A new characterization of Gromov hyperbolicity for negatively curved surfaces

Publ. Mat. 50 (2006), 249–278 A NEW CHARACTERIZATION OF GROMOV HYPERBOLICITY FOR NEGATIVELY CURVED SURFACES Jos´ e M. Rodr´ ıguez(1) and Eva Tour´ ıs(1)(2) Abstract 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. 1. Introduction To understand the connections between graphs and Potential Theory on Riemannian manifolds (see e.g. [ARY], [CFPR], [FR2], [HS], [K1], [K2], [K3], [R1], [R2], [So]) Gromov hyperbolic spaces are a useful tool. Besides, the concept of Gromov hyperbolicity grasps the essence of negatively curved spaces, and has been successfully used in the theory of groups (see e.g. [GH], [G1], [G2] and the references therein). 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 Poincar´e 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 Z2-covering of a torus with genus two) is not hyperbolic. 2000 Mathematics Subject Classification. 30F, 30F20, 30F45. Key words. Gromov hyperbolicity, hyperbolic Riemann surface, closed geodesic. (1)Research partially supported by a grant from DGI (BFM 2003-04870), Spain. (2)Research partially supported by a grant from DGI (BFM 2000-0022), Spain. 250 J. M. Rodr´ ıguez, E. Tour´ ıs We are interested in studying when Riemann surfaces equipped with their Poincar´e 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 Sis hyperbolic if and only if the c0-triangles contained in simple closed geodesics of Ssatisfy the Rips condition. By a c0-triangle we mean a triangle with continuous injective (1, c0)-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 δ0, then every geodesic triangle satisfy it with constant δ= max{11, δ0+ 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 Poincar´e 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 {Fn}nwith limn→∞ L(∂Fn) = ∞, 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. A New Characterization of Gromov Hyperbolicity 251 Notations. We denote by Xor Xngeodesic metric spaces. By dX, LXand BXwe 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,Sor S0Riemann surfaces. We assume that the metric defined on these surfaces is the Poincar´e metric, unless the contrary is specified. If Ω is a plane domain, we shall denote by λΩthe conformal density of the Poincar´e metric in Ω, i.e. the function such that ds =λΩ(z)|dz| is the Poincar´e metric in Ω. We denote by ℜzand ℑzthe real and imaginary part of z, respectively. Finally, we denote by l,cand ci, positive constants which can assume different values in different theorems. Acknowledgements. We would like to thank Professor J. L. Fern´andez 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 facts about these spaces. We refer to [GH] for more background and further results. Definition 2.1. Let us fix a point win a metric space (X, d). We define the Gromov product of x, y ∈Xwith respect to the point was (x|y)w:= 1 2d(x, w) + d(y, w)−d(x, y)≥0. We say that the metric space (X, d) is δ-hyperbolic (δ≥0) if (x|z)w≥min(x|y)w,(y|z)w−δ, for every x, y, z, w ∈X. We say that Xis hyperbolic (in the Gromov sense) if the value of δis not important. It is convenient to remark that this definition of hyperbolicity is not universally accepted, since sometimes the word hyperbolic refers to negative curvature or to the existence of Green’s function. However, in this paper we only use the word hyperbolic in the sense of Definition 2.1. Examples. (1) Every bounded metric space Xis (diam X)-hyperbolic. 252 J. M. Rodr´ ıguez, E. Tour´ ıs (2) Every complete simply connected Riemannian manifold with sectional curvature which is bounded from above by −k, with k > 0, is hyperbolic. (3) Every tree with edges of arbitrary length is 0-hyperbolic. We refer the reader to [BHK], [GH] and [CDP] for further examples. Definition 2.2. If γ: [a, b]−→ Xis a continuous curve in a metric space (X, d), we can define the length of γas L(γ) := sup (n X i=1 d(γ(ti−1), γ(ti)) : a=t0< t1<···< tn=b). We say that γis a geodesic if it is an isometry, i.e. L(γ|[t,s]) = d(γ(t), γ(s)) = |t−s|for every s, t ∈[a, b]. We say that γis a local geodesic if for every t∈[a, b] there exists ε > 0 such that the restriction of γto [t−ε, t +ε]∩[a, b] is a geodesic. We say that Xis a geodesic