CC HROMATIC NUMBERS OF HYPERBOLIC SURFACES
Hugo Parlier ** , Camille Petit †† Abstract.
This article is about chromatic numbers of hyperbolic surfaces. For a metric space,the d -chromatic number is the minimum number of colors needed to color the points of thespace so that any two points at distance d are of a different color. We prove upper boundson the d -chromatic number of any hyperbolic surface which only depend on d . In anotherdirection, we investigate chromatic numbers of closed genus g surfaces and find upperbounds that only depend on g (and not on d ). For both problems, we construct families ofexamples that show that our bounds are meaningful.
1. I
NTRODUCTION
The chromatic number of a graph is the minimum number of colors needed to color itsvertices so that any two adjacent vertices are colored differently. Given a metric space ( X , δ ) and a real number d >
0, one can associate a (possibly infinite) graph where vertices arepoints of X and two vertices are joined by an edge if they are exactly at distance d . Thechromatic number of this graph gives rise to a notion of chromatic number χ (( X , δ ) , d ) fora metric space. The particular case when X is the Euclidean plane (and d = d is equivalent) has attracted a particular amount of attention and is called theHadwiger-Nelson problem (see [77, 55, 88] and references therein). By exhibiting an explicitcoloring coming from a hexagonal tiling, it is not particularly difficult to prove that itis at most 7. A lower bound of 4 can be obtained by exhibiting a four-chromatic unitdistance graph in the plane, for instance the Moser spindle. Going beyond these two ratherelementary bounds is completely open and will probably require either great perseveranceor an inspired idea.Other metric spaces have been investigated including n -dimensional Euclidean space (seefor instance [66, 99]) and more recently the hyperbolic plane H [33]. Unlike Euclidean spaces,the hyperbolic plane is not invariant by homothety, so a priori the chromatic numberdepends on a choice of d . Bounds for χ ( H , d ) in function of d have been established by * Research supported by Swiss National Science Foundation grant number PP00P2 153024 † Research supported by Swiss National Science Foundation grant number 200021 153599
Primary: 05C15, 30F45. Secondary: 05C63, 53C22,30F10.
Key words and phrases: chromatic numbers, hyperbolic surfaces1 a r X i v : . [ m a t h . G T ] N ov loeckner, but it is not known whether or not there exists a uniform (independent of d )upper bound. A theorem of de Bruijn-Erd ¨os [22] says that any given infinite graph can becolored by k colors if and only if all of its finite subgraphs can as well. So, showing that thechromatic number of H can be made arbitrarily large amounts to exhibiting “subgraphs”of H (by which we mean geometric copies of finite graphs) with arbitrarily large chromaticnumber. For the moment, the best known lower bound is only 4.Our main focus is on the more general setup of hyperbolic surfaces (not necessarily theplane). We ask only that they be complete hyperbolic surfaces. Our first results is an upperbound on the chromatic number which only depends on d . Theorem 1.1.
There exists a constant C > such that for every number d > every completehyperbolic surface S satisfies χ ( S , d ) ≤ C e d .In first instance, our upper bound, exponential in d , seems particularly weak in comparisonwith the linear upper bound for the hyperbolic plane. But in fact we exhibit, for any d > d / . Theorem 1.2.
There exists a constant C > and a family of complete hyperbolic surfaces S d ,d > so that χ ( S d , d ) ≥ C e d / .The optimal constants have growth that are exponential in α d for some α in between and1. Determining the exact value for α could be an interesting problem. The construction andproofs are quite elementary and only require some basic tools about hyperbolic geometryand trigonometry.Given a metric space ( X , δ ) , there is a natural way of associating a chromatic number thatdoesn’t depend on d . One defines the chromatic number of ( X , δ ) as χ (( X , δ )) : = sup { χ (( X , δ ) , d ) : d > } .When ( X , δ ) is the Euclidean plane, this is simply the chromatic number discussed previ-ously. When ( X , δ ) is the hyperbolic plane, we’ve seen that it is unknown whether thisquantity is finite. In the particular case when ( X , δ ) is a compact Riemannian manifoldhowever, then by a compactness argument, this quantity is always finite. Now if one has anatural family of compact manifolds, one can study the supremum of this quantity over thewhole family. As an example, consider the following problem (which we don’t know theanswer to): among all 2 dimensional flat tori, which one has the largest chromatic number?2y the theorem of de Bruijn and Erd ¨os mentioned previously, this quantity is an upperbound for the chromatic number of the plane - in fact the chromatic number of any torus isan upper bound (this will be explained in the preliminaries).For closed hyperbolic surfaces, one can ask the same question. In this case, for each genus,we get a moduli space of isometry types of surfaces. Our second set of results are aboutbounds on the chromatic numbers that only depend on the genus and not on the individualgeometries. We begin with our upper bounds. Theorem 1.3.
There exists a constant C > such that for every integer g ≥ every closedhyperbolic surface S of genus g satisfies χ ( S ) ≤ C g .Again, one could ask whether there is not a universal upper bound (which doesn’t dependon genus) but we exhibit families of surfaces that provide the following lower bounds. Theorem 1.4.
