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Given any graph [MATH] , let us consider the natural length metric [MATH] where every edge has length 1. For any graph [MATH] , any vertex [MATH] and any [MATH] , let |
[MATH] and [MATH] denote the open and closed balls, respectively. The combinatorial Cheeger isoperimetric constant of [MATH] is defined to be |
[EQUATION] where [MATH] ranges over all non-empty finite subsets of vertices in [MATH] [MATH] and [MATH] denotes the cardinality of [MATH] |
We say that a Riemannian manifold or graph [MATH] satisfies the (Cheeger or linear) isoperimetric inequality (LII) if [MATH] , since in this case |
[EQUATION] for every bounded open set [MATH] if [MATH] is a Riemannian [MATH] -manifold, and [EQUATION] for every finite set [MATH] if [MATH] is a graph. |
Along the paper, we just consider manifolds and graphs [MATH] which are connected. This is not a loss of generality, since if [MATH] has connected components [MATH] , then [MATH] |
Let [MATH] and [MATH] be two metric spaces. A map [MATH] is said to be an [MATH] quasi-isometric embedding , with constants [MATH] , if for every [MATH] |
[EQUATION] The function [MATH] is [MATH] full if for each [MATH] there exists [MATH] with [MATH] A map [MATH] is said to be quasi-isometry , if there exist constants [MATH] such that [MATH] is an [MATH] -full |
[MATH] -quasi-isometric embedding. Two metric spaces [MATH] and [MATH] are quasi-isometric if there exists a quasi-isometry [MATH] One can check that to be quasi-isometric is an equivalence relation. |
A graph [MATH] is said to be [MATH] uniform if each vertex [MATH] of [MATH] has at most [MATH] neighbors, i.e., [EQUATION] If a graph [MATH] is [MATH] -uniform for some constant [MATH] we say that [MATH] is uniform |
A complete Riemannian [MATH] -manifold [MATH] is uniform if it has positive injectivity radius and a lower bound on its Ricci curvature. |
The injectivity radius inj [MATH] of [MATH] is defined as the supremum of those [MATH] such that [MATH] is simply connected or, equivalently, as half the infimum of the lengths of the (homotopically non-trivial) loops based at [MATH] The injectivity radius inj [MATH] |
of [MATH] is the infimum over [MATH] of inj [MATH] A celebrated theorem of Kanai in states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds and graphs. A main ingredient in Kanai’s proof is the following interesting fact: we can study the LII in a Riemannian manifold [MATH]... |
12 , Theorem 1.2] 23 , Theorem 5.1] A non-exceptional Riemann surface [MATH] is a Riemann surface whose universal covering space is the unit disk [MATH] , endowed with its Poincaré metric (also called the hyperbolic metric), i.e., the metric obtained by projecting the Poincaré metric of the unit disk |
[EQUATION] With this metric, [MATH] is a complete Riemannian manifold with constant curvature [MATH] The only Riemann surfaces which are left out (the exceptional Riemann surfaces) are the sphere, the plane, the punctured plane and the tori. |
A natural context to apply Kanai’s results are Riemann surfaces endowed with their Poincaré metrics, since they have constant negative curvature. However, these surfaces usually have isolated singularities which are cusps, and thus, injectivity radius equal to zero. That means that, unfortunately, it is not possible to... |
Our main result is Theorem 3.7 , that states that we can study the LII in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai’s graph, but it is different form it). Thus, Theorem 3.7 shows that the main tool in Kanai’s proof also works for a... |
Finally, we want to remark that 24 , Theorem 5.5] gives that every orientable and complete Riemannian surface with pinched negative curvature (with Gaussian curvature [MATH] satisfying [MATH] is bilipschitz equivalent to a non-exceptional Riemann surface (and therefore with constant negative curvature [MATH] ). |
24 , Theorem 5.5] shows that it suffices to work with surfaces of curvature [MATH] (instead of pinched negative curvature) in order to check LII, since this inequality is invariant by bilipschitz maps. This fact enlarges the scope of Theorems 3.7 and 4.12 |
