Embedding of metric graphs on hyperbolic surfaces

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Mar 7, 2017 - The central questions in this paper are the following: Question 1.1. ..... angled hyperbolic triangle ACD (see Formula (IV) of Theorem 2.2.2 in [6]) we get sinhx = cot. (π ... Further, if ϵ = 0 then we call it as λ-quasi-geodesic.
EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

arXiv:1703.02359v1 [math.GT] 7 Mar 2017

BIDYUT SANKI Abstract. An embedding of a metric graph (G, d) into a closed hyperbolic surface is called essential if it is isometric and each complementary region has a negative Euler characteristic. Note that, an embedding would mean isometric. We show, by construction, that every metric graph can be essentially embedded(up to scaling) in a closed hyperbolic surface. The essential genus ge (G) of a metric graph (G, d) is defined to the lowest integer among the genera of those surfaces on which the metric graph can be essentially embedded. In the next result we establish a formula to compute ge (G). Furthermore, we show that for every integer g > ge (G), (G, d) can be essentially embedded up to scaling on a surface of genus g. Next, we study minimal embedding, where each complementary region has Euler characteristic −1. We define a parameter gemax (G) to be the maximal essential genus of a graph (G, d) in minimal embedding. We compute an upper bound for gemax (G) and prove that for any integer g, ge (G) 6 g 6 gemax (G), there exists a closed surface on which (G, d) can be minimally embedded. Finally, we describe a method explicitly for an essential embedding of (G, d) where ge (G) and gemax (G) are realized.

1. Introduction A 2-cell embedding of a graph G on the closed oriented (topological) surface Sg of genus g is a cellular decomposition of Sg whose 1-skeleton is isomorphic to G (see [8]). In topological graph theory, the characterization of the closed surfaces on which a graph can be 2-cell embedded is a well studied and famous problem (see [17]). In this direction, Kuratowski was the first who shown that a graph is planar if and only if it does not contain K3,3 or K5 as a minor, where K3,3 is the complete bipartite graph with (3, 3) vertices and K5 is the complete graph with 5 vertices and hence, K3,3 , K5 are the only minimal non-planar graphs. The genus of a graph G is defined by g(G) = min{g(F )} where the minimum is taken over all those surfaces on which G is 2-cell embedded. The maximum genus gM (G) is similarly defined [10]. In [8], Duke have shown that, for every integer k, g(G) 6 k 6 gM (G), G has a 2-cell embedding on the surface of genus k. A 2-cell embedding of G on a surface F is called minimal (maximal) if g(G) = g(F ) (gM (G) = g(F ) respectively). In [8] (see Theorem 3.1), Duke obtained a sufficient condition for the non-minimality of a 2-cell embedding and provided an algorithm to transform such a non-minimal embedding into a 2-cell embedding of the same graph in a surface with lower genus. The maximum genus problem has been studied by Xuong in [10]. Xuong proved the theorem stated below: Date: March 8, 2017. Key words and phrases. Quasi-geodesic, δ-hyperbolic space, hyperbolic corridor, fat graph, girth, Betti number, Betti deficiency. The author is initially supported by the Post Doctoral Fellowship funded by a J. C. Bose fellowship of Prof. Mahan Mj. 1

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Theorem 1 (Xuong, [10]). The genus of a maximal 2-cell embedding of a graph G is given by γM (G) = 12 (β(G) − ζ(G)), where β(G), ζ(G) are the Betti number and Betti deficiency of G respectively. Furthermore, the number of faces in a maximal 2-cell embedding is 1 + ζ(G). For more results on 2-cell embedding we refer [17], [14]. The configuration geodesics and graphs on the hyperbolic surfaces have become increasingly important in the study of mapping class groups of hyperbolic surfaces and the moduli spaces of Riemann surfaces through the systole function, filling pair length function, in particular. The study of filling system has its origin in work of Thurston, and more recently has been studied by Parlier, Aougab, Schmutz Schaller, Pettet, Anderson and others (see [12], [4], [3]). Parlier produces upper and lower bounds on the numbers of systoles(minimal length essential curves) on hyperbolic surfaces (see [12]). The set consisting of those hyperbolic surfaces of genus g whose systoles fill called the Thurston set and is denoted by χg (see [18]). Anderson, Parlier and Pettet studied the thick part of the moduli space to understand the shape of χg . In [2], the authors have shown that any trivalent graph with the combinatorial metric is quasi-isometric to a hyperbolic surface whose systoles fill, allows to give a lower bound on the Hausdorff distance between χg and the set Yg of trivalent surfaces in the moduli space Mg (Section 3 in [2]). There is a natural connection between graphs and the surfaces. In particular, the union of all systolic curves is a so called fat graph on the surface, well studied in [4], [12], [16]. In [4], Balacheff and Parlier study the geometry and topology of Riemann surface with a Riemann metric by embedding a suitable graph into the surface which captures some of its geometric and topological properties. In this paper, we are interested in graphs on hyperbolic surfaces. Recall that, a graph on a hyperbolic surface F is a pair of sets G = (V, E) where V is a finite set of points in F , called the set of vertices, and E is a set of geodesic arcs with ends at the vertices, called the set of edges. Further, the edges may meet only at mutually incident vertices. Such a graph has a metric where the distances between points on the graph are measured along the shortest path on the graph using the hyperbolic metric on the surface. We call G is essential on F if each component of F r G has negative Euler characteristic. The central questions in this paper are the following: Question 1.1. Given a metric graph (G, d), where G is a graph and d is a metric on the set of undirected edges of G. (1) Does there exist a closed hyperbolic surface on which (G, d) can be essentially embedded? (2) Characterize the closed oriented hyperbolic surfaces on which (G, d) can be essentially embedded. (3) What is the lowest genus of such surfaces? An embedding of a metric graph would mean isometric, i.e., an injective map Φ : (G, d) → F which preserves the lengths of the edges. Definition 1.1 (Essential embedding). An isometric embedding Φ : (G, d) → F is called essential if each component of F r Φ(G) have negative Euler characteristic.