There exists C > and a family of closed hyperbolic surfaces S g , where S g hasgenus g, such that χ ( S g ) ≥ C √ g .So again, we show that there exists a constant β , lying somewhere in between and 1 suchthat the optimal upper bound on chromatic numbers behaves like g α . Whether there isrelation between the constant α described above and this constant β remains to be seen.Interestingly, our results rely on the celebrated result of Ringel and Youngs about the genusof complete graphs (which already provided an answer to another graph coloring problem,namely the Heawood conjecture). We also mention that one could ask the same questions,and apply some of the same techniques, to other moduli spaces, such as hyperbolic surfaceswith punctures but for the sake of clarity, we’ve restricted our study to closed surfaces.One by-product of our lower bounds in the above theorem is an example of hyperbolicsurface which has infinite chromatic number. It is not literally a corollary of the theorembut can be directly obtained using the same building blocks as the ones we need in theproof of Theorem 1.41.4. Corollary 1.5.
There exists a hyperbolic surface Z such that χ ( Z ) = ∞ .This article is organized as follows. After a preliminary section in which we introduce toolson the geometry of hyperbolic surfaces, we have two main sections. The first of these isabout bounds of d -chromatic numbers in terms of d and the second about closed surfacesand bounds on chromatic numbers in terms of the genus.3 cknowledgement. Both authors would like to thank Bill Balloon for inspiration.
2. P
RELIMINARIES
We’ll use this preliminary section to introduce definitions and notations, and also to givea short description of how we’re thinking about hyperbolic surfaces and some of theproperties we’ll use throughout the paper.
For a metric space ( X , δ ) we define its chromatic number χ (( X , δ ) , d ) relative to a distance d > d -chromatic number) as the minimal number of colors needed to colorall points of X such that any x , y ∈ X with δ ( x , y ) = d are colored differently. We call a d -coloring a coloring of ( X , δ ) where any x , y ∈ X with δ ( x , y ) = d are colored differently.When the metric space consists in the vertices of a graph, distance to edge distance and d =
1, this is the usual definition of the chromatic number of the graph.Equivalently, one can define the chromatic number of a metric space using the usualchromatic number of graphs by associating a graph to the metric space as follows. Givena metric space ( X , δ ) and a real number d >
0, we construct a graph whose vertices arepoints of X and we place an edge between points if they are exactly at distance d .For certain metric spaces, the choice of d is crucial; for others, such as n -dimensionalEuclidean space, any choice of d is equivalent. This prompts the following definition. Wedefine the chromatic number χ (( X , δ )) of a metric space ( X , δ ) to be the quantity χ (( X , δ )) : = sup { χ (( X , δ ) , d ) : d > } .As examples, for the Euclidean and hyperbolic planes, the following inequalities are known:4 ≤ χ ( R ) ≤ ≤ χ ( H ) ≤ ∞ .The theorem of de Bruijn and Erd ¨os [22] , mentioned in the introduction, is the following: Theorem 2.1.
Any infinite graph G can be colored by k colors if and only if all of its finite subgraphscan be colored by k colors.
As an example of an application of this theorem, consider any Euclidean flat torus T ofdimension n . Then we claim that χ ( T ) ≥ χ ( R n ) .4o see this, apply the de Bruijn-Erd ¨os theorem to R n . There is thus a finite set of points of R n that realize χ ( R n ) . They can be made to lie in a ball of arbitrarily small size. For a fixed T , one can find any Euclidean ball of sufficiently small size that is isometrically embeddedinside T . Thus the chromatic number of T is at least the chromatic number of R n . Whetheror not these quantities are equal - or finding an upper bound on χ ( T ) in function of χ ( R n ) -could be an interesting problem. In particular, understanding the behavior ofsup { χ ( T ) : T is a flat n -dimensional torus } in function of n .Problems for n -dimensional tori can often be translated to analogous questions to hyperbolicsurfaces of genus g ≥ M g the modulispace of genus g ≥ g ≥ χ ( S ) of a hyperbolicsurface S of genus g ≥
2, one can studysup { χ ( S ) : S ∈ M g } as a function of g . Investigating this quantity is one of the main goals of the article. Inthe next section we begin by properly defining which types of hyperbolic surfaces we’reinterested in and some of the tools we’ll need in the sequel. A hyperbolic surface is a surface locally isometric to the hyperbolic plane H . A surface issaid to be complete if it geodesically complete as a Riemannian manifold. We will generallybe concerned with complete hyperbolic surfaces but to construct them we will sometimesuse surfaces with (simple) geodesic boundary. We denote d S the distance function for asurface S .The simplest such surface is a funnel: topologically an infinite half-cylinder. Such a surfacecan be obtained as follows: we quotient H by a hyperbolic element to obtain an infinitecylinder. Such a cylinder has a unique simple closed geodesic (the quotient of the axis ofthe hyperbolic element by its action) which separates the cylinder into two half cylinders.One of these half cylinders is the funnel we’re talking about.These funnels provide a way of going from a surface with simple geodesic boundary toa complete surface by pasting funnels of the appropriate boundary length to the surfacewith boundary. Note that the original surface with boundary is a convex subset of the full5urface and is sometimes referred to as the convex core. From a dynamical point of view,everything interesting on the surface happens within the convex core.On hyperbolic surfaces, there is a unique geodesic representative in every isotopy class of asimple closed curve (which does not surround a disk or a cusp). Simple closed geodesics ofhyperbolic surface have an associated collar which is a tubular neighbourhood around thegeodesics which can be described as follows. Lemma 2.2 (Collar lemma) . Let γ be a simple closed geodesic on a complete hyperbolic surface S.Then the set C ( γ ) : = { x ∈ S : d S ( x , γ ) ≤ w ( γ ) } where w ( γ ) = arcsinh (cid:16) (cid:96) ( γ ) (cid:17) is an embedded cylinder isometric to [ − w ( γ ) , w ( γ )] × S with the Riemannian metricds = d ρ + (cid:96) ( γ ) cosh ( ρ ) dt .This version of the collar lemma, and many of the basic facts we state, can be found in [11].Furthermore, if any two simple closed geodesics are disjoint, then so are their collars.Immediate consequences of the collar lemma include the fact that any two simple closedgeodesics of length less than 2 arcsinh ( ) are always disjoint. One place collars naturallyappear is when a surface is decomposed into its thick and thin parts. For a given ε >
0, wecan separate a surface S into its ε -thick part, namely (cid:98) S ε : = { x ∈ S : injrad ( x ) > ε } where injrad ( x ) is the injectivity radius of S at the point x and its ε -thin part S \ (cid:98) S ε : = { x ∈ S : injrad ( x ) ≤ ε } .Because any two simple closed geodesics of length less than 2 arcsinh ( ) cannot intersect,when ε < arcsinh ( ) the set S \ (cid:98) S ε (if it is not empty!) consists of a collection of cylinders.These cylinders are either collars of a certain width around a simple closed geodesic or areneighbourhoods of cusps.Consider one of them, say C , that contains a simple closed geodesic γ . Any point on one ofits boundary curves is the base point of an embedded geodesic loop of length 2 ε . From thiswe can deduce that the two boundary curves, say γ + and γ − , of C are smooth curves, both6f equal length, and both parallel lines to the unique closed geodesic in their homotopyclass γ . To see this, observe that as these loops are all of equal length, and all parallel to γ ,the angle formed by any of these loops at the base point must always be the same. This issimply because the length of γ can be computed as a function of this angle and the lengthof the loop (which is always 2 ε ). Similarly, the distance in between any of the boundarypoints and γ is always equal. As such, for any C , there exists a K C which only depends on C (or alternatively ε and (cid:96) ( γ ) ) such that ∂ C = { x ∈ S : d S ( x , γ ) = K C } = γ + ∪ γ − .Let us compute the value of K C in function of (cid:96) ( γ ) and ε . Fix a point on the boundaryof C and consider a distance path η to γ . Cutting along the loop of length 2 ε , γ and η gives hyperbolic quadrilateral such as Figure 11. This quadrilateral can be divided into two η γ (cid:96) ( η ) (cid:96) ( γ ) εγ + Figure 1: Computing K C quadrilaterals with three right angles as in the figure with we can compute. By standardhyperbolic trigonometry we havesinh ( ε ) = sinh (cid:18) (cid:96) ( γ ) (cid:19) cosh ( (cid:96) ( h )) from which we obtain K C = (cid:96) ( η ) = arccosh sinh ( ε ) sinh (cid:16) (cid:96) ( γ ) (cid:17) .It’s interesting to compare this value to the width of the collar from the collar lemma. Inparticular note that the difference w ( γ ) − K C = arcsinh (cid:16) (cid:96) ( γ ) (cid:17) − arccosh sinh ( ε ) sinh (cid:16) (cid:96) ( γ ) (cid:17) (1)7s a positive number because ε < arcsinh ( ) . Furthermore it reaches its minimum in (cid:96) ( γ ) =
0. When ε = arcsinh ( ) , this minimum is exactly 0.The geodesic γ also divides C into two parts, C + , C − , whose other boundary curves are γ + and γ − . K C w ( γ ) γ + γ − γ Figure 2: The collar around a simple closed geodesic γ We now put a further restriction on ε which will be useful in what follows. Although C maynot be convex, provided ε > C + , C − will be. We suppose now that ε ≤ arcsinh (cid:18) √ (cid:19) and we’ll see that this condition ensures convexity. By computing the limit when (cid:96) ( γ ) = log ( ) . What will be crucial in the argument that follows is that twice this value is less thanarcsinh (cid:16) √ (cid:17) .We now turn our attention to the convexity of C + . To see this, we begin by observing thatthe convex hull of either of the boundary curves of C lies in its respective half: for examplefor any x , y ∈ γ + , the shortest geodesic between x , y lies in C + . The shortest path inside C + between x and y is of length strictly less than half (cid:96) ( γ + ) . So if this is not the minimalgeodesic path, then there is another shorter path between them.To show this never occurs, we fix a point x ∈ γ + and move the point y ∈ γ + away from x until this occurs for the first time. The path obtained lies entirely outside the interiorof C + and these two simple geodesic paths form a curve (see the left most configurationof Figure 33). On a hyperbolic surface there are no geodesic bigons so the correspondingcurve is non-trivial and has a geodesic representative ˜ γ which in turn has its own collar. By8 y γ + x y γ + γ x y γ + γ Figure 3: Three types of potential distance pathsconstruction the two collars C ( γ ) and C ( ˜ γ ) intersect, which is impossible by the versionof the collar lemma described in the preliminaries. We have reached a contradiction andshown that the convex hull of γ + is entirely contained in C + .Now the only way C + can be non-convex is if there is a distance path which leave C + andreturns through C − (see the middle case of Figure 33). As such it will have length at least thewidth of C − which is equal to K C . In addition it will have spent some time in the thick partof S : by the estimates given above this will add at least log ( ) to its length. So it is of lengthat least K C + log ( ) . The shortest path between the two points that lies entirely inside C + is of length at most K C + (cid:96) ( γ ) ≤ K C + arcsinh (cid:18) √ (cid:19) < K C + log ( ) and this provides a contradiction.
