Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics
Peter Hästö, Henri Linden, Ana Portilla, Jose M. Rodriguez, E. Touris
aa r X i v : . [ m a t h . C V ] M a y GROMOV HYPERBOLICITY OF DENJOY DOMAINS WITHHYPERBOLIC AND QUASIHYPERBOLIC METRICS
PETER H ¨AST ¨O ∗ , HENRI LIND´EN, ANA PORTILLA † , JOS´E M. RODR´IGUEZ †‡ ,AND EVA TOUR´IS †‡ Abstract.
We obtain explicit and simple conditions which in many cases allow onedecide, whether or not a Denjoy domain endowed with the Poincar´e or quasihyperbolicmetric is Gromov hyperbolic. The criteria are based on the Euclidean size of thecomplement. As a corollary, the main theorem allows to deduce the non-hyperbolicityof any periodic Denjoy domain. Introduction
In the 1980s Mikhail Gromov introduced a notion of abstract hyperbolic spaces, whichhave thereafter been studied and developed by many authors. Initially, the researchwas mainly centered on hyperbolic group theory, but lately researchers have shownan increasing interest in more direct studies of spaces endowed with metrics used ingeometric function theory.One of the primary questions is naturally whether a metric space (
X, d ) is hyperbolicin the sense of Gromov or not. The most classical examples, mentioned in every textbookon this topic, are metric trees, the classical Poincar´e hyperbolic metric developed in theunit disk and, more generally, simply connected complete Riemannian manifolds withsectional curvature K − k < j -metric (see [12]) is Gromov hyper-bolic; and that the Vuorinen j -metric (see [12]) is not Gromov hyperbolic except in thepunctured space. Also, in [14] the hyperbolicity of the conformal modulus metric µ andthe related so-called Ferrand metric λ ∗ , is studied.Since the Poincar´e metric is also the metric giving rise to what is commonly knownas the hyperbolic metric when speaking about open domains in the complex plane or inRiemann surfaces, it could be expected that there is a connection between the notionsof hyperbolicity. For simply connected subdomains Ω of the complex plane, it followsdirectly from the Riemann mapping theorem that the metric space (Ω , h Ω ) is in fact Date : October 26, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Poincar´e metric, hyperbolic metric, quasihyperbolic metric, Gromov hyper-bolic, Denjoy domain. ∗ Supported in part by the Academy of Finland. † Supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 andMTM 2006-26627-E), Spain. ‡ Supported in part by a grant from U.C.III M./C.A.M. (CCG06-UC3M/EST-0690), Spain.
Gromov hyperbolic. However, as soon as simple connectedness is omitted, there is noimmediate answer to whether the space h Ω is hyperbolic or not. The question has latelybeen studied in [2] and [18]–[24].The related quasihyperbolic metric has also recently been a topic of interest regard-ing the question of Gromov hyperbolicity. In [8], Bonk, Heinonen and Koskela foundnecessary and sufficient conditions for when a planar domain D endowed with the quasi-hyperbolic metric is Gromov hyperbolic. This was extended by Balogh and Buckley,[4]: they found two different necessary and sufficient conditions which work in Euclideanspaces of all dimensions and also in metric spaces under some conditions.In this article we are interested in Denjoy domains. In this case either the result of [8]or [4] implies that the domain is Gromov hyperbolic with respect to the quasihyperbolicmetric if and only if the domain is inner uniform (see Section 3). Although this is a vreynice characterization, it is somewhat difficult to check that a domain is inner uniform,since we need to construct uniform paths connecting every pair of points.In this paper we show that it is necessary to look at paths joining only a very small(countable) number of points when we want to determine the Gromov hyperbolicity.This allows us to derive a simple and very concrete conditions on when the domain isGromov hyperbolic. Much more importantly, our methods also suggest correspondingresults for the hyperbolic metric, which are also proven. To the best of our knowledge,this is the first time that Gromov hyperbolicity of any class of infinitely connecteddomains has been obtained from conditions on the Euclidean size of the complement ofthe domain.The main results in this article are the following: Theorem 1.1.
Let Ω be a Denjoy domain with Ω ∩ R = ( −∞ , ∪ S ∞ n =1 ( a n , b n ) , b n a n +1 for every n , and lim n →∞ a n = ∞ . (1) The metrics k Ω and h Ω are Gromov hyperbolic if lim inf n →∞ b n − a n a n > . (2) The metrics k Ω and h Ω are not Gromov hyperbolic if lim n →∞ b n − a n a n = 0 . It is interesting to note that in the case of Denjoy domains many of the results seemto hold for both the hyperbolic and the quasihyperbolic metrics. In fact, we know ofno planar domain which is Gromov hyperbolic with respect to one of these metrics, butnot the other.In the previous theorems, the boundary components had a single accumulation point,at ∞ , and the accumulation happened only from one side. It turns out that if this kindof domain is not Gromov hyperbolic, then we cannot mend the situation by adding someboundary to the other side of the accumulation point, as the following theorem shows. Theorem 1.2.
Let Ω be a Denjoy domain with ( −∞ , ⊂ Ω and let F ⊆ ( −∞ , be closed. If k Ω is not Gromov hyperbolic, then neither is k Ω \ F ; if h Ω is not Gromovhyperbolic, then neither is h Ω \ F . We also prove the non-hyperbolicity of any periodic Denjoy domain:
ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 3
Corollary 1.3.