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Scaling of metric. Given a metric graph (G, d) and a positive real number t, define dt : E → R>0 by dt (e) = td(e) for all e ∈ E. Then dt is said to be the metric obtained from d scaling by t. Perhaps, the more natural question is to ask all these up to scaling. An obstruction in the un-scaled case is the Margulis lemma (see Corollary 13.7 in [9]). Therefore, the general question is the following: Given a metric graph (G, d), does there exist a t ∈ R>0 such that (G, dt ) can be embedded essentially on a closed hyperbolic surface? From now and on, by an embedding we mean essential embedding upto scaling. The first major result, we obtain is stated below: Theorem 1.1 (B). Given a metric graph (G, d) with degree of each vertex at least three, there exists a closed hyperbolic surface S such that G can be embedded isometrically (up to scaling) on S. Furthermore, the embedding is essential. Idea of proof. There are five steps in the proof of Theorem 1.1. Step 1: We consider a fat graph structure on the given graph. Step 2: Corresponding to each vertex we construct a hyperbolic regular convex polygon and corresponding to each edge of the graph we construct a hyperbolic corridor. Step 3: We glue the sides of the polygons and the geodesic sides of the corridors according to the fat graph structure on the graph using hyperbolic isometry and obtain the hyperbolic surface with boundary (not geodesics). Step 4: We cut the surface, obtained in Step 3, along the unique geodesic representative in the free homotopy classes of boundary curves and obtain the hyperbolic surface with totally geodesic boundary. Step 5: Finally, we cap the surface, obtained in Step 4, to construct the closed hyperbolic surface. Notation. S(G, d) denotes the set of all hyperbolic surfaces that satisfy Theorem 1.1 for a given metric graph (G, d). Now, we focus on the genera of those hyperbolic surfaces on which a metric graph can be essentially embedded. Definition 1.2 (Essential genus). The essential genus of a metric graph (G, d), denoted by ge (G), is defined by ge (G) = min{g(X)| X ∈ S(G)} where g(X) denotes the genus of X. We prove the theorem stated below which computes the essential genus of a metric graph. Theorem 1.2 (B). The essential genus of a metric graph (G, d) is given by: 1 ge (G) = (β(G) − ζ(G)) + 2q + r 2 where β(G), ζ(G) are the Betti number and Betti deficiency of G respectively and q, r are the unique integers satisfying ζ(G) + 1 = 3q + r, 0 6 r < 3. The immediate consequence of Theorem 1.2 is the following: Corollary 1.3 (B). Let (G, d) be a metric graph with essential genus ge (G). Then for any given g > ge (G), the graph can be essentially embedded on a closed hyperbolic surface of genus g. An essential embedding of (G, d) on a hyperbolic surface S is simplest if each complementary region have least negative Euler characteristic, i.e., −1 and hence, we define the following:

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Definition 1.3 (Minimal embedding). An essential embedding Φ : G → F of a metric graph (G, d) on a hyperbolic surface F is called minimal if for each component Σ of the complement F r Φ(G), we have χ(Σ) = −1. The essential genus of a graph is realized by a minimal embedding. The set of closed hyperbolic surfaces on those (G, d) can be minimally embedded in denoted by Sm (G, d). It is straightforward to see that the genera of the surfaces in Sm (G, d) are bounded below by ge (G) and this bound is sharp. The theorem stated below provides an upper bound of the genera of the surfaces in Sm (G, d). We define gemax (G) = max{g(X)| X ∈ Sm (G, d)}. Theorem 1.4 (B). For a given metric graph (G, d) we have   1 2|E| max ge (G) 6 β(G) + 1 + , 2 T (G) where |E| and T (G) are the number of edges and the girth of G respectively. Furthermore, given any integer g satisfying ge (G) 6 g 6 gemax (G), there exists a closed hyperbolic surface of genus g on which (G, d) can be minimally embedded. Now we focus on explicit constructions of minimal (and maximal) embedding metric graphs because these are preferred over random constructions. To embed a graph on a surface minimally(or maximally), the crucial part is to find a suitable fat graph structure which gives the minimum(or maximum) number of boundary components among the all fat graph structures on the graph. For a fat graph structure σ0 on G, the number of boundary components in (G, σ0 ) is denoted by #∂(G, σ0 ). The questions about the fat graph structures we focus on are stated below: Question 1.2. Let σ0 =

Q

σv be a fat graph structure on a graph G.

v∈V

(1) What is the necessary and sufficient condition on σ0 such that the minimum number of boundary component of G realized by the fat graph σ0 ? (2) If the number of boundary component in (G, σ0 ) is not minimum among all the fat graph structures on G, how can we find a new fat graph structure σ00 such that #∂(G, σ 0 ) 6 #∂(G, σ)? We can ask the similar question in the context of maximality. The following theorem answers the above questions. Theorem 1.5. Let G = (E, ∼, σ1 ) be any graph with degree of each vertex at least three. Suppose, σ0 = {σv |v ∈ V } be a fat graph structure on G such that there is a vertex v which is common in m(> 3) boundary components. Then there is a rotation σv0 such that in the new fat graph structure σ00 obtained by replacing σv by σv0 in σ0 satisfy #∂(G, σ 0 ) = #∂(G, σ) − 2.