3. B
OUNDING THE CHROMATIC NUMBER IN FUNCTION OF d In this section we prove Theorem 1.11.1 from the introduction. We begin by proving a universalupper bound on the chromatic number which only depends on the parameter d . Then, forevery d >
0, we exhibit a surface S d which has chromatic number at least C e d / for someuniversal constant C >
0. 9 .1. Upper bounds
To prove our upper bounds, we will need to lift any complete hyperbolic surface S = H / Γ to its universal cover H and then construct a Γ invariant coloring of H .To begin, for a fixed value r >
0, we consider a maximal set ∆ r ⊂ S of points such that if x , y ∈ ∆ r and x (cid:54) = y then d S ( x , y ) > r . By construction this set satisfies the following twoproperties:For x , y ∈ ∆ r , x (cid:54) = y , B r / ( x ) ∩ B r / ( y ) = ∅ . S = ∪ x ∈ ∆ r B r ( x ) Here, B r ( x ) denotes the closed ball center at x of radius r . Note that we don’t ask that thesets B r / ( x ) be embedded balls in S .We remark however that for any ρ , a set B ρ ( x ) lifts to a union of embedded balls on H (notnecessarily disjoint - for instance if ρ is larger than the diameter of S , then the lift is H ).Furthermore, if B , B (cid:48) are two disjoint balls on S , then any of their lifts are also disjoint andthe distance d S ( B , B (cid:48) ) is simply the minimum of the distances of their lifts in H . This is allessentially in the definitions of a cover or the universal cover but we emphasize it as it willbe crucial in what follows.Now given d > S by balls of radius r so we set r : = min { d , arcsinh ( ) } .In particular each ball is of diameter strictly less than d . We then consider a ∆ r as describedabove.We now endow the set ∆ r with a graph structure G as follows. Vertices are points of ∆ r and two vertices x , y share an edge if there exists x (cid:48) ∈ B r ( x ) and y (cid:48) ∈ B r ( y ) such that d S ( x (cid:48) , y (cid:48) ) = d . Our strategy is to bound the degree of G by a function of d and r .To do this we lift a point x ∈ ∆ r to the universal cover. We denote ˜ x ∈ π − ( x ) , where π : H → S is the covering map. We observe that for any ρ > B ρ ( ˜ x ) (which lies in H ) covers the set B ρ ( x ) (and of course belongs to its preimage).For x ∈ ∆ r we want to bound the degree deg ( x ) of x in G . We compute an upper boundon this cardinality using ˜ x : it is bounded by the number of ˜ y ∈ ˜ ∆ r : = π − ( ∆ r ) such thatthere exists x (cid:48) ∈ B r ( ˜ x ) and y (cid:48) ∈ B r ( ˜ y ) which satisfy d H ( x (cid:48) , y (cid:48) ) = d .The balls of radius r / around any such ˜ y are disjoint and must lie entirely in the annulus A centered at ˜ x , of inner radius d − r and outer radius d + r . The area of a ball of radius ρ
10n the hyperbolic plane is 4 π sinh (cid:16) ρ (cid:17) so we have4 π |{ ˜ y ∈ ˜ ∆ r : B r / ( ˜ y ) ⊂ A }| sinh (cid:16) r (cid:17) < π (cid:32) sinh (cid:32) d + r (cid:33) − sinh (cid:32) d − r (cid:33)(cid:33) .Using this we deduce the following bound on |{ ˜ y ∈ ˜ ∆ r : B r / ( ˜ y ) ⊂ A }| which in turnbounds the degree of any point of G :deg ( G ) ≤ |{ ˜ y ∈ ˜ ∆ r : B r / ( ˜ y ) ⊂ A }| ≤ sinh (cid:16) d + r (cid:17) − sinh (cid:16) d − r (cid:17) sinh (cid:0) r (cid:1) = sinh ( r ) sinh (cid:0) r (cid:1) sinh ( d ) .Recalling the definition of r , we obtain deg ( G ) ≤ φ ( d ) , where φ ( d ) : = sinh ( d ) sinh ( d / ) , d ≤
10 arcsinh ( ) sinh (
10 arcsinh ( )) · sinh ( d ) , d ≥
10 arcsinh ( ) .We can now deduce our upper bounds. We use Brooks’ theorem on graph coloring whichasserts that a graph of degree at most D can be colored with at most D + G with at most φ ( d ) + d -coloring of S asfollows. We color each ball B r ( x ) , x ∈ ∆ r with the color corresponding to the vertex x in G . If a point belongs to several balls we choose one of the colors of the balls it belongs toarbitrarily as its color. This proves the upper bound χ ( S , d ) ≤ φ ( d ) +
1, which in particulargives χ ( S , d ) ≤ C e d for some constant C >
0, as stated in Theorem 1.11.1. Note that our d -coloring of S lifts to a Γ invariant d -coloring of H . The goal is to give, for any d >
0, a surface S d with a d -chromatic number that satisfies thelower bound of Theorem 1.21.2.Before going to the general construction, we begin by constructing a family of surfaces S d N for a discrete set of d N s with d N → ∞ as N → ∞ . The general construction that followsretains many of the key properties of this simpler construction.11or any integer N ≥