Let E ⊂ [0 , t ) be closed, t > , set E n := E + tn for n ∈ N or n ∈ Z ,and Ω := C \ ∪ n E n . Then h Ω and k Ω are not Gromov hyperbolic. Definitions and notation By H we denote the upper half plane, { z ∈ C : Im z > } , by D the unit disk { z ∈ C : | z | < } . For D ⊂ C we denote by ∂D and D its boundary and closure, respectively.For z ∈ D ( C we denote by δ D ( z ) the distance to the boundary, min a ∈ ∂D | z − a | .Finally, we denote by c , C , c j and C j generic constants which can change their valuefrom line to line and even in the same line.Recall that a domain Ω ⊂ C is said to be of hyperbolic type if it has at least twofinite boundary points. The universal cover of such domain is the unit disk D . In Ω wecan define the Poincar´e metric, i.e. the metric obtained by pulling back the metric ds =2 | dz | / (1 −| z | ) of the unit disk. Equivalently, we can pull back the metric ds = | dz | / Im z of the the upper half plane H . Therefore, any simply connected subset of Ω is isometricto a subset of D . With this metric, Ω is a geodesically complete Riemannian manifoldwith constant curvature −
1, in particular, Ω is a geodesic metric space. The Poincar´emetric is natural and useful in complex analysis; for instance, any holomorphic functionbetween two domains is Lipschitz with constant 1, when we consider the respectivePoincar´e metrics.The quasihyperbolic metric is the distance induced by the density 1 /δ Ω ( z ). By λ Ω wedenote the density of the Poincar´e metric in Ω, and by k Ω and h Ω the quasihyperbolicand Poincar´e distance in Ω, respectively. Length (of a curve) will be denoted by thesymbol ℓ d, Ω , where d is the metric with respect to which length is measured. If it is clearwhich metric or domain is used, either one or both subscripts in ℓ d, Ω might be left out.The subscript Eucl is used to denote the length with respect to the Euclidean metric.Also, as most of the proofs apply to both the quasihyperbolic and the Poincar´e metrics,we will use the symbol κ also as a “dummy metric” symbol, where it can be replaced byeither k or h .We denote by λ Ω the density of the hyperbolic metric in Ω. It is well known that forevery domain Ω λ Ω ( z ) δ Ω ( z ) ∀ z ∈ Ω , ℓ h, Ω ( γ ) ℓ k, Ω ( γ ) ∀ γ ⊂ Ω , and that for all domains Ω ⊂ Ω we have λ Ω ( z ) > λ Ω ( z ) for every z ∈ Ω .If Ω is an open subset of Ω, in Ω we always consider its usual quasihyperbolic orPoincar´e metric (independent of Ω). If D is a closed subset of Ω, we always consider in D the inner metric obtained by the restriction of the quasihyperbolic or Poincar´e metricin Ω, that is d Ω | D ( z, w ) := inf (cid:8) ℓ κ, Ω ( γ ) : γ ⊂ D is a continuouscurve joining z and w (cid:9) > d Ω ( z, w ) . It is clear that ℓ Ω | D ( γ ) = ℓ Ω ( γ ) for every curve γ ⊂ D . We always require that ∂D is a union of pairwise disjoint Lipschitz curves; this fact guarantees that ( D, d Ω | D ) is ageodesic metric space.A geodesic metric space ( X, d ) is said to be
Gromov δ -hyperbolic , if d ( w, [ x, z ] ∪ [ z, y ]) δ P. H ¨AST ¨O, H. LIND´EN, A. PORTILLA, J. M. RODR´IGUEZ, AND E. TOUR´IS for all x, y, z ∈ X ; corresponding geodesic segments [ x, y ] , [ y, z ] and [ x, z ]; and w ∈ [ x, y ].If this inequality holds, we also say that the geodesic triangle is δ -thin , so Gromovhyperbolicity can be reformulated by requiring that all geodesic triangles are thin.A Denjoy domain Ω ⊂ C is a domain whose boundary is contained in the real axis.Hence, it satisfies Ω ∩ R = ∪ n ∈ Λ ( a n , b n ), where Λ is a countable index set, { ( a n , b n ) } n ∈ Λ are pairwise disjoint, and it is possible to have a n = −∞ for some n ∈ Λ and/or b n = ∞ for some n ∈ Λ.In order to study Gromov hyperbolicity, we consider the case where Λ is countablyinfinite, since if Λ is finite then h Ω and k Ω are easily seen to be Gromov hyperbolic byProposition 3.5, below.3. Some classes of Denjoy domains which are Gromov hyperbolic
The quasihyperbolic metric is traditionally defined in subdomains of Euclidean n -space R n , i.e. open and connected subsets Ω ( R n . However, a more abstract settingis also possible, as was shown in the article [8] by Bonk, Heinonen and Koskela. Thereit is shown that if ( X, d ) is taken to be any metric space which is locally compact,rectifiably connected and noncomplete, the quasihyperbolic metric k X can be definedas usual, using the weight 1 / dist( x, ∂X ).Given a real number A >
1, a curve γ : [0 , → Ω is called A -uniform for the metric d if ℓ d ( γ ) A d ( γ (0) , γ (1)) andmin { ℓ d ( γ | [0 , t ]) , ℓ d ( γ | [ t, } A dist d ( γ ( t ) , ∂ Ω) , for all t ∈ [0 , . Moreover, a locally compact, rectifiably connected noncomplete metric space is saidto be A -uniform if every pair of points can be joined by an A -uniform curve. Theabbreviations “ A -uniform” and “ A -inner uniform” (without mention of the metric) mean A -uniform for the Euclidean metric and Euclidean inner metric, respectively.Uniform domains are intimately connected to domains which are Gromov hyperbolicwith respect to the quasihyperbolic metric (see [8, Theorems 1.12, 11.3]). Specifically,for a Denjoy domain Ω these results imply that k Ω is Gromov hyperbolic if and only ifΩ is inner uniform.Here we will use the generalized setting in [8] to show that for Denjoy domains itactually suffices to consider the upper (or lower) intersection with the actual domain,as can be done for the Poincar´e metric: Lemma 3.1.