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Given a metric graph (G, d), we chose a fat graph structure σ0 on G. Then applying Theorem 1.5 repeatedly to the fat graph, after finitely many steps we can obtain a fat graph structure on G which provide the realization of a minimal embedding. Finally, we remark that applying Theorem 1.5 in the reverse direction to increase the number of boundary components we can obtain a fat graph structure on G for the realization of gemax (G). Organization of the paper. This paper is organized as follows. In Section 2, we recall some definition and well known results on fat graphs, quasi-geodesics, hyperbolic polygons and corridors. Also, we prove a lemma on hyperbolic polygon which will be used in the subsequent section. In Section 3, we study essential embedding of metric graphs and prove Theorem 1.1. In Section 4, we focus on minimum genus problem and prove Theorem 1.2. We conclude this section with the proof of Corollary 1.3. Next, we study minimal essential embedding and prove Theorem 1.4 in Section 5. Finally, in Section 6, we prove Theorem 1.5. We conclude this section with Remarks 6.2 which provides an algorithm for minimal embedding with minimum and maximum genus. 2. Preliminaries In this section, we recall some geometric, graph theoretic notions and some results which are essential is the subsequent sections. 2.1. Fat graph. Before going to the formal definition of fat graph we would like to recall the definition of graph and a few graph parameters. The definition of graph we use here is not the standard one which is used in ordinary graph theory. It is straightforward to see that this definition is equivalent to the standard definition. Definition 2.1. A graph is a triple G = (E1 , ∼, σ1 ), where (1) E1 is a finite non-empty set with even number of elements. (2) σ1 is a fixed-point free involution on E1 . (3) ∼ is an equivalence relation on E1 . In ordinary language, the quotient V = E1 / ∼ is the set of vertices. For ~e ∈ E1 we say that ~e is emanating from the vertex [~e], the equivalence class of ~e. If ~e1 ∼ ~e2 then we say that they have the same initial vertex. The degree of a vertex v ∈ V is defined by deg(v) = |v|. The set E1 is the set of directed edges. The involution σ1 maps a directed edge ~e to its reverse edge e~, i.e., σ1 (~e) = e~. The elements of the quotient E = E1 /σ1 are called undirected edges. We recall the following graph parameters: Definition 2.2 (Girth). The girth T (G) of a graph G is the length of a shortest non-trivial simple cycle where the length of a cycle is the number of edges it contains. Definition 2.3 (Betti number). Given a graph G with |V | vertices and |E| edges. Then the Betti number of the graph G, denoted by β(G), is defined by β(G) = −|V | + |E| + 1. Definition 2.4 (Betti deficiency). The Betti deficiency of G, denoted by ζ(G), is defined by (2.1)

ζ(G) = min{ζ(C)| C is a co-tree of G}

where ζ(C) is the number of components with odd number of edges in a co-tree C of G.

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Now we are ready to define fat graph. Informally, a fat graph is a graph equipped with a cyclic order on the set of directed edges emanating from each vertex. If the degree of a vertex is less than three then the cyclic order is trivial. So, we consider the graphs with degree of each vertex at least three. Definition 2.5. A fat graph is a quadruple G = (E1 , ∼, σ1 , σ0 ), where (E1 , ∼, σ1 ) is a graph and σ0 is a permutation on E1 such that each cycle of σ0 is a cyclic order on some v ∈ V . For a vertex v with degree k, σv = (ev,1 , ev,2 , . . . , ev,k ) represents a cyclic order on v. A fat graph structure on a graph we denote as a set σ0 = {σv |v ∈ V } of cyclic orders on the Q vertices or equivalently σ0 = σv . Given a fat graph G = (E1 , ∼, σ1 , σ0 ), we can construct a v∈V

oriented topological topological surface by thickening the edges, denote by Σ(G). The number of boundary components in Σ(G) is the number of orbits of σ1 ∗ σ0−1 (see [15], Section 2.1). For more detail on fat graphs we refer [15], [16], [13]. 2.2. Hyperbolic Polygon. Let P (k, l) denote the regular k-sided hyperbolic convex polygon with the length of each side l, where k(> 3) ∈ Z and l ∈ R>0 . For the existence of such a polygon we refer Proposition 5.18 in [1]. We prove the following lemma, computes the length of the perpendicular from the center to a side of P (k, l), will be used in the proof of Theorem 1.1. Lemma 2.1. The length x of the perpendicular geodesic from the center to a side of P (k, l) is given by      π l (2.2) x = arcsinh cot tanh . k 2 Proof. Let C denote the center of P (k, l). The perpendicular from C to a side AB meets at the mid point D (see Figure 1). We are interested to compute the length x of the geodesic arc C

x

B

D

l 2

A

Figure 1. CD. Consider the hyperbolic triangle ACD. The interior angles at D, C are π2 , πk respectively. The length of the side AD is 2l . Therefore, applying hyperbolic trigonometry on the right angled hyperbolic triangle ACD (see Formula (IV) of Theorem 2.2.2 in [6]) we get   π  l sinh x = cot tanh k 2 which gives equation (2.2).