3, there is a unique ideal regular hyperbolic polygon with N sides. Ithas a well defined center point and a number of self isometries including rotations of angle π N around this center point. It also has a unique maximally embedded disk (we’ll computeits radius R N in the sequel), centered in the center point. By symmetry the disk toucheseach of the N sides in points which we’ll call the midpoints of the sides.For any N we consider N + N sides and glue the sides in pairs (see Figure 44). The only thing we ask of this gluing isthat it must obey the following two rules: every two distinct polygons share exactly oneside and the sides are pasted in their midpoints. The result is a connected finite area surfacewith at least one cusp. v j v k d N Figure 4: Constructing S d N We denote v , . . . , v N the center points of the polygons. These will be the vertices of anembedded complete graph of N vertices.Our first claim is that on the resulting surface d ( v j , v k ) = R N for k (cid:54) = j . This is simply because the distance between a center point v k and any of the sidesof the polygon is R N . So any path between two distinct vertices must pass through one ofthe sides of the polygons and as such has at least length 2 R N . Between any two distinctvertices there is a unique path of length 2 R N given by the concatenation of the radii andthis proves the claim. So we have geometric embedding of K N with edge length 2 R N .Let us set d N : = R N and compute its value. We consider a triangle formed by any twodistinct vertices v j , v k and an ideal point as in Figure 55.12 j v k d N π / N π / N Figure 5: Computing d N It has angles 0, π N , π N so by standard hyperbolic trigonometry the following holds:cosh ( d N ) = + cos ( π / N ) sin ( π / N )= ( π / N ) − d N = arccosh (cid:18) ( π / N ) − (cid:19) .We observe that d N grows asymptotically like 2 log ( N ) as N goes to infinity. Since fromour construction we obtain an embedded complete graph with N vertices formed by thecenters { v , · · · , v N + } and with edges length d N , we get the lower bound χ ( S d N , d N ) ≥ N ,which in turn gives χ ( S d N , d N ) ≥ C · e dN for some constant C >
0. So this example provides the correct lower bound but only worksfor a discrete set of values d N .We now adapt this construction to construct a surface for every d ≥ d . To do this we replacethe ideal polygons in the above construction with semi-regular right angled 2 N -gons withevery second side of length t for some t >
0; we ask that they have a rotational symmetryof order N which permutes the sides of length t (and thus the N remaining sides as well),as in Figure 66. These N remaining sides will be of some length s which only depends on t (for fixed N ). We can see the ideal polygon as the limit case of these polygons when t → t play the part of theideal points. More precisely we take N copies of the above polygon and we arbitrarily pastethe polygons along their N sides of length s where the only rule is that the resulting surface13 j ts d N ( t ) Figure 6: A semi-regular right angled 2 N -gonis orientable and any two distinct polygons are pasted along a single side. Here we don’thave to worry about the “shear” parameter as we are pasting two equal segments together.For each N we get a family of surfaces (with parameter t ) which has boundary curves. Weadd hyperbolic funnels to the boundary curves to get complete hyperbolic surfaces M N ( t ) .As before we obtain an embedded complete graph with N vertices formed by the centers ofthe polygons. The distances between these centers now depends on the parameter t . Forthe same reason as in the ideal case, the unique distance paths between the centers is theconcatenation of the radial distance paths from the centers to the sides of each polygonpasted together. We denote this distance d N ( t ) .We only need two facts about d N ( t ) : first of all lim t → + d N ( t ) = d N where d N is as definedabove; secondly d N ( t ) is a continuous (monotonous) function satisfying d N ( t ) → ∞ as t → ∞ . The first fact is by construction and the second is a direct consequence of thefact that if t becomes arbitrarily large, s becomes arbitrarily close to 0 so d N ( t ) becomesarbitrarily large. This can be seen more explicitly by hyperbolic trigonometry by relating t and d N ( t ) : cosh (cid:18) t (cid:19) = cosh (cid:18) d N ( t ) (cid:19) sin (cid:16) π N (cid:17) .We can now explain how we can associate one of these examples to any d >
0. We begin bychoosing the smallest integer N such that d < d N + .14n particular either N = d N ≤ d < d N + for some N ≥
3. We’ve already constructed the examples for d = d N so we suppose that d > d N . The properties of the functions d N ( t ) explained above imply that there exists a t d > d N ( t d ) = d .We set the example surface to be S d : = M N ( t d ) which has d -chromatic number boundedbelow by N by construction. From our previous computations for the ideal surfaces weknow d < d N + = arccosh (cid:18) ( π / N + ) − (cid:19) .from which we can deduce the lower bound of Theorem 1.21.2, that is, χ ( S d , d ) ≥ C e d for some constant C >
4. B
OUNDING THE CHROMATIC NUMBER IN FUNCTION OF THE GENUS
In this section we prove Theorems 1.31.3 and 1.41.4 from the introduction. We begin by provinga universal upper bound on the chromatic number which only depends on the genus. Wethen construct a family of surfaces to prove the lower bound.