metric space if for every x, y ∈Xthere exists a geodesic joining xand y; we denote by [x, y] any such geodesic (since we do not require uniqueness of geodesics, this notation is ambiguous, but it is convenient). Definition 2.3. If Xis a geodesic metric space and Jis a polygon whose sides are J1, J2,...,Jn, with Jj⊆X, we say that Jis δ-thin if for every x∈Jiwe have that d(x, ∪j6=iJj)≤δ. If x1, x2, x3∈X, a geodesic triangle T={x1, x2, x3}is the union of three geodesics J1:= [x1, x2], J2:= [x2, x3] and J3:= [x3, x1]. The space Xis δ-thin (or satisfies the Rips condition with constant δ) if every geodesic triangle in Xis δ-thin. Remark. Every geodesic quadrilateral in a δ-thin geodesic space is 2δ-thin. To see this, it is enough to divide the quadrilateral in two triangles. In general, every geodesic polygon of nsides is (n−2)δ-thin. If we have a triangle with two identical vertices, we call it a “bigon”; obviously, every bigon in a δ-thin space is δ-thin. A fundamental result is that hyperbolicity is equivalent to the Rips condition: Theorem A ([GH, p. 41]).Let us consider a geodesic metric space X. (1) If Xis δ-hyperbolic, then it is 4δ-thin. (2) If Xis δ-thin, then it is 4δ-hyperbolic. We present now the class of maps which play the main role in the theory. A New Characterization of Gromov Hyperbolicity 253 Definition 2.4. A function between two metric spaces f:X−→ Yis aquasi-isometry if there are constants a≥1, b≥0 with 1 adX(x1, x2)−b≤dY(f(x1), f(x2)) ≤a dX(x1, x2) + b, for every x1, x2∈X. Such a function is called an (a, b)-quasi-isometry. An (a, b)-quasigeodesic in Xis an (a, b)-quasi-isometry between an interval of Rand X. Let us observe that a quasi-isometry does not have to be continuous (for instance, the map f:R−→ Zsuch that f([n, n + 1)) = nfor every integer nis a (1,1)-quasi-isometry). Quasi-isometries are important since they are maps which preserve hyperbolicity: Theorem B ([GH, p. 88]).Let us consider an (a, b)-quasi-isometry between two geodesic metric spaces f:X−→ Y. If Yis δ-hyperbolic, then Xis δ′-hyperbolic, where δ′is a constant which depends only on δ, aand b. Definition 2.5. Let us consider H > 0, a metric space X, and subsets Y, Z ⊆X. The set VH(Y) := {x∈X:d(x, Y )≤H}is called the H-neighbourhood of Yin X. The Hausdorff distance of Yto Zis defined by H(Y, Z) := inf{H > 0 : Y⊆VH(Z), Z ⊆VH(Y)}. The following is a beautiful and useful result: Theorem C ([GH, p. 87]).For each δ, b ≥0and a≥1, there exists a constant H=H(δ, a, b)with the following property: Let us consider a δ-hyperbolic geodesic metric space Xand an (a, b)- quasigeodesic gjoining xand y. If γis a geodesic joining xand 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 X0is a subset connected by rectifiable paths of a metric space (X, d), then we associate to it the inner or intrinsic distance dX0(x, y) := dX|X0(x, y) := infL(γ) : γ⊂X0is a continuous curve joining xand y ≥dX(x, y). 254 J. M. Rodr´ ıguez, E. Tour´ ıs 3. Results in metric spaces We are interested in studying when non-exceptional Riemann surfaces equipped with their Poincar´e metric are Gromov hyperbolic. However, we have proved several results on hyperbolicity for general metric spaces, which are interesting by themselves and have consequences for Riemann surfaces (see Section 5). 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 Xand ε > 0. If γ is a continuous curve joining x, y ∈Xwith LX(γ)≤dX(x, y) + ε, then γis a (1, ε)-quasigeodesic with its arc-length parametrization. Proof: Let us consider γwith its arc-length parametrization γ: [0, l]−→ X. Since γis continuous, it is clear that dX(γ(t), γ(s)) ≤LX(γ([t, s])) = |t−s|. Let us show now |t−s| ≤ dX(γ(t), γ(s))+ε. We assume that there are 0 ≤t, s ≤lwith |t−s|> dX(γ(t), γ(s))+ε. Without loss of generality we can assume t < s. We define a curve γ0as a concatenation of three curves: γ([0, t]), a geodesic ηconnecting γ(t) with γ(s), and γ([s, l]). Since γ0is a continuous curve connecting xwith y, we have that dX(x, y)≤LX(γ0) = LX(γ)−LX(γ([t, s])) + dX(γ(t), γ(s)) =LX(γ)−|t−s|+dX(γ(t), γ(s)) < LX(γ)−ε≤dX(x, y), which is a contradiction. Corollary 3.1. Let us consider a geodesic metric space Xand ε > 0. If γis a continuous curve with LX(γ)≤ε, then γis a (1, ε)-quasigeodesic with its arc-length parametrization. Lemma 3.2. Let us consider a metric space Xwith a closed geodesic g of length l. If γis a continuous injective (1, c)-quasigeodesic in Xwith its arc-length parametrization, and it is contained in g, then L(γ)≤ (l+c)/2. A New Characterization of Gromov Hyperbolicity 255 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. 