It follows from Lemma 2.1 that  π   l = 2 arctanh tan sinh x . k

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2.3. Quasi-geodesic. Now, we recall some definitions and facts on quasi-geodesic. A path γ : I → X in a geodesic metric space (X, d) is called a (λ, )-quasi-geodesic, where λ > 1,  > 0, if 1 l(γ|[t1 ,t2 ] ) −  6 d(γ(t1 ), γ(t2 )). λ Further, if  = 0 then we call it as λ-quasi-geodesic. If the restriction of γ on any sub-interval of length at most L is a (λ, )-quasi-geodesic then it is called L-locally (λ, )-quasi-geodesic. Now we recall two theorems which are essential in the subsequent sections. The first theorem says that local quasi-geodesics are global quasi-geodesics and second theorem says that a quasi geodesic is close to a geodesic. Theorem 2 (J. W. Cannon [7], Theorem 4). Given λ > 1,  > 0, there exists a L large enough and λ0 > 1, 0 > 0 having the following property. If γ is a path in H such that each subpath of γ of length 6 L is a (λ, )-quasi-geodesic then γ is a (λ0 , 0 )-quasi-geodesic. Theorem 3 ([5], Chapter III.H, Theorem 1.7). For all δ > 0,  > 1,  > 0 there exists a constant R = R(δ, λ, ) with the following property: If X is a δ-hyperbolic space, γ is a (λ, )-quasi-geodesic in X and [p, q] is a geodesic segment joining the end points of γ then the Hausdorff distance between [p, q] and the image of γ less than R. 2.4. Hyperbolic corridor. Let γ be a geodesic in the hyperbolic plane H and ργ denote the orthogonal projection of H onto γ. For I ⊂ γ and w > 0, the w-corridor about I along γ is defined by w(γ, I) = {z ∈ H| ργ (z) ∈ I, dH (z, γ) 6 w/2} . Now, for two preassigned positive real numbers L and w, we define corridor C(L, w) by the following: Let γ be the positive imaginary axis and I be the geodesic arc joining the points i and ieL in H. Then C(L, w) is the copy of w(γ, I) (see Figure 2). The geodesic I is called the central geodesic of C(L, w) which has length L. The corridor C(L, w) has two geodesic sides each of length w, called the width of C(L, w).

ieL

c

d b

i

a

Figure 2. The corridor C(L, w) 3. Essential embedding of metric graph In this section, we prove Theorem 1.1. Suppose G = (E1 , ∼, σ1 ) is a graph equipped with a metric d and v ∈ V is a vertex with degree 2. Then there are two directed edges ~e1 , ~e2 ∈ E1 such that v = {~e1 , ~e2 }. Let ei = {~ei , σ1 (~ei )} be the undirected edges corresponding to ~ei where i = 1, 2. We obtain the new metric graph G0 = (E10 , ∼0 , σ10 ) from G, with metric d0 , by

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removing the vertex v and replacing two edges e1 , e2 by a single edge e (see Figure 3). The graph G0 = (E10 , ∼0 , σ10 ) is described below: (1) E10 = (E1 r {~ei , e~i |i = 1, 2}) ∪ {~e, e~}. (2) ∼0 is uniquely determined by the partition V 0 of E 0 given by V 0 = (V r {v, [σ1 (~e1 )], [σ1 (~e2 )]}) ∪ {v10 , v20 }, where v10 = ([σ1 (~e1 )] r {σ1 (~e1 )}) ∪ {~e} and v20 = ([σ1 (~e2 )] r {σ1 (~e2 )}) ∪ {e~}. (3) σ10 is described by σ10 (x) = σ(x) for x ∈ E10 r {~e, e~} and σ10 (~e) = e~. v1

e1

e2 v

v2

v1

e1

e2 v e

v2

Figure 3. Replacement of two edges by a single edge and removal of a vertex. The set of undirected edges of the graph G0 is E 0 = (E r {e1 , e2 }) ∪ {e} and the metric d0 : E10 → R+ is defined by: d0 (x) = d(x), for all x ∈ E 0 r {e} and d1 (e) = d(e1 ) + d(e2 ). Definition 3.1 (Geometric graph). A metric graph is called geometric if it has an essential embedding, up to scaling, into a closed hyperbolic surface. Lemma 3.1. The graph (G, d) is geometric if and only if (G0 , d0 ) is geometric. Moreover, the essential genus of the graphs are the same, i.e., ge (G) = ge (G0 ). In light of Lemma 3.1, from now and on we assume that the degree of each vertex of the graph G is at least three, i.e., deg(v) > 3, for all v ∈ V. For a given metric graph (G, d), G = (E, ∼, σ1 ), we define   π (3.1) L = min{d(e)| e ∈ E} and θ0 = min v∈V deg(v) where V = G/∼ is the set of all vertices and E = E1 /σ1 is the set of undirected edges of G. Our goal is to show that there exists a t > 0 such that (G, dt ) can be essentially embedded on a closed oriented hyperbolic surface. In the first step, we consider a fat graph structure σ0 = {σv | v ∈ V } on the graph G. Before going to the further steps of the construction, we would like to define few objects and develop some lemmas which are useful for the construction. Let us consider a polygonal path γ in the hyperbolic plane H such that the interior angles at the vertices are bounded below and above by θ0 (> 0) and π3 respectively. Moreover, we assume that the lengths of the sides of γ are bounded below by tL where t is some arbitrary but fixed positive constant and L is given in equation (3.1). Then γ is a tL-locally λ(θ0 )quasi-geodesic where λ(θ0 ) = sin1θ0 + tan1 θ0 + 1. Further, there exists a t0 such that for all t > t0 , we have some λ > 1 and  > 0 such that γ is a (λ, )-quasi-geodesic (see Theorem 2).