Let ∅ 6 = E ⊂ R be a closed set, and denote D = C \ E and D = D ∩ { z ∈ C | Im z > } = D ∩ H . Then the metric space D , with the restrictionof the Poincar´e or the quasihyperbolic metric in D , is δ -Gromov hyperbolic, with someuniversal constant δ .Proof. We deal first with the quasihyperbolic metric. As the upper half-plane is uniformin the classical case, the same curve of uniformity (which is an arc of a circle orthogonalto R ) can be shown to be an A -uniform curve in the sense of [8] for the set D . Hence D is A -uniform. By [8, Theorem 3.6] it then follows that the space ( D, k D ) is Gromovhyperbolic.We also have that D is hyperbolic with the restriction of the Poincar´e metric h D ,since it is isometric to a geodesically convex subset of the unit disk (in fact, there is ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 5 just one geodesic in D joining two points in D ). Therefore, D has log (cid:0) √ (cid:1) -thintriangles, as does the unit disk (see, e.g. [3, p. 130]). (cid:3) Definition 3.2.
Let Ω be a Denjoy domain. Then we have Ω ∩ R = ∪ n > ( a n , b n ) forsome suitable intervals. We say that a curve in Ω is a fundamental geodesic if it is ageodesic joining ( a , b ) and ( a n , b n ), n >
0, which is contained in the closed halfplane H = { z ∈ C : Im z > } . We denote by γ n a fundamental geodesic corresponding to n . The next result was proven for the hyperbolic metric in [2, Theorem 5.1]. In viewof Lemma 3.1 one can check that the same proof carries over to the quasihyperbolicmetric.By a bigon we mean a closed polygon with two edges. Theorem 3.3.
Let Ω be a Denjoy domain and denote by κ Ω the Poincar´e or quasihy-perbolic metric. Then the following conditions are equivalent: (1) κ Ω is δ -hyperbolic. (2) There exists a constant c such that for every choice of fundamental geodesics { γ n } ∞ n =1 we have κ Ω ( z, R ) c for every z ∈ ∪ n > γ n . (3) There exists a constant c such that for a fixed choice of fundamental geodesics { γ n } ∞ n =1 we have κ Ω ( z, R ) c for every z ∈ ∪ n > γ n . (4) There exists a constant c such that every geodesic bigon in Ω with vertices in R is c -thin.Furthermore, the constants in each condition only depend on the constants appearingin any other of the conditions. Note that the case Ω ∩ R = ∪ Nn =0 ( a n , b n ) is also covered by the theorem. Corollary 3.4.
Let Ω be a Denjoy domain and denote by κ Ω the Poincar´e or quasi-hyperbolic metric. If there exist a constant C and a sequence of fundamental geodesics { γ n } n > with ℓ κ, Ω ( γ n ) C , then κ Ω is δ -Gromov hyperbolic, and δ just depends on C . If Ω has only finitely many boundary components, then it is always Gromov hyper-bolic, in a quantitative way:
Proposition 3.5.
Let Ω be a Denjoy domain with Ω ∩ R = ∪ Nn =1 ( a n , b n ) , and denoteby κ Ω the Poincar´e or quasihyperbolic metric. Then κ Ω is δ -Gromov hyperbolic, where δ is a constant which only depends on N and c = sup n κ Ω (cid:0) ( a n , b n ) , ( a n +1 , b n +1 ) (cid:1) . Note that we do not require b n a n +1 . Proof.
Let us consider the shortest geodesics g ∗ n joining ( a n , b n ) and ( a n +1 , b n +1 ) in Ω + :=Ω ∩ H . Then ℓ Ω ( g ∗ n ) ℓ Ω ( g n ) c for 0 n N − c , which onlydepends on c and N , such that κ Ω ( z, R ) c for every z ∈ ∪ Nn =1 γ n .For each 0 n N −
1, let us consider the geodesic polygon P in Ω + , with thefollowing sides: γ n , g ∗ , . . . , g ∗ n − , and the geodesics joining their endpoints which arecontained in ( a , b ) , . . . , ( a n , b n ). Since (Ω + , κ Ω ) is δ -Gromov hyperbolic, where δ is aconstant which only depends on c , by Lemma 3.1, and P is a geodesic polygon in Ω + with at most 2 N + 2 sides, P is 2 N δ -thin. Therefore, given any z ∈ γ n , there existsa point w ∈ ∪ N − k =0 g ∗ k ∪ R with κ Ω ( z, w ) N δ . Since ℓ Ω ( g ∗ k ) c for 0 k N − P. H ¨AST ¨O, H. LIND´EN, A. PORTILLA, J. M. RODR´IGUEZ, AND E. TOUR´IS there exists x ∈ R with κ Ω ( x, w ) c /