2. If γis a geodesic, it is clear that L(γ)≤l/2; Lemma 3.2 generalizes this fact. Proof: Let us consider γwith its arc-length parametrization γ: [0, l0]−→ X. Assume that l0>(l+c)/2; then l−l0< l0−c. Observe that d(γ(0), γ(l0)) ≤l−l0, since g\γis a continuous curve of length l−l0 joining γ(0) and γ(l0) (γis continuous and injective). Hence, we have that l0−c≤d(γ(0), γ(l0)) ≤l−l0< l0−c, which is a contradiction. Lemma 3.3. Every (a, b)-quasigeodesic triangle in a δ-hyperbolic geodesic metric space Xis (4δ+ 2H(δ, a, b))-thin, where His the constant in Theorem C. Proof: Given an (a, b)-quasigeodesic triangle in Xof sides q1,q2, q3, Theorem C gives that there exist geodesics g1,g2,g3, such that gihas the same end points as qiand H(gi, qi)≤H=H(δ, a, b). If {i, j, k}is any permutation of {1,2,3},and x∈qi, then there is a point x′∈gi with d(x, x′)≤H. Since Xis 4δ-thin, we can find y′∈gj∪gkwith d(y′, x′)≤4δ. We also have a point y∈qj∪qkwith d(y′, y)≤H. Consequently d(x, qj∪qk)≤d(x, y)≤4δ+ 2H. 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 {Xn}n∈Λ are a (c1, c2)-regular decomposition of the geodesic metric space Xif they verify the following conditions: (a) X=∪n∈ΛXnand Xn∩Xm=ηnm, where for each n∈Λ, {ηnm}m∈Λ\{n}are pairwise disjoint closed subsets of Xn(ηnm =∅ is allowed); furthermore any geodesic in Xwith finite length meets at most a finite number of ηnm’s. (b) For any n, m ∈Λ, diamXn(ηnm)≤c1and if ηnm 6=∅, then X\ηnm is not connected and a,bare in different connected components of X\ηnm for any a∈Xn\ηnm,b∈Xm\ηnm. 256 J. M. Rodr´ ıguez, E. Tour´ ıs (c) For each n∈Λ there exist disjoint sets An, Bn⊆Λ, verifying the following properties: if m /∈An∪Bn, then ηnm =∅; diamXn(∪m∈Anηnm)≤c2; and every geodesic joining two points in Xncannot escape from Xnacross a ηnm with m∈Bn. Remarks. 1. The sets Λ, Anand Bndo not need to be countable. 2. The hypothesis on X\ηnm guarantees that the graph R= (V, E) constructed in the following way is a tree: V=∪n∈Λ{vn}and [vn, vm]∈Eif and only if ηnm 6=∅. 3. We can think that the hypothesis “a geodesic joining two points in Xncannot escape from Xnacross a ηnm with m∈Bn”, is very restrictive; however, Lemma 5.5 below gives a very simple condition which allows one to assure this hypothesis. 4. If Xis a Riemann surface, {Xn}n∈Λare bordered Riemann surfaces and ηnm ⊂∂Xn∩∂Xm, condition “a,bare in different components of X\ηnm for any a∈Xn\ηnm,b∈Xm\ηnm” in (b), is a consequence of “X\ηnm is not connected”. 5. We wish to emphasize that condition diamXn(ηnm)≤c1is 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 ηnm) and wide parts, the problem is more difficult and interesting. Theorem 3.1. Let us consider a (c1, c2)-regular decomposition {Xn}n∈Λ of the geodesic metric space X. If there exists a constant δ0such that Xnis δ0-thin for every n∈Λ, then Xis δ-thin with δ= 20δ0+max{c1+ c2/2, c2}. Proof: Let us consider a geodesic triangle T={a, b, c}in X. If T⊆Xn for some n, then Tis δ0-thin, by hypothesis. We assume now that Tintersects several Xn’s. We intend to study Tin each of those Xn’s separately. Let us take z∈T. If zbelongs to two sides of T, there is nothing to prove; if zonly belongs to one side of T, we denote by Athe union of the sides of Twhich do not intersect z. Let us fix n∈Λ. We assume first that the three sides of Tintersect Xn. We construct a geodesic polygon Pnin Xnmodifying T∩Xnin the following way: Let us consider a side γi(i= 1,2,3) of T. If γi⊆Xn, we define gi:= γi. If γiis not contained in Xn, then we consider A New Characterization of Gromov Hyperbolicity 257 γi: [0, l]−→ X. Let us define ti 1:= min{0≤t≤l:γi(t)∈Xn}, ti 4:= max{0≤t≤l:γi(t)∈Xn}. If γi([ti 1, ti 4]) ⊆Xn, we consider gi:= γi([ti 1, ti 4]). In other case, we define ti 2:= min{0≤t≤l:γi(t)∈ ∪m∈Anηnm}, ti 3:= max{0≤t≤l:γi(t)∈ ∪m∈Anηnm}, and gi:= γi([ti 1, ti 