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Let Γ be the geodesic in H connecting the end points of γ. Then by Theorem 3, there exists a positive real number l0 such that γ ⊂ N bd(Γ, l0 /2) where N bd(Γ, l0 /2) = {z ∈ H| dH (z, Γ) 6 l0 /2}.

(3.2)

For such a l0 , motivated by Lemma 2.1, for each vertex v of the graph we define (3.3)

xv = arcsinh [cot (π/ deg(v)) tanh (l0 /2)] .

Further, we define, x = max{xv | v ∈ V }.

(3.4)

We prove the lemma below, provides us a upper bound of x. Lemma 3.2. Let ABC be an ideal right angled hyperbolic triangle. If the interior angles at A, B, C are π2 , 0, θ respectively, then the length of the side AC is given by dH (A, C) = log (cot(θ/2)) . Proof. Let us consider the half circle in H, centred at 0 and radius 1, and the vertical line L on the right side of the positive imaginary axis which intersect the half circle at an angle θ, denote the intersection point by C. Then the coordinate of C is (cos θ, sin θ) and hence, L intersect the real axis at (cos θ, 0). We draw the half circle in H with centred at (cos θ, 0) and radius (1 − cos θ). Then the triangle ABC (see Figure 4) is as in Lemma. Thus we have

C = (cos θ, sin θ) A θ

i sin θ i(1 − cos θ)

θ cos θ

B

Figure 4. ∆ABC   sin θ dH (A, C) = dH (i(1 − cos θ), i sin θ) = log = log (cot(θ/2)) . 1 − cos θ  Corollary 3.3. We have the bound on x below   θ0 0 < x < log cot 2 where θ0 is given in equation (3.1). We choose t(> t0 ) sufficiently large such that    tL π (3.5) 0 < xv < min , log cot 2 2 deg(v) for all vertex v, where xv is given in equation (3.1). For such a t and each edge e with endpoints on the vertices u, v we define (3.6)

Lte = dt (e) − (xu + xv ).

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Remark 3.4. By the definitions it follows that Lte > 0. Now we are ready to describe the construction. 3.1. Construction of (labelled) Polygons. Corresponding to each vertex v we construct a labelled regular convex hyperbolic polygon, described by following: Let v = {ev,1 , ev,2 , . . . , ev,k } be a vertex, where k = deg(v), with the cyclic order σv on v given by (ev,1 , ev,2 , . . . , ev,k ). We construct the regular hyperbolic deg(v)-sided convex polygon with each side of length l0 (see Equation (3.2) for the definition of l0 ), denoted by Pv (l0 ). Next, we label the sides of the polygon by ev,1 , ev,2 , . . . , ev,k counted anti-clockwise starting at an arbitrary edge. The perpendicular geodesic from the centre cv of Pv (l0 ) to the side ev,j , we denote by δv,j ; j = 1, 2, . . . , deg(v). Therefore, by construction the length of δv,j is xv , defined in equation (3.3). 3.2. Construction of (labelled) Corridors. For each edge e and the constant t(> t0 ), we construct labelled hyperbolic corridor, denoted by Ce (t), which is the identical copy of C(Lte , l0 ) (see Subsection 2.4) and denote the central geodesic by γe which has length Lte . Suppose the edge e is incident on the vertices u and v. Then we have e = eu,i and e = ev,j for some eu,i ∈ u and ev,j ∈ v. We label the geodesic sides of the corridor Ce (t) by eu,i and ev,j . 3.3. Embedding on surface with boundary. Now, we put the things together. We glue the sides of the polygons with the geodesic sides of the corridors by hyperbolic isometry to obtain a hyperbolic surface with boundary. If ev,j is a geodesic side of some corridor Ce (t) then there is a side of Pv (l0 ) which has the same label. Moreover, they have the same hyperbolic length l0 . Therefore, we glue them by a hyperbolic isometry, so that the end point of the central geodesic γe and the foot of the perpendicular δv,j are identified. In this way we obtain the hyperbolic surface denoted by Σ1 (G, dt , σ0 ). If e is an edge of the graph with incident vertices u, v then by the construction, the geodesic arc segments δu,i , γe , δv,j project to the geodesic arc joining cu and cv , the centres of the polygons Pu (l0 ) and Pv (l0 ) respectively, which is denoted by e˜. The length of e˜ is Lte + xu + xv = dt (e). We define the map Φ

:

G → Σ1 (G, dt , σ0 ) by

Φ(v) = cv for all v ∈ V and Φ(e) = e˜ for all e ∈ E. The image of Φ in Σ1 (G, dt , σ0 ) is the spine of the surface. By construction, we arrive at the lemma stated below: Lemma 3.5. The metric graph (G, dt ) is isometrically embedded on Σ1 (G, dt , σ0 ). 3.4. Embedding on surface with totally geodesic boundary. In this subsection, We cut the surface Σ1 (G, dt , σ0 ) along the simple closed geodesics in the free homotopy classes of the boundary components to obtain the surface with totally geodesic boundary. Let β 0 be a boundary component of Σ1 (G, dt , σ0 ). Then there is a polygonal simple closed curve βs0 in the spine Φ(G) ⊂ Σ1 (G, dt , σ0 ) which is freely homotopic to β 0 . Consider a lift β˜s0 of βs0 on the universal cover H. The angle at each corner point of β˜s0 is bounded below by θ0 and above by π3 . Moreover, the geodesic segments β˜s0 are bounded below by tL. Therefore, we

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have β˜s0 is a tL-local λ(θ0 )-quasi-geodesic, where λ(θ0 ) = sin1θ0 + tan1 θ0 + 1 (see Lemma 6.4 ˜ Then β˜ projects to the in [16]). The geodesic joining the end points of β˜s0 we denote by β. simple closed geodesic, denoted by β, which is unique simple closed geodesic representative in the free homotopy class of β 0 . Furthermore, the choice of l0 and the acute interior angle at each corner ensure that β lies in the topological cylinder enclosed by β 0 and βs0 . Therefore, by cutting the surface Σ1 (G, dt , σ0 ) along these simple closed geodesic representatives in the free homotopy classes of boundary components we obtain the hyperbolic surface, denoted by Σ0 (G, dt , σ0 ), hyperbolic surface with totally geodesic boundary. 3.5. Embedding on closed hyperbolic surface. In this subsection, we cap the surface Σ0 (G, dt , σ0 ) by hyperbolic surfaces with boundary to obtain closed surface, equivalently we embed Σ0 (G, dt , σ0 ) isometrically and essentially on a closed hyperbolic surface and thus we complete the construction. First, we describe two gluing procedures: 3.5.1. Glue I. In this gluing procedure we assume that Σ0 (G, dt , σ0 ) has at least three boundary components. Assume that β1 , β2 , β3 are three boundary components. Then we take the hyperbolic pair of pants, denoted by Y , with boundary geodesics β10 , β20 , β30 of lengths l(β1 ), l(β2 ), l(β3 ) respectively. Then we glue βi with βi0 , i = 1, 2, 3 by hyperbolic isometries and with arbitrary twist parameters. In this gluing the resultant surface have genus two more than the genus of Σ0 (G, dt , P ) and number of boundary components three less than that of Σ0 (G, dt , P ). 3.5.2. Glue II. Assume that β is a geodesic boundary of the surface Σ0 (G, dt , σ0 ). We consider a hyperbolic surface, denoted by S1,1 , of genus one and a single boundary component β 0 with length l(β). Then we identify β with β 0 by a hyperbolic isometry. The resultant surface will have genus one more than that of Σ0 (G, dt , σ0 ) and boundary components one less than that of Σ0 (G, dt , σ0 ). Now, assume that Σ0 (G, dt , σ0 ) has b geodesic boundary components. Then by division algorithm there are unique integers q, r such that b = 3q + r where 0 6 r 6 2. Then following the gluing procedure Glue I (see Subsection 3.5.1) for q times and Glue II (see Subsection 3.5.2) for r times we obtain the desired closed hyperbolic surface denoted by S(G, dt , σ0 ). This completes the proof of Theorem 1.1. Remark 3.6. The genus of S(G, dt , σ0 ) depends on the fat graph structure σ0 . 4. Minimum Genus problem Let (G, d) be a metric graph with degree of each vertex at least three and χ(G) denote the Euler characteristic of G. We consider a fat graph structure σ0 = {σv | v ∈ V } on G and Σ0 (G, dt , σ0 ), the hyperbolic surface with geodesic boundary obtained in Subsection 3.4. As G is a spine of Σ0 (G, dt , σ0 ), we have (4.1)

χ(Σ0 (G, dt , σ0 )) = χ(G),

where χ(Σ0 (G, dt , σ0 )) denotes the Euler characteristic of Σ0 (G, dt , σ0 ). Lemma 4.1. χ(Σ0 (G, dt , σ0 )) < 0.

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Proof. Because of equation (4.1), it suffices to show that χ(G) < 0. Each edge e ∈ E contributes degree two to the vertices. Hence total degree of the graph is 2|E|. Therefore, P 2|E| = deg(v) > 3|V | and hence v∈V