2. Hence, κ Ω ( z, R ) κ Ω ( z, x ) N δ + c / κ Ω is δ -Gromov hyperbolic. (cid:3) Theorem 3.6.
Let Ω be a Denjoy domain with Ω ∩ R = ∪ ∞ n =0 ( a n , b n ) , ( a , b ) = ( −∞ , and b n a n +1 for every n . Suppose that b n > Ka n for a fixed K > and every n . Then h Ω and k Ω are δ -Gromov hyperbolic, with δ depending only on K .Proof. Fix n and consider the domainΩ n = 1 a n Ω = (cid:26) xa n | x ∈ Ω (cid:27) . If we define D := C \ [0 , ∪ [ K, ∞ ), then D ⊂ Ω n , and ℓ k, Ω n ( γ ) ℓ k,D ( γ ) for everycurve γ ⊂ Ω n . The circle σ := S (0 , (1 + K ) /
2) goes around the boundary component[0 ,
1] in D and has finite quasihyperbolic length: ℓ k,D ( σ ) Z σ | dz | ( K − / π K + 1 K − . Consider the shortest fundamental geodesics joining ( a , b ) with ( a n , b n ), with thePoincar´e and the quasihyperbolic metrics, γ hn and γ kn , respectively. Then, ℓ k, Ω ( γ kn ) = ℓ k, Ω n (cid:16) a n γ kn (cid:17) ℓ k, Ω n ( σ ) ℓ k,D ( σ ) π K + 1 K − ,ℓ h, Ω ( γ hn ) ℓ h, Ω ( γ kn ) ℓ k, Ω ( γ kn ) π K + 1 K − . Therefore h Ω and k Ω are δ -Gromov hyperbolic (and δ depends only on K ), by Corol-lary 3.4. (cid:3) Proof of Theorems 1.1(1).
If lim inf n →∞ ( b n − a n ) /a n >
0, then we can choose
K > b n − a n ) /a n > K − n , whence b n > Ka n . Thus the previous theoremimplies the claims. (cid:3) Some classes of Denjoy domains which are not Gromov hyperbolic
The following function was introduced by Beardon and Pommerenke [6].
Definition 4.1.
For Ω ( C , define β Ω ( z ) as the function β Ω ( z ) := inf n(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12) z − ab − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : a, b ∈ ∂ Ω , | z − a | = δ Ω ( z ) o . The function β Ω has a geometric interpretation. We say that an annulus { z ∈ C : r < | z − a | < R } separates E ⊂ C if { z ∈ C : r < | z − a | < R } ∩ E = ∅ , { z ∈ C : | z − a | r } ∩ E = ∅ and { z ∈ C : | z − a | > R } ∩ E = ∅ . We say that E is uniformly perfect if there exists a constant c such that R/r c for every annulus { z ∈ C : r < | z − a | < R } separating E (see [6, 16, 17]). Now we see that β Ω is boundedprecisely when Ω is uniformly perfect.Thus it follows from the next theorem, that λ Ω and 1 /δ Ω are comparable if and onlyif Ω is uniformly perfect. Theorem 4.2 (Theorem 1, [6]) . For every domain Ω ⊂ C of hyperbolic type and forevery z ∈ Ω , we have that − / λ Ω ( z ) δ Ω ( z ) ( k + β Ω ( z )) π/ , where k = 4 + log(3 + 2 √ . ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 7
Lemma 4.3.
Let γ be a curve in a domain D ⊂ R n from a ∈ D with Euclidean length s . Then: (1) ℓ k,D ( γ ) > log (cid:0) sd D ( a ) (cid:1) . (2) If D is a Denjoy domain and a ∈ ( a n , b n ) , with b n − a n r , then ℓ h,D ( γ ) > − / log (cid:0) k − log (cid:0) sr (cid:1)(cid:1) , with k as in Theorem 4.2.Proof. Let z ∈ ∂D be a point with δ D ( a ) = | a − z | . Without loss of generality weassume that z = 0. By monotonicity ℓ k,D ( γ ) > ℓ k, R n \{ } ( γ ). Further, it is clear that ℓ k, R n \{ } ( γ ) > ℓ k, R n \{ } ([ | a | , | a | + s ]), whence the first estimate by integrating the density1 / | x | .We then prove the second estimate. Without loss of generality we assume that b n = 0. By monotonicity ℓ h,D ( γ ) > ℓ h, C \{ a n , } ( γ ). By [15, Theorem 4.1(ii)] we havethat λ C \{ a n , } ( z ) > λ C \{ a n , } ( | z | ) and by [15, Theorem 4.1(i)] that λ C \{ a n , } ( r ) is a de-creasing function in r ∈ (0 , ∞ ); hence, ℓ h, C \{ a n , } ( γ ) > ℓ h, C \{ a n , } ([ | a n | , | a n | + s ]) = ℓ h, C \{− , } ([1 , s/ | a n | ]). By Theorem 4.2 ℓ h,D ( γ ) > ℓ h, C \{− , } ([1 , s/ | a n | ]) > Z s/ | a n | − / dxx (cid:0) k + log x (cid:1) = 2 − / log (cid:16) k − log (cid:16) s | a n | (cid:17)(cid:17) > − / log (cid:16) k − log (cid:16) sr (cid:17)(cid:17) . (cid:3) Proof of Theorem 1.1(2), for the quasihyperbolic metric.