2]) ∪[γi(ti 2), γi(ti 3)] ∪γi([ti 3, ti 4]), where we choose a geodesic [γi(ti 2), γi(ti 3)] in Xn. This minimum and this maximum exist since γiis a continuous function in a compact interval and γi∩(∪m∈Anηnm) is a compact set: each ηnm is a closed set and γimeets at most a finite number of ηnm’s. It is possible that g1∪g2∪g3is not a polygon, since there can exist gaps between two gi’s. Since diamXn(ηnm)≤c1and X\ηnm is not connected for any m∈Λ, we can find three geodesics h1,h2,h3in Xn of length less or equal than c1such that g1∪h1∪g2∪h2∪g3∪h3is a geodesic polygon Pnin Xn(some hican be a point). It is clear that Pnhas at most 12 sides, and then it is 10δ0-thin. Without loss of generality we can assume that z∈g1. In order to simplify the notation, we define xj:= γ1(t1 j) for 1 ≤j≤4. If g1:= γ1([t1 1, t1 4]) = [x1, x4], then there exists w′∈Pn\int g1 with dXn(z, w′)≤10δ0, where int g1denotes g1without its end points. If w′∈A, then dX(z, A)≤10δ0; if w′/∈A, then there exists w∈ Pn∩Awith dXn(w, w′)≤max{c1, c2/2}, and therefore dX(z, A)≤ 10δ0+ max{c1, c2/2}. Let us assume now that g1:= [x1, x2]∪[x2, x3]∪[x3, x4]. Recall that [x1, x2]∪[x3, x4]⊆γi⊆T, and LX([x2, x3]) ≤c2. We denote by a1∈[x1, x2] the point farther of x2such that dXn(a1,[x2, x3]) ≤10δ0; in a similar way, we define a2∈[x3, x4] as the point farther of x3such that dXn(a2,[x2, x3]) ≤10δ0; then dXn(a1, a2)≤20δ0+c2. Let us consider b1∈[a1, x1] the point farther of a1such that dXn(b1,[x3, x4]) ≤10δ0(if this b1does not exist, we take b1:= a1) and b2∈[a2, x4] the point farther of a2such that dXn(b2,[x1, x2]) ≤10δ0(if this b2does not exist, we take b2:= a2). If b16=a1, then dX(b1, x3) = LX([b1, x3]) = dX(b1,[x3, x4]) ≤10δ0; in a similar way, if b26=a2, then dX(b2, x2)≤10δ0. We consider now the next four possibilities: 264 J. M. Rodr´ ıguez, E. Tour´ ıs to a point; since T1is homotopic to a point, the above argument implies that T1is δ0-thin. Given x∈[a, B] there is some y∈[a, C]∪[B, C] with dAl(x, y)≤δ0; if y∈[a, C], then dAl(x, [a, C]) ≤δ0; if y∈[B, C], we have dAl(x, [a, C]) ≤dAl(x, y) + dAl(y, C)≤δ0+l. If x∈[a, C], we obtain a similar result. Let us consider the quadrilateral Q1={b, c, C, B}, where we choose as [B, C] the local geodesic gi⊂gsuch that [b, c]∪[c, C]∪[C, B]∪[B, b] is homotopic to a point; since Q1is homotopic to a point, the above argument implies that Q1is 2δ0-thin. In a similar way to the case of T1, given any point in T∩Q1there is a point y∈T∩Q1(in other side of T) with dAl(x, y)≤2δ0+l. Then Tis (2δ0+l)-thin. Let us assume now that l > 0 and T∩g=∅. Next, we find an upper bound for dAl(T, g). Given a point wof T, we denote by w0 the point in gwith dAl(w, w0) = dAl(w, g). If T={a, b, c}, we have that dAl(a0, b0) + dAl(b0, c0) + dAl(c0, a0) = l. Hence, without loss of generality we can assume that dAl(a0, b0)≥l/3. Let us consider the point x∈[a, b] with dAl(x, g) = dAl([a, b], g). We consider first the case x∈(a, b). We can assume that t:= dAl(a0, x0)≥l/6. We consider now the geodesic quadrilateral Q:= {a, a0, x0, x}with three right angles (known as Lambert quadrilateral). If s:= dAl(x0, x) and φis the angle of [a, a0] and [a, x] in a, the trigonometric formulas give sinh ssinh t= cos φ(see e.g. [B, p. 157], [C, p. 263]). Then sinh s=cos φ sinh t<1 sinh t≤1 sinh(l/6). Therefore, we have that (5.1) dAl(T, g)<Arcsinh 1 sinh(l/6). If x=aor x=b, a similar argument with t:= dAl(a0, b0) gives sinh ssinh t < 1, and we obtain sinh s < 1/sinh(l/3), which also implies (5.1). Without loss of generality we can assume that dAl(T, g) = dAl(x, g) = dAl(x, x0) = s. Let us consider the local geodesic gxstarting and finishing in x, which is freely homotopic to g. We consider first the case x∈(a, b). We denote by 2dxthe length of gxand by ythe point in gxat distance dxof x. We consider the geodesic quadrilateral R:= {x, x0, y0, y}with three right angles. Since dAl(x0, y0) = l/2, the trigonometric formulas give A New Characterization of Gromov Hyperbolicity 265 (see e.g. [F, p. 88]) sinh dx= sinh(l/2) coshs= sinh(l/2)p1 + sinh2s <sinh(l/2)q1 + cosech2(l/6) = sinh(l/2) cotanh(l/6). Let us assume now that l= 0, i.e. that we deal with the case A0=D∗; then Tis freely homotopic to the puncture. We consider the