2. Background and technical results geodesic domain in a non-exceptional Riemann surface [MATH] is a domain [MATH] (which is not simply or doubly connected) such that [MATH] consists of finitely many simple closed geodesics, and [MATH] is finite. |
[MATH] does not have to be relatively compact since it may “surround” finitely many cusps. We can think of a cusp as a boundary geodesic of zero length. Recall that if [MATH] is a closed curve in [MATH] and [MATH] denotes its free homotopy class in [MATH] , then there is a unique simple closed geodesic of minimal lengt... |
In 21 , Lemma 1.2] it was proved the following. Lemma 2.1 If a non-exceptional Riemann surface [MATH] satisfies [EQUATION] for some constant [MATH] and every geodesic domain [MATH] in [MATH] , then [MATH] has LII. |
In fact, if we define [MATH] as [EQUATION] where [MATH] ranges over all geodesic domains in [MATH] , then 28 , Theorem 7] gives [EQUATION] |
Definition 2.2 Given a non-exceptional Riemann surface [MATH] , a subset [MATH] and a positive constant [MATH] , let us denote by [MATH] the set of points [MATH] such that the connected component of [MATH] containing [MATH] has length at least [MATH] A non-exceptional Riemann surface [MATH] is [MATH] regular |
if there exists a positive constant [MATH] such that every geodesic domain [MATH] in [MATH] satisfies that [MATH] (where [MATH] denotes the cardinality of the set of connected components of [MATH] ). We say that [MATH] is regular if it is [MATH] -regular for some [MATH] |
Theorem 2.3 If a non-exceptional Riemann surface is not regular, then it does not have LII. Proof. Since the non-exceptional Riemann surface [MATH] is not regular, for each [MATH] there exists a geodesic domain [MATH] in [MATH] with |
[EQUATION] Hence, since [MATH] , we have [EQUATION] If [MATH] [MATH] and [MATH] denote the cardinality of the simple closed geodesics in [MATH] , the cusps surrounded by [MATH] and the genus of [MATH] , respectively, it is well-known that Gauss-Bonnet Theorem gives [MATH] and so, [MATH] Note that [MATH] We are going to... |
[EQUATION] If [MATH] , then [MATH] Assume that [MATH] If [MATH] , then [MATH] If [MATH] , then, since [MATH] , we have [MATH] and [MATH] If [MATH] , then, since [MATH] , we have [MATH] and [MATH] Therefore, ( 2.3 ) holds. |
Now, ( 2.2 ) and ( 2.3 ) give [EQUATION] Thus, [MATH] does not have LII. Remark 2.4 Theorem 2.3 shows that, in order to study LII in non-exceptional Riemann surfaces, it suffices to consider regular surfaces. |
Next, we need to consider also bordered Riemann surfaces whose boundary is a finite union of disjoint simple closed curves. They will always arise as closed subsets of some open non-exceptional Riemann surface, and we give them the metric induced from the Poincaré metric of the host open surface. |
Such induced metric has of course curvature [MATH] An example of such a bordered Riemann surface is the closure of any geodesic domain. |
Y-piece is a compact bordered Riemann surface which is topologically a sphere without three open disks and whose boundary curves are simple closed geodesics. Given three positive numbers [MATH] , there is a unique (up to conformal mapping) Y-piece such that their boundary curves have lengths [MATH] (see, e.g., 36 , p.4... |
generalized Y-piece is a bordered or non-bordered Riemann surface which is topologically a sphere without [MATH] open disks and [MATH] points, with integers [MATH] such that [MATH] so that the [MATH] boundary curves are simple closed geodesics and the [MATH] deleted points are cusps. Observe that a generalized Y-piece ... |
The following example shows that regularity is not a sufficient condition in order to have LII. Example 2.5 Let [MATH] be a non-exceptional Riemann surface built as follows: Consider the generalized [MATH] -pieces [MATH] with one cusp and such that [MATH] with [MATH] Let [MATH] be the surface obtained by identifying [M... |
collar in a non-exceptional Riemann surface [MATH] about a simple closed geodesic [MATH] is a doubly connected domain in [MATH] “bounded” by two Jordan curves (called the boundary curves of the collar) orthogonal to the pencil of geodesics emanating from [MATH] such collar is equal to [MATH] for some positive constant ... |