We use the characterization ofBonk, Heinonen and Koskela [8]. Hence it suffices to show that the domain in not inneruniform. So, suppose for a contradiction that the domain is A -inner uniform for somefixed A > s n := max m n ( b m − a m ). It is clear that s n is an increasing sequenceand lim n →∞ s n /a n = 0. If we define g n := p s n /a n , then b m − a m a n g n for every1 m n and lim n →∞ g n = 0.Since g n >
0, we can choose a subsequence { g n k } with g n k > g m for every m > n k ;consider a fixed n from the sequence { n k } . Set c n = b n + a n , the mid-point of ( a n , b n ). Wedefine x n = c n + ic n g n and y n = c n − ic n g n . Since [ x n , y n ] ⊂ Ω, we have ℓ Eucl , Ω ([ x n , y n ]) =2 c n g n . Let γ be an A -inner uniform curve joining x n and y n , and let z ∈ γ ∩ R . Since | x n − z | , | y n − z | > c n g n , we conclude by the uniformity of the curve that δ Ω ( z ) > c n g n A .On the other hand, the uniformity of γ also implies that | z − c n | Ac n g n .We may assume that n is so large that c n > Ac n g n . Then z lies in the positivereal axis, which means that z ∈ ( a m , b m ) for some m >
1. If m n , then we have b m − a m s n = a n g n < c n g n . For m > n we have b m − a m g m a m g n a m . However,since a m < z c n + 2 Ac n g n < c n , so for every m we have b m − a m < c n g n .Since δ Ω ( z ) < b m − a m , we conclude that c n g n A < c n g n . Since g n → A is a constant,this is a contradiction. Hence the assumption that an A -inner uniform curve exists wasfalse, and we can conclude that the domain is not Gromov hyperbolic. (cid:3) For the proof in the hyperbolic case we need the following concepts. A functionbetween two metric spaces f : X −→ Y is an ( a, b ) -quasi-isometry , a > b >
0, if1 a d X ( x , x ) − b d Y ( f ( x ) , f ( x )) ad X ( x , x ) + b , for every x , x ∈ X. An ( a, b )- quasigeodesic in X is an ( a, b )-quasi-isometry between an interval of R and X .For future reference we record the following lemma: P. H ¨AST ¨O, H. LIND´EN, A. PORTILLA, J. M. RODR´IGUEZ, AND E. TOUR´IS
Lemma 4.4.
Let us consider a geodesic metric space X and a geodesic γ : I −→ X ,with I any interval, and g : I −→ X , with d ( g ( t ) , γ ( t )) ε for every t ∈ I . Then g isa (1 , ε ) -quasigeodesic.Proof. We have for every s, t ∈ Id ( g ( s ) , g ( t )) > d ( γ ( s ) , γ ( t )) − d ( γ ( s ) , g ( s )) − d ( γ ( t ) , g ( t )) > | t − s | − ε. The upper bound is similar. (cid:3)
Proof of Theorem 1.1(2), for the hyperbolic metric.
We consider two cases: either { b m − a m } m is bounded or unbounded. We start with the latter case.As in the previous proof, we define s n := max m n ( b m − a m ) and g n := p s n /a n .Then b m − a m a n g n for every 1 m n and lim n →∞ g n = 0. Since g n >
0, wecan choose a subsequence { g n k } with g n k > g m for every m > n k . Since { b m − a m } m is not bounded we may, moreover, choose the sequence so that g n = ( b n − a n ) /a n forevery n ∈ { n k } . Fix now n from the sequence { n k } . As before, we conclude that b m − a m a n g n for m n and b m − a m a m g m a m g n for m > n . θ Sx0
Figure 1.
The set S Consider x ∈ ( a n , b n ) which lies on the shortest fundamental geodesic γ n joining( −∞ ,
0) with ( a n , b n ). Define an angle θ = arc tan g n ∈ (0 , π/
2) and a set S = [ x + ixg n , x + ixg n ] ∪ { x + ixg n + te πiθ | t > } . The set S is shown in Figure 1. Notice that any point ζ ∈ S satisfies g n Re ζ Im ζ g n Re ζ . It is clear that γ n hits the set S ∪ [ x + ixg n , x ]. We claim that it in fact hits S . Assume to the contrary that this is not the case. Then it hits [ x + ixg n , x ]. Let γ ′ denote a part of γ n connecting x and this segment which does not intersect S . SinceΩ is a Denjoy domain, we conclude that b λ Ω ( a + ib ) is decreasing for b > ℓ h, Ω ( γ ′ ) > ℓ h, Ω ([ x + ixg n , x + ixg n ]). Since the gap size in[ x, x ] is at most a n g n , we have δ Ω ( w ) p x g n + a n g n √ xg n . Since the gap sizeis smaller than the distance to the boundary, it follows from Theorem 4.2 that λ Ω ( w ) > Cδ Ω ( w ) > Cxg n for w ∈ [ x + ixg n , x + ixg n ]. Multiplying this with the Euclidean length x of thesegment gives ℓ h, Ω ( γ n ) > ℓ h, Ω ([ x + ixg n , x + ixg n ]) > Cg n . ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 9
We next construct another path σ and show that it is in the same homotopy classas the supposed geodesic, only shorter. Let z be the midpoint of gap n and let σ bethe curve [ z, z + iz ] ∪ [ z + iz, − z + iz ] ∪ [ − z + iz, − z ]. Using b n − a n = a n g n we easilycalculate ℓ h, Ω ( σ ) ℓ k, Ω ( σ ) (cid:16) za n g n (cid:17) + C (cid:16) g n (cid:17) + C with an absolute constant C . The curve σ joins ( −∞ ,