universal covering map π:U−→ D∗, given by π(z) = exp(2πiz). It is clear that πmaps bijectively U0:= {z∈U: 0 ≤ ℜz < 1}) in D∗. Without loss of generality we can assume that π(z1) = a,π(z2) = band π(z3) = c, with ℜz1= 0 and 1/3≤ ℜz2≤ ℜz3<1. Since ℜ(z2−z1)≥1/3, there exists a point z∈[z1, z2] with ℑz > 1/6; then max ℑz:π(z)∈T>1/6. We denote by z0a point of U0in which this maximum is attained. Let us consider the local geodesic g0in D∗starting and finishing in π(z0), which is freely homotopic to the puncture; if we denote by 2dπ(z0) the length of g0, (4.1) gives that sinh2dπ(z0)= sinh2dU(z0,1 + z0) 2<sinh2dU(i/6,1 + i/6) 2= 9, and consequently dπ(z0)<Arcsinh 3. 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∈Tsuch that the local geodesic gp in Alstarting and finishing in p, which is freely homotopic to gor to the puncture, has length 2dp<2d(l). Let us assume first that pis not a vertex of T; without loss of generality we can assume also that p∈[a, c]. Since gpis freely homotopic to T, we have a geodesic pentagon P′:= {a′, b′, c′, p′ 1, p′ 2}in D, which is isometric to the pentagon Pmade of [a, b], [b, c], [c, p], gpand [p, a], if we identify p′ 1with p′ 2(we have chosen P′such that dD(a′, b′) = dAl(a, b), dD(b′, c′) = dAl(b, c), dD(c′, p′ 1) = dAl(c, p), dD(p′ 1, p′ 2) = LAl(gp) and dD(p′ 2, a′) = dAl(p, a)). It is clear that if x′,y′, are the corresponding points in P′to the points x, y ∈P, we have dAl(x, y)≤dD(x′, y′). Now we use a similar argument to the one in the proof of Theorem 3.1. Since P′is a geodesic pentagon in D, we have that it is 3δ0-thin. Let us consider the point α′ 1in the oriented geodesic [p′ 1, c′], defined by α′ 1:= max{z∈[p′ 1, c′] : dD(z, [p′ 1, p′ 2]) ≤3δ0}, and the point α′ 2 in the oriented geodesic [p′ 2, a′], defined by α′ 2:= max{z∈[p′ 2, a′] : dD(z, [p′ 1, p′ 2]) ≤3δ0}. 266 J. M. Rodr´ ıguez, E. Tour´ ıs If αjis the corresponding point in Pto α′ j, we have that LAl([α1, α2])= dAl(α1, α2)≤6δ0+d(l), since dAl(αj, gp)≤3δ0and diamAl(gp)≤dp< d(l). We define now β′ 1:= max {α′ 1}∪{z∈[p′ 1, c′] : dD(z, [p′ 2, a′]) ≤3δ0}, β′ 2:= max {α′ 2} ∪ {z∈[p′ 2, a′] : dD(z, [p′ 1, c′]) ≤3δ0}. Let us denote by βjthe corresponding point in Pto β′ j. If β16=α1, then dAl(β1, p) = LAl([β1, p]) = dAl(β1,[p, a]) ≤3δ0; in a similar way, if β26=α2, then dAl(β2, p)≤3δ0. We consider now the next four possibilities: •If β1=α1and β2=α2, we have seen that dAl(β1, β2)≤6δ0+d(l). •If β16=α1and β26=α2, then dAl(β1, β2)≤dAl(β1, p)+dAl(p, β2)≤ 6δ0. •If β16=α1and β2=α2, then there is a point z0∈[p′ 1, p′ 2] with dD(β′ 2, z0)≤3δ0; since there is some p′ iwith dD(p′ i, z0)≤d(l), we obtain that dAl(β1, β2)≤dD(β′ 1, β′ 2)≤dD(β′ 1, p′ i) + dD(p′ i, z0) + dD(z0, β′ 2)≤6δ0+d(l). •If β1=α1and β26=α2, we obtain in a similar way that dAl(β1, β2)≤ 6δ0+d(l). Therefore, in the four situations we have dAl(β1, β2)≤6δ0+d(l). If x∈[β1, c]∪[β2, a], then dAl(x, [a, b]∪[b, c]) ≤3δ0. If x∈[β1, β2], we can take βiwith dAl(x, βi)≤3δ0+d(l)/2; since dAl(βi,[a, b]∪[b, c]) ≤3δ0, we obtain dAl(x, [a, b]∪[b, c]) ≤6δ0+d(l)/2. If x∈[a, b], there exists a point y′∈P′\(a′, b′) with dD(x′, y′)≤3δ0. If y′/∈[p′ 1, p′ 2], then dAl(x, [b, c]∪[c, a]) ≤3δ0. If y′∈[p′ 1, p′ 2], there is p′ i with dD(y′, p′ i)≤d(l), and hence dD(x′, p′ i)≤3δ0+d(l). Since p∈[a, c], we have that dAl(x, [b, c]∪[c, a]) ≤3δ0+d(l). A similar result is true if x∈[b, c]. These facts give that Tis max{3δ0+d(l),6δ0+d(l)/2}-thin. If pis 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 Awith simple closed geodesic γ. A generic triangle Tin Ais determined by the coordinates of three points, i.e., by six real coordinates; however, a generic triangle T0 in the simple closed geodesic γis determined by three real coordinates. Therefore Theorem 5.1 is a remarkable improvement of Rips condition in the context of Riemann surfaces. A New Characterization of Gromov Hyperbolicity 267 Definition 5.2. By a c0-triangle we mean a triangle with continuous injective (1, c0)-quasigeodesic sides, with its arc-length parametrization. We define the constants c0:= log(5 + 2√6) <2.2925, K:= 2 log(1 + √2) + log(5 + 2√6) + log √6 + √10 2<5.0869. Theorem 5.1. Let us consider a non-exceptional Riemann surface S (with or without boundary); if Shas boundary, we also require that ∂S is the union of local geodesics (closed or non-closed). Then Sis hyperbolic if and only if every c0-triangle contained in a simple closed geodesic in S is δ0-thin. Furthermore, if every c0-triangle contained in a simple closed geodesic in Sis δ0-thin, then Sis δ-thin, with δ= max{δ(4c0), δ0+K}, where δ(t)is the constant in Lemma 5.4 (it verifies δ(4c0)<10.9325). Remarks. 