χ(Σ2 (Σ0 (G, dt , σ0 ))) = |V | − |E| 6

−|E| < 0. 3 

Lemma 4.2. Let σ0 and σ00 be two fat graph structures on (G, d). Then the difference between the number of boundary components of Σ0 (G, dt , σ0 ) and Σ0 (G, dt , σ00 ) is an even integer, i.e., #∂(Σ0 (G, dt , σ0 )) − #∂(Σ0 (G, dt , σ00 )) is divisible by 2. For a surface F , the number of boundary components and genus of F are denoted as #∂F and g(F ) respectively. Further, for a fat graph (G, σ0 ), by the genus of the fat graph we mean the genus of the associated surface and is denoted by g(G, σ0 ). Similarly, we define the number of boundary components of a fat graph and is denoted by ∂(G, σ0 ). Proposition 4.3. Let σ0 and σ00 be two fat graph structures on a metric graph (G, d) such that #∂(Σ0 (G, dt , σ0 )) − #∂(Σ0 (G, dt , σ00 )) = 2. Then we have g(S(G, dt , σ00 )) 6 g(S(G, dt , σ0 )). Proof. Suppose the genus and the number of boundary components of Σ0 (G, dt , σ0 ) are g and b respectively. Then by Euler’s formula and equation (4.1) we have 2 − 2g − b = χ(G) which implies that b = 2 − 2g − χ(G). For the integer b, by division algorithm, there exist unique integers q and r such that b = 3q + r, where 0 6 r < 3. Therefore, by the construction we have the genus of S(G, dt , σ0 ) given by g(S(G, dt , σ0 )) = g + 2q + r. Let us assume that the genus and number of boundary components of Σ0 (G, dt , σ00 ) are g 0 and b0 respectively. Then the Euler’s formula and equation (4.1) give b0 = 2 − 2g 0 − χ(G). The condition b0 = b − 2 of the proposition implies g 0 = g + 1. Now, we compute the genus of closed surface S(G, dt , σ00 ). There are three cases to be consider as b0 = b − 2 = 3q + r − 2 with r ∈ {0, 1, 2}. Case 1. r = 0. In this case b0 = 3(q−1)+1. Thus the genus of S(G, dt , σ00 ) is g 0 +2(q−1)+1 = g+2q which is equal to the genus of S(Σ, dt , σ0 ). Therefore, the proposition holds with equality. Case 2. r = 1. In this case b0 = 3(q − 1) + 2. Therefore, genus of S(G, d, σ00 ) is g + 1 + 2(q − 1) + 2 = g + 2q + 1 which is equal to the genus of S(Σ, dt , σ0 ). Therefore, the proposition holds with equality. Case 3. The remaining possibility is r = 2. In this case the genus of S(G, dt , σ0 ) is g + 2q + 2. Now, b0 = b − 2 = 3q implies that the genus of S 0 (G, dt ) is g 0 + 2q = g + 1 + 2q. Therefore, we have g(S(G, dt , σ00 )) = g(S(G, dt , σ0 )) − 1 < g(S(G, dt , σ0 )). 

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Proof of Theorem 1.2. To find the essential genus of (G, d), we consider a fat graph structure σ0 on G which gives maximum genus of Σ0 (G, dt , σ0 ), equivalently minimum number of boundary components (see Proposition 4.3). For such a fat graph structure σ0 , the genus of Σ0 (G, dt , σ0 ) is 21 (β(G) − ζ(G)) which follows from Theorem 1. Moreover, the number of boundary component of the fat graph (G, σ0 ) is 1 + ζ(G) which is equal to the number of boundary component of Σ0 (G, dt , σ0 ). By division algorithm, for the integer 1 + ζ(G), there are unique integers q, r such that 1 + ζ(G) = 3q + r, where 0 6 r < 3. Therefore, the genus of S(G, dt , σ0 ) is ge (G) = 21 (β(G) − ζ(G)) + 2q + r.



Corollary 4.4. For given any metric graph (G, d) and any integer g > ge (G) there exists a hyperbolic surface of genus g on which (G, d) can be essentially embedded. Proof. Let us consider a fat graph structure σ0 on G such that g(S(G, dt , σ0 )) = ge (G) where t is a suitably chosen sufficiently large positive real number. We define g 0 = g − ge (G). Now, there are two possibilities. Case 1. If the number of boundary components of Σ0 (G, dt , σ0 ) is divisible by 3 then we have a Y -piece, denoted by Y (β10 , β20 , β30 ), attached to Σ0 (G, dt , σ0 ) along the boundary components β1 , β2 , β3 by hyperbolic isometries. We replace this Y -piece from S(G, dt , σ0 ) by a hyperbolic surface Fg0 ,3 of genus g 0 and three boundary components, again denoted by β10 , β20 , β30 , of lengths l(β1 ), (β2 ) and l(β3 ) respectively and denote the new surface by Sg0 (G, dt , σ0 ). Case 2. In this case we consider the number of boundary components of Σ0 (G, dt , σ0 ) is not divisible by 3. Then there is a subsurface F1,1 with genus 1 and a single boundary component β 0 which we attached to Σ0 (G, dt , σ0 ) along the boundary component β to obtain S(G, dt , σ0 ). Now, we replace F1,1 by Fg0 +1,1 , a hyperbolic surface of genus g 0 + 1 and a single boundary component β 0 of length l(β), in S(G, dt , σ0 ) and the thus obtained new surface denote by Sg0 (G, dt , σ0 ). Then the closed hyperbolic surface Sg0 (G, dt , σ0 ) has genus g on which (G, dt ) isometrically embedded. 