The first idea for the upper bound is surprisingly simple and is close to what did in aprevious section. Let S be a surface of genus g .We begin by fixing a constant r > d > r .Recall the r -thick part (cid:98) S r / of S is defined by: (cid:98) S r / : = (cid:110) x ∈ S : injrad ( x ) > r (cid:111) .We will choose r such that S \ (cid:98) S r / is a collection of cylinders. On (cid:98) S r / we consider ∆ r amaximal set of points on (cid:98) S r / such that if x , y ∈ ∆ r and x (cid:54) = y then d S ( x , y ) > r . As the setis maximal we have (cid:98) S r / ⊂ ∪ x ∈ ∆ r B r ( x ) .15nd for x , y ∈ ∆ r , x (cid:54) = y , B r / ( x ) ∩ B r / ( y ) = ∅ . This allows us to bound the number ofpoints in ∆ r . We haveArea (cid:16) ∪ x ∈ ∆ r B r ( x ) (cid:17) < Area ( (cid:98) S r / ) ≤ Area ( S ) = π ( g − ) from which, using the formula for the area of a ball in H , we deduce4 π | ∆ r | sinh (cid:16) r (cid:17) < π ( g − ) and thus | ∆ r | < g − (cid:0) r (cid:1) .We color points of (cid:98) S r / by giving each ball centered in a point of ∆ r and of radius r adifferent color. If a point belongs to several balls we choose one of the colors of the balls itbelongs to arbitrarily as its color. As balls are of diameter < d , it provides a d -coloring of (cid:98) S r / .What remains to be colored are points lying in the cylinders comprising S \ (cid:98) S r / . Considera cylinder C that lies in this set. To color C we will divide it into sections of diameter lessthan d and color the sections.Any point on one of its boundary curves is the base point of an embedded geodesic loopof length r . As seen in the preliminaries, the two boundary curves γ + and γ − of C aresmooth curves, both of equal length, and both parallel lines to the unique closed geodesic γ in their homotopy class. There also exists a constant K C which only depends on C such that ∂ C = { x ∈ S : d S ( x , γ ) = K C } .The geodesic γ divides C into two convex subsets C + , C − . Using the convexity of thehalf-collars, we can define our sections. Sections are slices of the half-collars delimited bylines parallel to γ in the following way. We want each section to be of diameter less than d but very close to d . We define the height of each section to be the distance between theboundary curves. Note that the diameter of a section is realized by pairs of points, oneon each of the boundaries of the section, similarly to diametrically opposite points on aEuclidean cylinder (see Figure 77).For C + (we proceed analogously for C − ) the first section we construct is the one with γ + asthe top boundary curve. Denote d (cid:48) its diameter with d − d (cid:48) (arbitrarily) small. There is atriangle with sides of length h , (cid:96) ( γ + ) and d (cid:48) as in Figure 77. Thus the height h satisfies thefollowing inequality: h + (cid:96) ( γ + ) > d (cid:48) γ + d (cid:48) h (cid:96) ( γ + ) d (cid:48) Figure 7: A section and its diameter d (cid:48) From this and the fact that (cid:96) ( γ + ) ≤ r we have h > d (cid:48) − r r ≤ d we can choose d (cid:48) so that h > d C + in sections of diameter close to d iteratively from the top down (seeFigure 88). Although we’ve only proved that we can make the height greater than d for the γ γ + h Figure 8: Slicing a half cylinder into sectionsfirst section, to show this was a bound on the “width” (half the length of a boundary curve).As we move closer and closer to γ , the boundary curves become smaller and smaller andso the above argument continues to work. Thus we can make the subsequent all of heightat least d . It stops working once we reach γ so we don’t get a lower bound on the height ofthe last section (but we won’t need one).From these sections we create a graph: each section is a vertex and we relate two verticesby an edge if there exist two points, one in each corresponding section, at distance d . By theabove properties a section is related to at most two subsequent sections and two precedingsections so the graph is of degree at most 4. From a coloring of the graph we obtain a17 -coloring of C + by coloring points in each section in the color of the corresponding vertex(boundary points between two sections can be colored by either of the two colors). As thegraph is of degree at most 4, at most 5 colors are required to d -color C + . Analogously, wecan d -color C − with 5 as well. As we are only interested in the rough growth of the numberof colors, although we can clearly d -color C with a total of at most 10 colors.Now, there is at most 3 g − d -color the totality of the cylinders with atmost 10 ( g − ) colors.We can now conclude that at most g − (cid:0) r (cid:1) + ( g − ) colors are sufficient to d -color any S . The upper bound in Theorem 1.31.3 can be deduced by asimple manipulation of the above term setting r = ( ) for d ≥ ( ) andfrom Theorem 1.11.1 when d < ( ) . The goal is to obtain lower bounds by constructing