1. Although one can think of quasigeodesic triangles as an artificial technical device, the example after the proof of Theorem 5.1 shows that they are essential. 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, c0)-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 Tin Swith a c0-triangle contained in a simple closed geodesic γ in S. In some way, we can consider Tand γas “subsets” of the annulus Al (with l:= LS(γ)). The geodesic triangles in the simple closed geodesic of Alare (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 lis big we need a narrow metric relationship between Tand γ. If Sis hyperbolic, Lemma 3.3 guarantees that every c0-triangle in S is δ0-thin. Let us assume that every c0-triangle contained in a simple closed geodesic in Sis δ0-thin. First, we want to remark that if Shas boundary, 268 J. M. Rodr´ ıguez, E. Tour´ ıs 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 Rby pasting to Sa funnel in each simple closed geodesic, and a half-disk in each non-closed simple geodesic. Since Sis geodesically convex in R(every geodesic connecting two points of Sis contained in S), then dR(z, w) = dS(z, w) for every z, w∈S, and any simple closed geodesic in Ris contained in S. Let us consider a geodesic triangle Tin S. By Lemma 2.1 in [RT1], we can assume that Tis a simple closed curve. We have three possibilities: Tis homotopic to a point, Tis homotopic to a puncture, or Tis freely homotopic to a simple closed geodesic in S. This is well known if Shas no boundary; if Shas boundary, it is enough to apply the result to R, since Rhas not additional topological obstacles (the fundamental groups of Sand Rare isomorphic). If Tis 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 Tis log(1 + √2)-thin, since Dis log(1 + √2)-thin (see [An, p. 130]). If Tis 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 Tis δ(0)-thin, with δ(0) the constant in Lemma 5.4. Since every geodesic triangle in Dis isometric to some geodesic triangle in D∗, we have that log(1+√2)≤δ(0). In other case, Tis freely homotopic to a simple closed geodesic γin S. If L(γ)<4c0, let us consider the annulus AL(γ)with a simple closed geodesic gof length L(γ). We have that AL(γ)is δ(L(γ))-thin, with δ(L(γ)) the constant in Lemma 5.4. Since d=d(l) = Arcsinh sinh(l/2) sinh(l/6) cosh(l/6), if l > 0 and d(0) = liml→0d(l), we have that d=d(l) is an increasing function for l≥0; then we also have that δ(0) ≤δ(L(γ)) < δ(4c0), with δ(4c0) = max 4c0+ 2 log(1 + √2), Arcsinh sinh(2c0) cotanh(2c0/3)+ 3 log(1 + √2), 1 2Arcsinh sinh(2c0) cotanh(2c0/3)+ 6 log(1 + √2) = 4c0+ 2 log(1 + √2) <10.9325. A New Characterization of Gromov Hyperbolicity 269 In this case, the closed set in Sbounded by Tand γis, with its intrinsic distance, isometric to a set in AL(γ), bounded by gand a triangle T0. These facts give that Tis δ(4c0)-thin. We consider now the case L(γ)≥4c0. First, we assume that γ∩T=∅. If ηis a side of T, we associate to it two curves η′,η′′, in the following way. We consider a simply connected locally geodesic quadrilateral Qin Swith pairwise disjoint sides A,C,Band η, of lengths a,c,band l0, respectively, with the following conditions: (i) C⊂γ, (ii) Cmeets orthogonally the sides A and B.Qis uniquely determined by these conditions. If c≥c0, the arc η′:= A∪C∪Bis a continuous injective (1, c0)-quasigeodesic with its arc-length parametrization by Lemmas 3.1 and 5.1. If c < c0, we take η′:= η, which is a geodesic. (Observe that we have c < c0for at most one side of T, since L(γ)≥4c0; in other case, Twould not be a geodesic triangle.) In both cases, we define η′′ := C⊂γ. We have that η′′ is always a continuous injective (1, c0)-quasigeodesic with its arclength parametrization: this is clear if c≥c0(since η′′ ⊂η′), and it is a consequence of Corollary 3.1 if c < c0. If Tis the union