Let [MATH] be a non-exceptional Riemann surface with a cusp [MATH] (if [MATH] , every isolated point in [MATH] is a cusp). collar in [MATH] about [MATH] is a doubly connected domain in [MATH] |
“bounded” both by [MATH] and a Jordan curve (called the boundary curve of the collar) orthogonal to the pencil of geodesics emanating from [MATH] It is well-known that the length of the boundary curve is equal to the area of the collar (see, e.g., |
). A collar of area [MATH] about [MATH] is called a [MATH] -collar. We denote by [MATH] the [MATH] -collar of the cusp [MATH] with area [MATH] |
We will use several times the following result known as Collar Lemma (see ). Lemma 2.6 If [MATH] is a simple closed geodesic in a non-exceptional Riemann surface [MATH] , then there exists a collar about |
[MATH] of width [MATH] , where [MATH] Remark 2.7 Along this paper, [MATH] will denote a simple closed geodesic in [MATH] and [MATH] the width of the collar of [MATH] , where |
[MATH] Denote by [MATH] the collar of [MATH] of width [MATH] and by [MATH] the collar of [MATH] of width [MATH] It is well-known that if [MATH] and [MATH] are disjoint simple closed geodesics, then [MATH] |
For each cusp there exists a [MATH] -collar and [MATH] -collars of different cusps are disjoint. Besides, the collar [MATH] of the simple closed geodesic [MATH] does not intersect the [MATH] -collar of a cusp (see |
and 10 , Chapter 4] ). We will use the thick-thin decomposition of Riemann surfaces given by Margulis Lemma (see, e.g., , p.107] ). Concretely, for any [MATH] any Riemann surface, [MATH] , can be partitioned into a thick part, |
[MATH] , and a thin part, [MATH] whose connected components are either collars of cusps or collars of simple closed geodesics of length less than [MATH] In fact, 12 , Lemma 4.9] gives the following. |
Lemma 2.8 Let [MATH] be a non-exceptional Riemann surface and [MATH] If [MATH] , then the shortest geodesic loop [MATH] with base point [MATH] is contained either in the [MATH] -collar of a cusp or in the collar [MATH] of a simple closed geodesic [MATH] |
We collect below a well-known hyperbolic trigonometric formula (see, e.g., 10 , p.454] which will be useful. Proposition 2.9 The following formula holds for polygons on the unit disk (and then for simply connected polygons on any non-exceptional Riemann surface). |
Let us consider a geodesic quadrilateral with three right angles and let [MATH] the other angle. If [MATH] are the lengths of the sides which meet with angle [MATH] |
and [MATH] is the length of the opposite side to the side with length [MATH] , then [MATH] Lemma 2.10 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] a connected component of [MATH] By Margulis Lemma, [MATH] is a collar. |
[MATH] If [MATH] is a collar of a simple closed geodesic [MATH] , then [MATH] and [MATH] [MATH] If [MATH] is a collar of a cusp [MATH] , then [MATH] and [MATH] |
Proof. Assume first that [MATH] is a collar of a simple closed geodesic [MATH] Fix [MATH] Since [MATH] , Lemma 2.8 gives that the shortest geodesic loop |
[MATH] with base point [MATH] is contained in [MATH] Let [MATH] By Proposition 2.9 , we have [EQUATION] Collar Lemma gives that there exists a collar about [MATH] of width [MATH] , where [MATH] Thus, |
[EQUATION] [MATH] and [MATH] is contained in [MATH] Assume now that [MATH] is a collar of a cusp [MATH] Fix [MATH] Since [MATH] , Lemma 2.8 gives that the shortest geodesic loop |
[MATH] with base point [MATH] is contained in [MATH] As usual, consider a fundamental domain for [MATH] in the upper half-plane [MATH] |
contained in [MATH] and such that [MATH] corresponds to [MATH] Thus, [MATH] corresponds to [MATH] Without loss of generality we can assume that [MATH] corresponds to [MATH] Thus, we can represent [MATH] in the upper half-plane by means of a geodesic with endpoints [MATH] and [MATH] We have |
[EQUATION] [MATH] and [MATH] is contained in [MATH] Lemma 2.11 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] a connected component of [MATH] If [MATH] is a connected component of [MATH] , then [MATH] |
Proof. By Margulis Lemma, [MATH] is a collar. Assume first that [MATH] is a collar of a simple closed geodesic [MATH] Lemma 2.10 gives that [MATH] and [MATH] with [MATH] We have |