geometric embeddings of completegraphs with small genus.We begin with the Ringel and Youngs ([44]) topological embedding of K n into a surface M n of genus g n where g n = (cid:22) ( n − )( n − ) (cid:23) (2)and g n is the smallest possible genus in which we could embed K n . This embedding ϕ : K n → M n has the following property: for all n (cid:54) = n ≡ M n \ ϕ ( K n ) is a collection of triangles (see [1010]). This topological embedding will serve as a blue printfor constructing hyperbolic surfaces with chromatic number roughly root of the genus. Asfor our lower bounds of Theorem 1.21.2, we begin by a slightly easier construction beforeshowing how to make it work in the general case; specifically we first construct a family ofsurfaces with growing genus before constructing a family with a surface in every genus.Fix an integer N such that N + ≡ M N + \ ϕ ( K N + ) by equilateral hyperbolic triangles with all three anglesequal to π N . These triangles are our building blocks and they are what will change in themore general construction which will follow.18here is a unique such triangle and its three equal side lengths can be directly computedusing hyperbolic trigonometry. (cid:96) N = arccosh (cid:32) cos ( π N ) + cos ( π N ) sin ( π N ) (cid:33) .This construction gives a family of well defined smooth hyperbolic surfaces S g N + of genus g N + for a family of N → ∞ . What we claim is that χ ( S g N + , (cid:96) N ) ≥ N +
1. This will followfrom the geometric embedding of K N + with edge distance (cid:96) N .To show that the embedding is indeed geometric we need to show that the images of thevertices of K N + are all at pairwise distance at least (cid:96) N (by construction they are at distanceat most (cid:96) N ). Consider a simple geodesic path between two distinct vertices of ϕ ( K N + ) thatis not the side of one of the triangles. We orient this path and look at it as a concatenationof simple segments that each pass through individual triangles. What we claim is that thefirst of the segments (and hence by symmetry the last) has length strictly greater than (cid:96) N .To see this we look at the geometry of the individual triangle. The segment leaves from avertex and so, as it is geodesic, must leave the triangle through the opposite side. It’s lengthis then at least the minimal distance between a vertex and the opposite side. This distance,in any equilateral hyperbolic triangle, is strictly greater than half of the length of one of thesides (see Figure 99). (cid:96) N Figure 9: An equilateral triangleFrom this, the length of any geodesic path between distinct vertices which is not the side ofa triangle is strictly greater than (cid:96) N . So the embedding is geometric and from Equation (22)with n = N + χ ( S g N + , (cid:96) N ) ≥ N + ≥ (cid:112) g N + + S g N + has the lower bound on chromatic number that we arelooking for but doesn’t have a surface in every genus. We shall need to modify the aboveconstruction to get a surface in every genus.We begin by changing the building blocks. From equilateral triangles we revert to “equilat-eral triangles” with a single interior boundary curve of length t . More precisely, for giveninteger N and given t > t and another boundary curveconsisting of a triangle with three angles equal to π N . One constructs such a triangle bygluing three copies of a certain quadrilateral: this quadrilateral is the unique quadrilateralwith two right angles and a side between them of length t and two other angles equal to π N and a central symmetry as is Figure 1010. π / N π / N t Figure 10: The quadrilateral used to build the one holed triangleNow by taking three copies of this quadrilateral and pasting them as in Figure 1111 oneobtains the desired building block.Figure 11: The one holed triangleWe now analyse the geometry of this one holed triangle in more detail. We begin by fixing t to some small value; exactly which value we choose is of no importance but it will becrucial that it be sufficiently small to satisfy a certain property we shall exhibit in whatfollows. The fact that we can choose a uniform t independently of N will be important in20ur infinite genus surface we construct at the very end.We begin by looking, in the one holed triangle, at the distance a between any of the threevertices and the center hole. We’re interested in how this distance relates to the side length,which we denote (cid:96) . In particular, we claim that, provided t is small enough, then a > (cid:96) .Indeed, by looking the quadrilateral highlighted in Figure 1212, the values a , (cid:96) and t satisfythe following equality: sinh (cid:18) (cid:96) (cid:19) = sinh (cid:18) t (cid:19) cosh ( a ) . (cid:96) ac Figure 12: a , (cid:96) and c From this we can deduce that a = arccosh (cid:32) sinh ( (cid:96) ) sinh ( t ) (cid:33) Since N ≥