of the geodesics η1,η2,η3, we consider the (1, c0)- quasigeodesic triangle T′defined as the union of the (1, c0)-quasigeodesics η′ 1,η′ 2,η′ 3. We consider also the (1, c0)-quasigeodesic triangle T′′ ⊂γdefined as the union of the (1, c0)-quasigeodesics η′′ 1,η′′ 2,η′′ 3. By hypothesis, T′′ is δ0-thin. We prove now that T′is δ1-thin, with δ1:= max{δ0,2 log(1 + √2)}+c0. If η′ i6=ηi, for i= 1,2,3,then T′is δ0-thin, since every point in T′\T′′ belongs to two sides of T′. If it is not the case, there is only one iwith η′ i=ηi; we can assume η′ 1=η1. Let us consider the quadrilateral Q1with sides A1,C1,B1 and η1; we have that L(C1)< c0. Since Q1is simply connected, it is isometric to a quadrilateral in Dwhich is 2 log(1 + √2)-thin. Then for each z∈η′ 1=η1, there exists w∈A1∪C1∪B1with d(z, w)≤2 log(1+√2). If w∈A1∪B1, then d(z, η′ 2∪η′ 3)≤2 log(1+√2). If w∈C1, then there exists w′∈A1∪B1with d(w, w′)≤c0(since L(C1)< c0), and we have d(z, η′ 2∪η′ 3)≤2 log(1 + √2) + c0. If z∈η′ 2, we consider three cases. If z∈η′ 2∩γ=η′′ 2, then d(z, η′ 1∪ η′ 3)≤d(z, η′′ 3)≤d(z, η′′ 1∪η′′ 3) + c0≤δ0+c0. If z∈η′ 2∩η′ 3, then d(z, η′ 1∪η′ 3) = 0. In other case, z∈A1∪B1(we can assume that A1⊂η′ 2and B1⊂η′ 3); then there exists w∈B1∪C1∪η1with d(z, w)≤ 2 log(1 + √2); since L(C1)< c0, there exists w′∈B1∪η1⊂η′ 3∪η′ 1with d(w, w′)≤c0, and we have d(z, η′ 1∪η′ 3)≤d(z, w′)≤2 log(1 + √2) + c0. 270 J. M. Rodr´ ıguez, E. Tour´ ıs Consequently, T′is δ1-thin, with δ1:= max{δ0,2 log(1 + √2)}+c0. The case z∈η′ 3is similar to z∈η′ 2. We show now that Tis δ2-thin, with δ2:= δ1+ 2 log(1 + √2) + c2, and c2:= Arcsinh cotanh c0 2= log √6 + √10 2. Let us consider x∈T; we can assume that x∈η1. If η16=η′ 1, then η1∪η′ 1is a simply connected geodesic quadrilateral, and therefore there exists x′∈η′ 1with d(x, x′)≤2 log(1 + √2). If η1=η′ 1, we take x′=x. Then there exists y′∈η′ 2∪η′ 3with d(x′, y′)≤δ1; without loss of generality we can assume that y′∈η′ 2. If η26=η′ 2, Lemma 5.3 gives that there exists y∈η2with d(y, y′)< c2. If η2=η′ 2, we take y=y′. Consequently we have that d(x, y)< δ2:= δ1+ 2 log(1 + √2) + c2= max{δ0,2 log(1 + √2)}+K. Therefore Tis δ-thin, with δ:= max{δ(4c0), δ0+K, 2 log(1 + √2) + K}= max{δ(4c0), δ0+K}, since δ(4c0)>10 >2 log(1 + √2) + K. We assume now that γ∩T6=∅. If γ∩Thas only one connected component, the same argument works. If γ∩Thas 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 < c0. 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 Tin a triply connected Riemann surface S0such that T′′ is not geodesic. Given x0<Arcsinh 1, there exists y > 0 with sinh(x0+y)>cosh y. Then sinh(x+y)>cosh yfor any x0≤x < Arcsinh 1, and consequently we can choose some x < Arcsinh 1 such that sinh xsinh(x+y)>cosh y. If we define ε:=Arcsinh(1/sinh x)−x > 0, we have that sinh xsinh(x+ ε) = 1. Let us consider a geodesic quadrilateral Vwith 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 Y0with two punctures and a simple closed geodesic γwith L(γ) = 4(x+ε). We obtain S0by gluing Y0with a funnel Fwhose simple closed geodesic has length 4(x+ε). A New Characterization of Gromov Hyperbolicity 271 Let us denote by µ0the geodesic in Y0with L(µ0) = 2x, joining γ with itself which is not homotopic to any curve contained in γ. We denote by p′′,q′′ the end points of µ0. Let us consider the non bounded geodesic µin S0which contains µ0, and the two points p, q ∈µ∩Fat distance yof γ. Let us define the triangle Tas the union of the two geodesics α,β in Fjoining pand q(in fact, Tis a geodesic “bigon”). The length of the segment of µbetween pand qis 2x+ 2y; by [F, p. 88] we have sinh(L(α)/2) = sinh(x+ε) cosh y= cosh y/ sinh x < sinh(x+y); then we obtain L(α)<2x+ 2y, and consequently α,βare in fact geodesics in S0. However, T′′ ={p′′, q′′}is contained in γand then L(α′′) = L(β′′) = 2x+ 2ε > 2x=L(µ0); hence α′′,β′′ are not geodesics in S0. From now on we will obtain several consequences of Theorem 5.1. Corollary 5.1. The annulus Alsuch that its simple closed geodesic has length l≥4c0is (l/4 + K)-thin, with K < 5.0869 the constant in Theorem 5.1. The same is true for each funnel of Al. Remark. This bound of the hyperbolicity constant for the annulus is asymptotically sharp: we have that the best thin