0) and ( a n , b n ); therefore ℓ h, Ω ( γ n ) ℓ h, Ω ( σ ). But this contradicts the previously derived bounds for the lengths as g n → γ n does not intersect S was wrong, so we concludethat γ n ∩ S = ∅ . Let now ζ ∈ S ∩ γ n . We claim that h Ω ( ζ , R ) → ∞ , which means thedomain is not Gromov hyperbolic, by Theorem 3.3. Let ξ ∈ Ω ∩ R ; chose m so that ξ ∈ ( a m , b m ). Let α be a curve joining ξ and ζ .If 0 < m n , then the size of ( a m , b m ) is at most a n g n , so δ Ω ( ξ ) a n g n . Then α has Euclidean length at least Im ζ > xg n , so by Lemma 4.3, ℓ h, Ω ( α ) > c log log( C/g n ).As g n →
0, this bound tends to ∞ . If, on the other hand, m > n , then the Euclideanlength of α is at least d ( ξ, ζ ) > d ( ξ, S ) > ξ sin θ > ξ tan θ = ξg n , and the size of the gap is at most a m g n . By Lemma 4.3 this implies that ℓ h, Ω ( α ) > c log log( C/g n ). As g n →
0, this bound again tends to ∞ .It remains to consider m = 0, i.e., ξ <
0. We consider only the case ζ ∈ [ x + ixg n , x + ixg n ], since the other case is similar. Now the Euclidean length of α is at least x . Sincethe gap size in [0 , x ] is at most a n g n , we see that the boundary satisfies the separationcondition when | Im z | > a n g n in which case also δ Ω ( z ) > | Im z | > a n g n . Since λ Ω ( z ) isdecreasing in | Im z | (see [15, Theorem 4.1(i)]), we conclude that(4.5) λ Ω ( z ) > C max {| Im z | , a n g n } > C max { δ Ω ( z ) , a n g n } for the points on the curve with Re z ∈ (0 , x/ α − be the part of α on which δ Ω ( z ) < a n g n . If ℓ Eucl ( α − ) > xg / n , then ℓ h, Ω ( α ) > ℓ h, Ω ( α − ) > xg / n a n g n > g − / n . If ℓ Eucl ( α − ) xg / n , then ℓ Eucl ( α \ α − ) > x − xg / n . Hence we conclude (as in theproof of part (1) in Lemma 4.3) that Z α λ Ω ( z ) | dz | > C Z x/ δ Ω ( ζ )+ xg / n dtt > C log (cid:16) x/ √ a n g n + xg / n (cid:17) > C log (cid:16) g n (cid:17) − C. Hence in either case we get a lower bound which tends to infinity as g n → { b m − a m } m is unbounded. Assume next thatsup m ( b m − a m ) = M < ∞ . In this case it is difficult to work with bigons, sincewe do not get a good control on what the gedesics look like; the problem with theprevious argument is that we cannot choose g n k = ( b n k − a n k ) /a n k in our sequence, andconsequently we do not get a good bound on the length of the curve σ , as defined above. To get around this we consider a geodesic triangle. Assume for a contradiction that h Ω is δ -Gromov hyperbolic. By geodesic stability [9], there exists a number δ ′ so thatevery ( √ , δ ′ -thin.Fix R ≫ M and set w ± = ± iR . Let γ be the geodesic segment joining w + and w − . Choose t > h Ω ( γ , H t ) > δ ′ , where H t = { z ∈ C | Re z > t } . Let w ∈ Ω ∩ R be a point in H { t,R } , and let γ + ⊂ H be a geodesic joining w and w + .If γ + dips below the ray from w through w + , then we replace the part below the rayby a part of the ray. The resulting curve is denoted by ˜ γ + . Let us show that ˜ γ + isa quasigeodesic. We define a mapping f : γ + → ˜ γ + as follows. If x ∈ γ + ∩ ˜ γ + , then f ( x ) = x . If x ∈ γ + \ ˜ γ + then we set f ( x ) to equal the point on ˜ γ + with real part equalto Re x .Since Ω is a Denjoy domain, the function b λ Ω ( a + ib ) is decreasing for b > λ Ω ( f ( x )) λ Ω ( x ). The arc-length distance elementis the vertical projection of the distance element at x to the line through w and w + :specifically, the distance element ( dx, dy ) becomes ( dx, θdx ), where θ is the slope ofthe line. Thus the maximal increase in the distance element is √ θ . Since theslope of the line lies in the range [ − , γ + is a( √ , γ − and conclude that it is a ( √ , ζ ∈ ˜ γ + ∩ H max { t,R } with Im ζ = √ R . Since γ ∪ ˜ γ + ∪ ˜ γ − is a ( √ , ζ with some point in γ ∪ ˜ γ − using a pathof length δ ′ . By the definition of t , h Ω ( ζ , γ ) > δ ′ . If α is a path connecting ζ and γ − , then it crosses the real axis at some point ξ . If ξ lies in ( a m , b m ), m >
0, then ℓ h, Ω ( α ) > C log log √ RM , by Lemma 4.3. Otherwise, ξ ∈ ( −∞ , h Ω ( ζ , γ − ) > δ ′ provided R is large enough. But this means that Ω is not Gromowhyperbolic, as was to be shown. (cid:3) In Theorem 1.1(2) the gaps ( a n , b n ) and ( a n +1 , b n +1 ) are separated by a boundarycomponent [ b n , a n +1 ]. We easily see from the proofs that it would have made no differ-ence if this boundary component had some gaps, as long as they at most comparableto the lengths of the adjecent gaps, ( a n , b n ) and ( a n +1 , b n +1 ). Thus we get the followingstronger theorem by the same proofs. (In the proofs we can assume that ( −∞ , ⊂ Ω,by using Theorem 1.2).
Theorem 4.6.
