is homotopic to a puncture, or T is freely homotopic to a simple closed geodesic in S. This is well known if S has no boundary; if S has boundary, it is enough to apply the result to R, since R has not additional topological obstacles (the fundamental groups of S and R are isomorphic).
If T is homotopic to a point, then it is the boundary of a simply connected closed set E, and consequently E, with its intrinsic distance, is isometric to some subset of D; this implies that T is log(1 + √ 2 )-thin, since D is log(1 + √ 2 )-thin (see [An,p.130]).
If T is homotopic to a puncture, then it is the boundary of a closed doubly connected set, which is, with its intrinsic distance, isometric to some subset of D * := D \ {0}; this implies that T is δ(0)-thin, with δ(0) the constant in Lemma 5.4. Since every geodesic triangle in D is isometric to some geodesic triangle in D * , we have that log(1 + √ 2 ) ≤ δ(0).
In other case, T is freely homotopic to a simple closed geodesic γ in S.
If L(γ) < 4c 0 , let us consider the annulus A L(γ) with a simple closed geodesic g of length L(γ). We In this case, the closed set in S bounded by T and γ is, with its intrinsic distance, isometric to a set in A L(γ) , bounded by g and a triangle T 0 . These facts give that T is δ(4c 0 )-thin.
First, we assume that γ ∩ T = ∅. If η is a side of T , we associate to it two curves η , η , in the following way. We consider a simply connected locally geodesic quadrilateral Q in S with pairwise disjoint sides A, C, B and η, of lengths a, c, b and l 0 , respectively, with the following conditions: (i) C ⊂ γ, (ii) C meets orthogonally the sides A and B. Q is uniquely determined by these conditions.
If c ≥ c 0 , the arc η := A ∪ C ∪ B is a continuous injective (1, c 0 )-quasigeodesic with its arc-length parametrization by lemmas 3.1 and 5.1. If c < c 0 , we take η := η, which is a geodesic. (Observe that we have c < c 0 for at most one side of T , since L(γ) ≥ 4c 0 ; in other case, T would not be a geodesic triangle.) In both cases, we define η := C ⊂ γ. We have that η is always a continuous injective (1, c 0 )-quasigeodesic with its arc-length parametrization: this is clear if c ≥ c 0 (since η ⊂ η ), and it is a consequence of Corollary 3.1 if c < c 0 .
If η i = η i , for i = 1, 2, 3, then T is δ 0 -thin, since every point in T \ T belongs to two sides of T .
If it is not the case, there is only one i with η i = η i ; we can assume η 1 = η 1 . Let us consider the quadrilateral Q 1 with sides A 1 , C 1 , B 1 and η 1 ; we have that L(C 1 ) < c 0 . Since Q 1 is simply connected, it is isometric to a quadrilateral in D which is 2 log(1 + √ 2 )-thin.
The case z ∈ η 3 is similar to z ∈ η 2 .
If γ ∩ T has only one connected component, the same argument works.
If γ ∩ T has two connected components, the argument is similar, using now Lemma 5.2 instead of Lemma 5.1. The constant in this case is smaller, since 3 log 2 < c 0 .
The following example shows that the quasigeodesic triangle T in the proof of Theorem 5.1 does not need to be geodesic.
Example. There is a geodesic triangle T in a triply connected Riemann surface S 0 such that T is not geodesic.
If we define ε := Arcsinh(1/ sinh x) − x > 0, we have that sinh x sinh(x + ε) = 1. Let us consider a geodesic quadrilateral V with three right angles and an angle equal to zero, such that the two finite sides have length x and x + ε (see e.g. [B, p.157], [F, p.89]). If we paste four quadrilaterals isometric to V , we obtain a generalized Y -piece Y 0 with two punctures and a simple closed geodesic γ with L(γ) = 4(x + ε). We obtain S 0 by gluing Y 0 with a funnel F whose simple closed geodesic has length 4(x + ε).
Let us denote by µ 0 the geodesic in Y 0 with L(µ 0 ) = 2x, joining γ with itself which is not homotopic to any curve contained in γ. We denote by p , q the end points of µ 0 . Let us consider the non bounded geodesic µ in S 0 which contains µ 0 , and the two points p, q ∈ µ ∩ F at distance y of γ.
From now on we will obtain several consequences of Theorem 5.1.
Corollary 5.1. The annulus A l such that its simple closed geodesic has length l ≥ 4c 0 is (l/4 + K)thin, with K < 5.0869 the constant in Theorem 5.1. The same is true for each funnel of A l .
Remark. This bound of the hyperbolicity constant for the annulus is asymptotically sharp: we have that the best thin constant of A l is greater than or equal to l/4, since we have a geodesic triangle contained in the simple closed geodesic with sides of lengths l/2, l/4, l/4.
Proof. Let us observe that the last part of the proof of Theorem 5.1 gives that A l is δ 2 -thin, if l ≥ 4c 0 .
In this case the hypothesis "any continuous injective (1, c 0 )-quasigeodesic triangle contained in a simple closed geodesic in S is δ 0 -thin", can be changed by "any geodesic triangle contained in the simple closed geodesic γ of A l is δ 0 -thin", since T is a geodesic triangle in A l if T is a geodesic triangle in A l . Since the sides of any geodesic triangle contained in γ have length less than or equal to l/2, any geodesic triangle contained in γ is δ 0 -thin, with δ 0 = δ 0 (A l ) = l/4. Consequently, we obtain that A l is δ 2 -thin with since l/4 ≥ c 0 > 2 > 2 log(1 + √ 2 ). The same is true for each funnel of A l .