constant of Alis 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 Alis δ2-thin, if l≥4c0. In this case the hypothesis “any continuous injective (1, c0)-quasigeodesic triangle contained in a simple closed geodesic in Sis δ0-thin”, can be changed by “any geodesic triangle contained in the simple closed geodesic γof Alis δ0-thin”, since T′′ is a geodesic triangle in Alif Tis a geodesic triangle in Al. Since the sides of any geodesic triangle contained in γhave length less than or equal to l/2, any geodesic triangle contained in γis δ0-thin, with δ0=δ0(Al) = l/4. Consequently, we obtain that Alis δ2-thin with δ2= max l 4,2 log(1 + √2)+K=l 4+K, since l/4≥c0>2>2 log(1 + √2). The same is true for each funnel of Al. 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. 272 J. M. Rodr´ ıguez, E. Tour´ ıs Corollary 5.2. Let us consider a non-exceptional Riemann surface S (with or without boundary) of genus 0; if Shas boundary, we also require that ∂S is the union of local geodesics (closed or non-closed). If Sis of finite type, then it is hyperbolic. In fact, if every simple closed geodesic γ in Sverifies L(γ)≤l, then Sis δ-thin, with δ= max{δ(4c0), K + (l+ c0)/4}and c0, δ(4c0), K the constants in Theorem 5.1. Proof: The set of simple closed geodesics in Sis finite: {γ1,...,γk}, and we have L(γj)≤l. Every continuous injective (1, c0)-quasigeodesic with its arc-length parametrization g⊂γjverifies L(g)≤(l+c0)/2 by Lemma 3.2; hence d(z, ∂g)≤(l+c0)/4 for every z∈g. Then the hypothesis of Theorem 5.1 is verified with δ0:= (l+c0)/4. Hence Sis δ-thin with δ= max{δ(4c0), K + (l+c0)/4}. A consequence of this corollary is the following result. Corollary 5.3. Every generalized Y-piece Ywith L(γi)≤l, where γi (i= 1,2,3) are the simple closed geodesics in ∂Y , is δ-thin, with δ= max{δ(4c0), K + (l+c0)/4}. 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. Ahalf-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:ℜz≥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 {Fn}nwith limn→∞ L(∂Fn) = ∞, 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. A New Characterization of Gromov Hyperbolicity 273 Theorem 5.2. Let us consider a non-exceptional Riemann surface S (with or without boundary); if Shas boundary, we also require that ∂S is the union of local geodesics (closed or non-closed). Let us denote by Fthe union of some pairwise disjoint funnels and half-disks of S. Let S0be the bordered non-exceptional Riemann surface obtained by deleting from S the interior of F. Then Sis hyperbolic if and only if S0is hyperbolic. Furthermore, if Sis δ-thin (hyperbolic), then S0is δ-thin (hyperbolic); if S0is δ′-hyperbolic, then Sis δ-thin, with δ= max{δ(4c0),4δ′+ 2H(δ′,1, c0)+K},c0, δ(4c0), K the constants in Theorem 5.1, and Hthe constant in Theorem C. 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 Fnwith L(∂Fn)−→ ∞ as n−→ ∞ (see the examples in Section 4 of [RT2]). Proof: Let us assume that Sis δ-thin (hyperbolic). As S0is geodesically convex in S(every geodesic connecting two points of S0is contained in S0), then dS(z, w) = dS0(z, w) for every z, w ∈S0. Therefore S0is also δ-thin (hyperbolic). Let us assume now that S0is δ′-hyperbolic. By Lemma 3.3, every (1, c0)-quasigeodesic triangle Tin S0is (4δ′+ 2H(δ′,1, c0))-thin, where His the constant in Theorem C. Let us observe that any simple closed geodesic in Sis contained in S0. Since dS(z, w) = dS0(z, w) for every z, w ∈S0, every (1, c0)-quasigeodesic triangle in S(contained in a simple closed geodesic in S) is also a (1, c0)-quasigeodesic triangle in S0. Let us observe also that H≥1>log(1 + √2). Then Theorem 5.1 gives that Sis δ-thin, with δ= max{δ(4c0),4δ′+ 2H(δ′,1, c0) + K}. 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. Lemma 5.5. Let us consider a non-exceptional Riemann surface S(with or without boundary), a simple closed geodesic ηof Ssuch that S\ηis not connected, and the closure S0of a connected component of S\η. We define L0:= 4 Arccosh t0, where t0is the unique solution greater than 1 of the equation 2t3−2t−1 = 0: t0:= 3 s9 + √33 36 +1 3 3 s36 9 + √33 <1.1915. If L(η)< L0, then every geodesic connecting two points of S0is contained in S0, and consequently dS(z, w) = dS0(z, w)for every z, w ∈S0.