Let Ω be a Denjoy domain with Ω ∩ R = S ( a n , b n ) and lim sup n →∞ a n = ∞ . Suppose G : R + → R + is a function with lim x →∞ G ( x ) = 0 . If b n − a n a n G ( a n ) for every a n > , then κ Ω , the hyperbolic or quasihyperbolic metric, is not Gromovhyperbolic. The function G plays the role of g n in the proofs of Theorem 1.1(2). Remark . The condition Ω ∩ R = S ( a n , b n ) (without the hypothesis b n a n +1 forevery n ) allows any topological behaviour; for instance, ∂ Ω can contain a countablesequence of Cantor sets.Let E ⊂ [0 , t ) be closed, t >
0, set E n := E + tn for n ∈ N , and Ω := C \ ∪ n E n .Then Ω satisfies the hypotheses of Theorem 4.6 for G ( x ) = t/x . From this we deduceCorollary 1.3, the non-hyperbolicity of periodic Denjoy domain, in the case the indexset is N . The case with index set Z follows from this and Theorem 1.2. ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 11 On the far side of the accumulation point
Lemma 5.1.
Let Ω be a Denjoy domain with Ω ∩ R = ∪ ∞ n =0 ( a n , b n ) and a = −∞ . If h Ω is not Gromov hyperbolic, then for every N > there exist fundamental geodesics γ n k , n k > N , such that the hyperbolic distance of the endpoints of γ n k to ( −∞ , b ) isgreater than N , and points z k ∈ γ n k with lim k →∞ h Ω ( z k , R ) = ∞ .Proof. Let us choose fundamental geodesics { γ n } . Since h Ω is not Gromov hyperbolic,by Theorem 3.3 there exists points w k ∈ γ n k with n k > N and lim k →∞ h Ω ( w k , R ) = ∞ .Since lim x → b n h Ω ( x, ( −∞ , b )) = ∞ for every n , there exist x ∈ ( a , b ) and x n k ∈ ( a n k , b n k ), with h Ω ( x , ( −∞ , b )) , h Ω ( x n k , ( −∞ , b )) > N .Let us consider the fundamental geodesics γ n k joining x and x n k , as well as thebordered Riemann surface X := Ω ∩ H , which as in the proof of Theorem 3.1 can beshown to have log (cid:0) √ (cid:1) -thin triangles.Let Q k be the geodesic quadrilateral given by γ n k , γ n k and the two geodesics (con-tained in ( a , b ) and ( a n k , b n k )) joining their endpoints. Since Q k ⊂ X , it is 2 log (cid:0) √ (cid:1) -thin, and there exists z k ∈ γ n k ∪ R with h Ω ( z k , w k ) (cid:0) √ (cid:1) .Since lim k →∞ h Ω ( w k , R ) = ∞ , we deduce that z k ∈ γ n k for every k > k andlim k →∞ h Ω ( z k , R ) = ∞ . (cid:3) Lemma 5.2 (Lemma 3.1, [1]) . Consider an open Riemann surface S of hyperbolictype, a closed non-empty subset C of S , and set S ∗ := S \ C . For ǫ > we have < ℓ S ∗ ( γ ) /ℓ S ( γ ) < coth( ε/ , for every curve γ ⊂ S with finite length in S such that h S ( γ, C ) > ε . Given a Riemann surface S , a geodesic γ in S , and a continuous unit vector field ξ along γ orthogonal to γ , we define Fermi coordinates based on γ as the map Y ( r, t ) :=exp γ ( r ) tξ ( r ).It is well known that if the curvature is K ≡ −
1, then the Riemannian metric can beexpressed in Fermi coordinates as ds = dt + cosh t dr (see e.g. [10, p. 247–248]). Corollary 5.3.
Consider an open Riemann surface of hyperbolic type S , a closed non-empty subset C of S , and set S ∗ := S \ C . For ǫ > and C ε := { z ∈ S : h S ( z, C ) > ε } we have h S ( z, w ) h S ∗ ( z, w ) , for every z, w ∈ S ∗ ,h S ∗ ( z, w ) coth( ε/ h S | C ε ( z, w ) , for every z, w ∈ C ε . Furthermore, if S is a Denjoy domain and C is a component of S ∩ R then h S ∗ ( z, w ) cosh ε coth( ε/ h S ( z, w ) , for every z, w in the same component of C ε with Im z, Im w > .Proof. The first and second inequalities are direct consequences of Lemma 5.2. In orderto prove the third one, it is sufficient to prove that(5.4) h S | C ε ( z, w ) (cosh ε ) h S ( z, w ) , for every z, w in the same component of C ε with Im z, Im w > z, w in the same component Γ of C ε . Since Im z, Im w > γ ⊂ S ∩ H joining z with w .If γ ⊂ Γ, then h S | C ε ( z, w ) = h S ( z, w ). If γ is not contained in Γ, then it is sufficientto show that there exists a curve η joining z and w in Γ, with ℓ h,S ( η ) (cosh ε ) ℓ h,S ( γ ). In order to prove this, consider the geodesics γ z , γ w ⊂ S ∩ H joining z and w with C ,and the geodesic γ ⊂ C joining the endpoints of γ z , γ w (which are in C ).We denote by P the simply connected closed region with boundary γ ∩ γ z ∩ γ w ∩ γ .Since P is simply connected, we can identify it with a domain P ⊂ H using Fermicoordinates based on C .If g is the lift of γ , then g := g ∩ { ( r, t ) : 0 t ε } is the lift of γ \ C ε . If g ∩ { ( r, t ) : t = ε } = { ( r , ε ) , ( r , ε ) } (with r < r ), then we define g := { ( r, ε ) : r r r } and g := { ( r,
0) : r r r } . Notice that in order to prove (5.4) it is sufficient to showthat ℓ ( g ) (cosh ε ) ℓ ( g ). But this is a direct consequence of the facts ℓ ( g ) ℓ ( g )and ℓ ( g ) = (cosh ε ) ℓ ( g ). (cid:3) Proof of Theorem 1.2.