[EQUATION] Since the function [MATH] satisfies [MATH] for every [MATH] , we conclude [MATH] [MATH] and [MATH] for every [MATH] Since the function [MATH] satisfies |
[EQUATION] for every [MATH] , the function [MATH] is decreasing on [MATH] Consequently, [MATH] for every [MATH] , and [EQUATION] |
Assume now that [MATH] is a collar of a cusp [MATH] Lemma 2.10 gives that [MATH] and [MATH] Thus, [MATH] Lemma 2.12 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] a connected component of [MATH] that is a collar of a simple closed geodesic [MATH] Then |
[EQUATION] Furthermore, if [MATH] , then [EQUATION] Proof. Let [MATH] Lemma 2.10 gives [MATH] and [MATH] It is well-known that [MATH] , and so, |
[EQUATION] Since the function [MATH] is decreasing on [MATH] , we have [EQUATION] If [MATH] , then [EQUATION] Lemma 2.13 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] a connected component of [MATH] that is a collar of a simple closed geodesic [MATH] with |
[EQUATION] Then, [MATH] and [EQUATION] Proof. By Lemma 2.10 we have [MATH] Define [MATH] We also have [EQUATION] Besides, [MATH] and [MATH] We have |
[EQUATION] Denote by [MATH] the collar [MATH] Since [MATH] and [MATH] , we conclude [EQUATION] Finally, by Lemma 2.12 , since [EQUATION] |
we have [EQUATION] Lemma 2.14 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] two different connected components of [MATH] Let [MATH] be the collar given by the Collar Lemma corresponding to [MATH] Then |
[EQUATION] Furthermore, if [MATH] is a collar of a simple closed geodesic [MATH] , then [MATH] is an increasing function on [MATH] |
Proof. By Margulis Lemma, [MATH] and [MATH] are collars. Assume that [MATH] is a collar [MATH] for some simple closed geodesic [MATH] Collar Lemma gives that there exists a collar [MATH] about [MATH] of width [MATH] , where [MATH] Lemma 2.10 gives that [MATH] is contained in [MATH] and |
[EQUATION] If [MATH] , then [EQUATION] If [MATH] , then we define the function [EQUATION] Since [EQUATION] [MATH] is an increasing function in [MATH] In particular, [MATH] for every [MATH] Since |
[EQUATION] we consider [EQUATION] Thus, [EQUATION] Hence, [EQUATION] Assume now that [MATH] is a collar [MATH] for some cusp [MATH] Consider a fundamental domain for [MATH] in the upper half-plane [MATH] |
contained in [MATH] and such that [MATH] corresponds to [MATH] Thus, [MATH] corresponds to [MATH] , and Lemma 2.10 gives that [MATH] |
and [MATH] is contained in [MATH] Hence, [EQUATION] Since collars of geodesics and cusps are pairwise disjoint by Remark 2.7 , we conclude |
[EQUATION] Let [MATH] be a non-exceptional Riemann surface and [MATH] Consider the thick-thin decomposition given by Margulis Lemma: |
[MATH] and [MATH] Given the cusps [MATH] in [MATH] , let us consider for each [MATH] the collar in [MATH] about [MATH] with area [MATH] [MATH] , and let [MATH] |
Given the simple closed geodesics [MATH] in [MATH] with [MATH] let us consider for each [MATH] its collar [MATH] in [MATH] of width [MATH] |
By Lemma 2.10 , we have [EQUATION] The following result is well-known. Lemma 2.15 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] Then |
[EQUATION] Furthermore, if [MATH] , then [EQUATION] 3. Surfaces and graphs We are going to construct a graph associated to any non-exceptional Riemann surface. |
A subset [MATH] in a metric space [MATH] is called [MATH] -separated [MATH] , if [MATH] for any distinct [MATH] . Note that if [MATH] is maximal with this property, then the union [MATH] covers [MATH] . A maximal [MATH] -separated set [MATH] in a metric space [MATH] is called an [MATH] approximation of [MATH] |
Fix [MATH] Given any [MATH] -approximation [MATH] of [MATH] the graph [MATH] with [MATH] and [MATH] is called an [MATH] net Given any [MATH] satisfying |
[EQUATION] let [MATH] , where [MATH] denotes the index set of the collars about simple closed geodesics in the thick-thin decomposition from equation ( 2.4 ). Note that [MATH] and so, [MATH] Let [MATH] Therefore, [MATH] is the surface [MATH] without the collars [MATH] about the cusps and the collars [MATH] with [MATH] ... |