11, we can guarantee that (cid:96) has a certain length (indeed (cid:96) is certainly longer than (cid:96) N , the side length of the corresponding equilateral triangle with the same angles but no holes). Now for any t such that sinh (cid:0) t (cid:1) <
1, the quantityarccosh (cid:32) sinh ( (cid:96) ) sinh ( t ) (cid:33) − (cid:96) (cid:96) . Fix for instance t > (cid:18) t (cid:19) =
14 ,which garanties that a − (cid:96) > (cid:96) : indeed such a path must pass through thesegment marked c in Figure 1212. As such it is of length at least (cid:96) .21or fixed N , again using hyperbolic trigonometry, we can compute the length (cid:96) (cid:48) N of the sideof our one holed triangle: (cid:96) (cid:48) N = (cid:32) cosh ( t ) sin ( π N ) (cid:33) .We now return to the global construction. Again, we paste copies of the one holed triangleusing the blueprint provided by the embedding of K N + in M N + . We obtain a surface withboundary curves, one for each triangle, and all of length t . We shall complete the surfaceby gluing something on each of the boundary curves, but let us remark already that inwhatever fashion we do this, the properties of our building blocks imply that the resultingsurface has a geometric embedding of K N + with edge lengths (cid:96) (cid:48) N . Indeed, any simple pathbetween vertices that is not the side of a triangle can be decomposed into simple segments,each of which lies on a single one holed triangle. What we claim is that, if the path isoriented, the first and last segments are both of length greater than (cid:96) . Indeed it is either ageodesic path between the vertex and a side of the triangle - so is of length strictly greaterthan (cid:96) as shown above - or it is a path between the vertex and the hole, again of lengthstrictly greater than (cid:96) . We can conclude that the embedding is geometric.For given N , this construction gives us a surface of genus g N + with boundary curves withthe property that no matter how we complete it, the resulting surface has a geometricallyembedded copy of K N + for an appropriate edge length. We denote these surfaces F N .Let us describe what type of surfaces we can build by pasting this surface in different ways.We shall use it in two ways, the first of which is to construct closed surfaces of differentgenus to fill the gaps left in the earlier construction. Closing the gaps
The number of boundary curves of F N is exactly the number of one holed triangles used toconstruct it. We can compute this number T N using the Euler characteristic. We begin byobserving that T N must be even because the sides of the one holed triangles are pasted inpairs. There are N + N ( N + ) edges so N + − N ( N + ) + T N = − g N + .From this T N = − (cid:22) ( N − )( N − ) (cid:23) + N − N T N grows like N in function of N . The smallest genus closed surfacecontaining F N is obtained by pasting the boundary curves in pairs. As T N is even, this is22qual to g N + + T N so by the above formula is equal to N − N +
12 .By a simple topological argument, instead of constructing a minimal genus surface, we canconstruct a hyperbolic surface, containing F N , of any genus greater than this minimal genus.To do this, we just attach a hyperbolic surface with a single boundary curve of length t of the appropriate genus to one of the boundary curves and then complete the surface asabove.To synthesize the above construction, for any N + ≡ k ≥
0, thereexists a surface with a geometric embedding of K N + and genus N − N + + k .Now, for any g ≥
2, we want to construct a surface of genus g with a geometric embeddingof a K N + for N as large as possible. We choose the maximal N ≡ − g ≥ N − N + + k for some k ≥
0. Using the condition on the N we are allowed in our construction, we areguaranteed to find a suitable N satisfying g < ( N + ) − N + + ≤ ( N + ) N + ≥ (cid:112) g − Hyperbolic surfaces with infinite chromatic number
The building blocks F N can be used for another purpose - to construct a surface Z such thatlim sup d → ∞ χ ( Z , d ) = ∞ .We note that such a surface must necessarily be of infinite area and in fact the surfaces weshall describe are infinite genus as well. It is entirely possible that there be a much simplersurface with this property, namely H , but this is currently unknown.The only thing we require of our surface Z is that it contain copies of F N k for N k → ∞ as k → ∞ . This is easy to construct as the boundary curves of the F N all have the same length.One way to do this is to string together the sequence of F N k , for instance joining one ofthe boundary geodesics of F N k with F N k + for all k (each of them has at least two boundarycomponents so this is possible). We then paste together the remaining (infinite number) ofboundary curves in any way. As each surface F N k has chromatic number N k for some valueof d , this proves the result. 23 EFERENCES [1] Peter Buser,
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