Since κ Ω is not Gromov hyperbolic, by Proposition 3.5, we con-clude that Ω has countably infinitely many boundary components: Ω ∩ R = ∪ ∞ n =0 ( a n , b n ).Without loss of generality we can assume that ( −∞ , ⊆ ( a , b ).We first prove that (Ω \ F, k Ω \ F ) is not Gromov hyperbolic. Let us consider fundamen-tal geodesics γ n of k Ω joining the midpoint c of ( a , b ) with the midpoint c n of ( a n , b n )for n > γ n is contained in { z ∈ C : c Re z c n } ,and k Ω \ F = k Ω in { z ∈ C : Re z > inf n > a n } , we deduce that γ n is also a fundamentalgeodesic with the metric k Ω \ F .Since k Ω is not Gromov hyperbolic, there exist points z k ∈ γ n k with lim k →∞ k Ω ( z k , R ) = ∞ by Theorem 3.3. Since γ n k are also fundamental geodesics with the metric k Ω \ F , wededuce that lim k →∞ k Ω \ F ( z k , R ) > lim k →∞ k Ω ( z k , R ) = ∞ . Consequently, (Ω \ F, k Ω \ F )is not Gromov hyperbolic.We now prove that (Ω \ F, h Ω \ F ) is not Gromov hyperbolic. Choose ε >
0. Since h Ω is not Gromov hyperbolic, by Lemma 5.1 there exist fundamental geodesics γ n k of h Ω ,such that the hyperbolic distance of the endpoints of γ n k to ( −∞ , b ) is greater than ε ,and points z k ∈ γ n k with lim k →∞ h Ω ( z k , R ) = ∞ .Fix ε ∈ (cid:0) , min { ε , min k h Ω ( z k , R ) } (cid:1) . If we define U ε := { z ∈ Ω : h Ω ( z, ( −∞ , b )) > ε } , we see that z k ∈ γ n k ∩ U ε for every k . (Notice that γ n k ∩ ∂U ε has at most two points.)If γ n k ∩ ∂U ε is empty or a one-point set, we define g n k := γ n k . Since the endpoints of γ n k are in U ε , we conclude that g n k ⊂ U ε .Then assume that γ n k ∩ ∂U ε = { w , w } . If there is an arc α in ∂U ε joining w and w , we define a curve g n k joining ( a , b ) with ( a n k , b n k ) in U ε , by g n k := ( γ n k ∩ U ε ) ∪ α .Then γ n k and g n k have the same endpoints and are homotopic. If there is not an arcin ∂U ε joining w and w , there are still maximal arcs α, β in ∂U ε joining w and ω ∈ ( a m , b m ), and w and ω ∈ ( a m , b m ), respectively, and a geodesic η (withrespect to h Ω ) in Ω \ U ε joining ω and ω , such that if γ n k ∩ U ε = [ z , w ] ∪ [ z , w ], then[ z , w ] ∪ α ∪ η ∪ β ∪ [ z , w ] has the same endpoints as γ n k , and they are homotopic.Since ε < h Ω ( z k , R ), we have either z k ∈ [ z , w ] or z k ∈ [ z , w ]. Without loss ofgenerality we can assume that z k ∈ [ z , w ]. Then we define g n k := β ∪ [ z , w ] ⊂ U ε ,which is a curve joining ( a m , b m ) with ( a n k , b n k ).In any case, Lemma 4.4 gives that g n k is a (1 , ε )-quasigeodesic with respect to h Ω .Hence, for every t, s, we have | t − s | − ε h Ω (cid:0) g n k ( t ) , g n k ( s ) (cid:1) | t − s | + 2 ε. ROMOV HYPERBOLICITY OF HYPERBOLIC AND QUASIHYPERBOLIC METRICS 13
Since g n k is contained in U ε , Corollary 5.3 implies that | t − s | − ε h Ω (cid:0) g n k ( t ) , g n k ( s ) (cid:1) < h Ω \ F (cid:0) g n k ( t ) , g n k ( s ) (cid:1) h Ω \ ( −∞ , (cid:0) g n k ( t ) , g n k ( s ) (cid:1) cosh ε coth( ε/ h Ω (cid:0) g n k ( t ) , g n k ( s ) (cid:1) cosh ε coth( ε/ (cid:0) | t − s | + 2 ε (cid:1) , and hence g n k is a (cid:0) cosh ε coth( ε/ , ε cosh ε coth( ε/ (cid:1) -quasigeodesic with respect to h Ω \ F .To get a contradiction, assume that (Ω \ F, h Ω \ F ) is Gromov hyperbolic. Considerthe fundamental geodesic η n k of h Ω \ F with the same endpoints as g n k . Then there is aconstant C such that the Hausdorff distance of g n k and η n k is less than C . Hence, thereexist points w k ∈ η n k with h Ω \ F ( z k , w k ) C , and thuslim k →∞ h Ω \ F ( w k , R ) > lim k →∞ h Ω \ F ( z k , R ) − C > lim k →∞ h Ω ( z k , R ) − C = ∞ , which contradicts h Ω \ F being Gromov hyperbolic. (cid:3) References [1] Alvarez, V., Pestana, D., Rodr´ıguez, J. M., Isoperimetric inequalities in Riemann surfaces ofinfinite type,
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