Remark 3.1 From the definition of [MATH] , it follows that no pair of vertices in [MATH] can be adjacent in [MATH] By Lemma 2.14 [MATH] are pairwise disjoint sets in [MATH] |
We have the following result in 26 , Lemma 2.3] Lemma 3.2 Let [MATH] be a complete Riemannian [MATH] -manifold whose Ricci curvature is bounded from below by [MATH] |
[MATH] , and let [MATH] be an [MATH] -separated subset of [MATH] Then we have [MATH] for all [MATH] and for all [MATH] , where [MATH] . Consequently, every |
[MATH] -net in a complete Riemannian manifold whose Ricci curvature is bounded from below is uniform. Lemma 3.2 has the following consequence. |
Lemma 3.3 If [MATH] is a non-exceptional Riemann surface and [MATH] is a [MATH] -separated set in [MATH] , then there is some constant [MATH] such that [MATH] for every [MATH] |
Proposition 3.4 If [MATH] is a non-exceptional Riemann surface, [MATH] and [MATH] then [MATH] is uniform. Proof. Let [MATH] be any vertex in [MATH] If [MATH] , then its adjacent vertices [MATH] in [MATH] satisfy [MATH] and, by Lemma 3.3 , these are at most [MATH] Also, by Remark 3.1 [MATH] has at most one adjacent vert... |
If [MATH] is the vertex corresponding to a collar [MATH] then, by Lemma 2.11 [MATH] satisfies [MATH] Let [MATH] be a maximal [MATH] -separated set in [MATH] Then, [MATH] and for any vertex [MATH] adjacent to [MATH] , we have [MATH] |
[MATH] and so, [MATH] must be contained in some [MATH] Thus, by Lemma 3.3 , there are at most [MATH] vertices adjacent to [MATH] in [MATH] |
If [MATH] is the vertex corresponding to a collar [MATH] then, by Lemma 2.11 [MATH] and [MATH] Let [MATH] be a maximal [MATH] -separated set in [MATH] Then, [MATH] and for any vertex [MATH] adjacent to [MATH] , we have [MATH] |
Let [MATH] be a non-exceptional Riemann surface and [MATH] the associated graph with vertex set [MATH] Given a geodesic domain [MATH] in [MATH] , let [MATH] |
Remark 3.5 Let [MATH] be a non-exceptional Riemann surface and [MATH] the associated graph defined above. Given a geodesic domain [MATH] , then [MATH] is the set of vertices [MATH] satisfying one of the following conditions: |
a) [MATH] is adjacent to a vertex [MATH] and either [MATH] or [MATH] In particular, we have either [MATH] or [MATH] b) [MATH] is adjacent to a vertex [MATH] with [MATH] for some [MATH] and so, [MATH] |
Notice that [MATH] for every [MATH] with [MATH] since, by Lemma 2.14 , the distance from [MATH] to [MATH] is at least [MATH] Note also that, since the collars are pairwise disjoint, given a geodesic domain [MATH] and a collar [MATH] , we have either [MATH] or [MATH] |
Let [MATH] denote the set of vertices [MATH] satisfying condition [MATH] above. Remark 3.6 For every vertex [MATH] [MATH] Theorem 3.7 |
Given a [MATH] -regular non-exceptional Riemann surface [MATH] [MATH] and [MATH] [MATH] satisfies LII if and only if [MATH] satisfies LII. |
Proof. Suppose [MATH] satisfies LII. Consider any non-empty finite subset of vertices [MATH] in [MATH] . Let us define [EQUATION] |
Choose [MATH] satisfying [EQUATION] Then we have the following: for any pair of vertices [MATH] [MATH] for any vertex [MATH] and any [MATH] , we have [MATH] where [MATH] |
for any vertex [MATH] and any [MATH] , we have [MATH] Let [MATH] and [MATH] Lemmas 2.13 (with [MATH] ) and 2.15 give [MATH] and [MATH] for every [MATH] , and |
[EQUATION] Now, let [MATH] By Lemmas 2.15 and 2.11 [EQUATION] Since, by Proposition 3.4 [MATH] is [MATH] -uniform for some constant [MATH] , we have [MATH] Therefore, |
[EQUATION] Thus, since [MATH] satisfies LII, there is some constant [MATH] such that [MATH] and, hence, [EQUATION] and [MATH] satisfies LII. |
Assume now that [MATH] satisfies LII. By Lemma 2.1 , it suffices to check that [MATH] satisfies LII on every geodesic domain. Let [MATH] be any geodesic domain and consider the set [MATH] Then, since |
[EQUATION] we have, by Lemmas 2.15 2.10 and 2.12 [EQUATION] For any connected component [MATH] of [MATH] consider a maximal [MATH] -separated set [MATH] . Notice that [MATH] By Remark 3.6 , for every point [MATH] [MATH] and since [MATH] is a maximal [MATH] -separated set, there is some [MATH] such that [MATH] |
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