Definition 5.3. We say that a non-exceptional Riemann surface S (with or without boundary) is of finite type if its fundamental group is finitely generated.
Corollary 5.2. Let us consider a non-exceptional Riemann surface S (with or without boundary) of genus 0; if S has boundary, we also require that ∂S is the union of local geodesics (closed or nonclosed). If S is of finite type, then it is hyperbolic. In fact, if every simple closed geodesic γ in S verifies L(γ) ≤ l, then S is δ-thin, with δ = max{δ(4c 0 ), K +(l +c 0 )/4} and c 0 , δ(4c 0 ), K the constants in Theorem 5.1.
Remark. As usual we see a puncture as a simple closed geodesic with zero length.
In order to prove the following result we need one definition.
Definition 5.4. A half-disk is a bordered non-exceptional Riemann surface which is topologically a closed half-plane and whose boundary is a simple geodesic. Every half-disk is conformally equivalent to the subset {z ∈ D : z ≥ 0} of the hyperbolic disk D.
It is clear that a funnel contains infinitely many half-disks.
Two additional consequences which are important in the study of hyperbolicity of Riemann surfaces can be deduced from Theorem 5.1. The first one (see Theorem 5.2 below) allows us to simplify the topology: it assures that to delete funnels and half-disks does not change the hyperbolicity of a Riemann surface. Theorem 5.2 is a useful result which has several applications in [RT2] and [PRT2].
One can think of the following as a natural first result in order to study hyperbolicity: if a Riemann surface has a sequence of funnels {F n } n with lim n→∞ L(∂F n ) = ∞, then it is not hyperbolic. In [RT2] we prove that this reasonable result is false indeed, and an important tool in the proof is Theorem

5.2.
Our recent research let us expect that Theorem 5.2 will be a key tool in the characterization of hyperbolic Denjoy domains. Remark. We want to emphasize that there is no hypothesis about the length of the boundary curves of the funnels. This is an important fact since there are hyperbolic Riemann surfaces containing funnels F n with L(∂F n ) −→ ∞ as n → ∞ (see the examples in Section 4 of [RT2]).
Proof. Let us assume that S is δ-thin (hyperbolic). As S 0 is geodesically convex in S (every geodesic connecting two points of S 0 is contained in S 0 ), then d S (z, w) = d S 0 (z, w) for every z, w ∈ S 0 . Therefore S 0 is also δ-thin (hyperbolic).
The following result on geodesically convex subsets of Riemann surfaces is a consequence of the Collar Lemma. It will be useful in the proof of Theorem 5.3. If L(η) < L 0 , then every geodesic connecting two points of S 0 is contained in S 0 , and consequently d S (z, w) = d S 0 (z, w) for every z, w ∈ S 0 .
We take z, w ∈ S 0 . In order to prove the lemma, without loss of generality we can assume that z, w ∈ η; therefore d S0 (z, w) ≤ L/2.
Then we have 2 cosh L 0 4 sinh 2 L 0 4 < 1 , sinh L 0 4 sinh L 0 2 < 1 , sinh L 0 4 < 1 sinh L0 2 < sinh d 0 , and hence we obtain L < 4d 0 , which is a contradiction. If S has boundary, then it is contained in a Riemann surface R and d S = d R | S . If γ is a geodesic in S joining z, w, and not contained in S 0 , then there is a geodesic in R joining z, w, which is not contained in S 0 , and we have seen that it is a contradiction.
Remark. If we follow the proof of Lemma 5.5, we can deduce that if L(η) = L 0 , it is possible for γ to escape from S 0 , but then L(η) = 2d 0 = L/2, and we also have d S (z, w) = d S0 (z, w) for every z, w ∈ S 0 . Many Riemann surfaces can be decomposed in a union of funnels and generalized Y -pieces (see [FM,Theorem 4.1] and [AR]). The following result uses this decomposition in order to obtain hyperbolicity.
We denote by L i for i = 1, 2, 3, the three lengths of the simple closed geodesics in ∂Y n (L i = 0 if its corresponding "geodesic" is a puncture).
We are going to consider different cases according to the values of L i .
(1) If L 0 ≤ L i ≤ l for i = 1, 2, 3, then the distance between any two simple closed geodesics of ∂Y n is less than or equal to K 0 ; therefore diam Yn (∪ m η nm ) ≤ l/2 + K 0 + l/2 = l + K 0 . Then we are in the hypothesis of Theorem 3.1, with c 2 = l + K 0 and B n = ∅.
(2) If L 1 < L 0 ≤ L 2 , L 3 ≤ l, then the distance between the simple closed geodesics of ∂Y n of length L 2 , L 3 , (say η nm , η nk ) is less than or equal to K 0 ; then diam Y n (η nm ∪ η nk ) ≤ l + K 0 . Then we are in the hypothesis of Theorem 3.1, with c 1 = l/2, c 2 = l + K 0 and A n = {m, k}.
(3) If L 1 , L 2 < L 0 , then we are in the hypothesis of Theorem 3.1, with c 1 = l/2 and A n = ∅.
The case of F m is similar to (3), with c 1 = l/2 and A n = ∅.
In fact, if l < L 0 , we only need to consider (3), and then Corollary 3.2 gives that we can take δ := 4δ 0 + l/2.