5. Minimal embedding Proof of Theorem 1.4. Let us consider the given metric graph (G, d) with degree of each vertex at least three. Let |E| be the number of edges and T (G) be the girth of the graph G. Consider an arbitrary fat graph structure σ0 on G. Assume that b is the number of boundary components and f1 , . . . , fb are the boundary components of Σ0 (G, dt , σ0 ). Then we define l(fi ) to be the number of edges in the cycle of the spine of Σ0 (G, dt , σ0 ) freely homotopic to the boundary

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component fi . Then we have l(fi ) > T (G) for all boundary component fi . We have b X i=1

l(fi ) >

b X

T (G) = bT (G)

i=1

⇒ bT (G) 6 2|E| ⇒ f 6

2|E| . T (G)

We consider a fat graph structure which realized the genus g(G) of the graph. Recall that, the genus of a graph is defined by g(G) := min{g(G, σ0 )| σ0 is a fat graph structure of G}. Then by Euler formula we have, |V | − |E| + b = 2 − 2g(G) b + 2g(G) = 2 − |V | − |E| = 1 + β(G) 1 ⇒ g(G) + b 6 (β(G) + 1 + b) 2   1 2|E| max ge (G) = γ(G) + b 6 β(G) + 1 + . 2 T (G)  6. Algorithm: Minimal embedding with minimum/maximum genus Let us consider a trivalent fat graph (Γ, σ0 ) where σ0 = {σv |v ∈ V (Γ)}. Suppose it has a vertex v which is shared by three distinct boundary components. Then we construct a new fat graph structure by changing σv to reduce the number of boundary components by two. Lemma 6.1. Let (Γ, σ0 ) be a three regular fat graph. Suppose it has a vertex which is common for three boundary components, then one can get a new fat graph to reduce the number of boundary components of (Γ, σ0 ) by changing the ordering on the edges incident at the vertex. Proof. Let v be a vertex in Γ which is in three distinct boundary components. Assume that ei , i = 1, 2, 3 are the edges incident on v (see Figure 5, left) and ∂i ’s are the boundary components given by (see Figure 5, right) ∂1 = ~e1 ∗ P1 ∗ e~3 , ∂2 = ~e2 ∗ P2 ∗ e~1 , ∂3 = ~e3 ∗ P3 ∗ e~2 where Pi ’s are finite(possibly empty) paths. We replace the order σv = (e2 , e3 , e1 ) by σv0 = (e2 , e1 , e3 ) and we obtain the new fat graph structure σ00 . Then the boundary components of Γ0 = (E, ∼, σ1 , σ00 ) given by ∂Γ0 = (∂Γ r {∂i |i = 1, 2, 3}) ∪ {∂} where ∂ = ~e2 ∗ P2 ∗ e~1 ∗ ~e3 ∗ P3 ∗ e~2 ∗ ~e1 ∗ P1 ∗ e~3 = ∂2 ∗ ∂3 ∗ ∂1 . Therefore, the number of boundary components in Γ0 is the same as the number of boundary components in Γ minus two. 

EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

P2 e1 e2

e3

(e1 , e2 , e3 )

e~1

~e1

∂2

∂1

~e2

P2 e~1 ~e1

P1

e1

e~3 e2

∂3

e~2

~e3 P3

P1 e~3

~e2

e3 (e2 , e1 , e3 )

15

e~2

e~3 P3

Figure 5. Change of order Proof of Theorem 1.5. Let us consider v0 = {~e1 , ~e2 , . . . , ~ek } be a node which is in the three boundary components b1 , b2 , b3 . We assume that the rotation at v0 is given by σv0 = {~e1 , ~e2 , ~e3 , . . . , ~ei , ~ei+1 , . . . , ~ek } where k > 3, 3 6 i 6 k and ~ek+1 = ~e1 . Furthermore, we assume that ~e2 is an edge of b1 and e~2 is an edge of b2 . We write b1 = ~e2 P1 e~1 , b2 = ~e3 P2 e~2 , b3 = ~ei+1 P3 e~i where Pj ’s are some paths in the fat graph. Now we consider σv0 = (~e2 , ~e3 , . . . , ~ei , ~e3 , ~ei+1 , . . . , ~ek ). Then in the new fat graph structure σ00 the boundary components of (G, σ00 ) are (∂(G, σ0 ) r {b1 , b2 , b3 }) ∪ {b} where b = b1 ∗ b2 ∗ b3 . Therefore we have #∂(G, σ00 ) = #∂(G, σ0 ) − 2.  Remarks 6.2. (1) let us consider a fat graph (G, σ0 ). If v is a vertex shared by k(6 2) boundary components then it follows from Proposition 4.3 and Theorem 1.5 that there is no replacement of the cyclic order σv (Keeping the cyclic order on other vertices unchanged) to reduced the number of boundary components. Therefore, by repeated application of Theorem 1.5, we obtain a fat graph structure on the graph G that provides the essential genus ge (G) (2) Using the Theorem 1.5 in the reverse way, we can obtain a fat graph structure which provides the maximal genus of a minimal embedding, i.e., gemax (G). Namely, if there is a vertex v with cyclic order σv = (~e2 , ~e3 . . . , ~ei , ~e1 , ~ei+1 , . . . ~ek ) and a boundary component ∂ of the form ∂ = ~e2 P1 e~1 e~3 P2 e~2 ei+1 P3 e~i , where Pj ’s are some paths in G, then only, we can replace σ0 by σv0 = (~e1 , ~e2 , ~e3 , . . . , ~ei , ~ei+1 , . . . , ~ek ) to obtain a new fat graph structure σ00 such that #∂(G, σ00 ) = #∂(G, σ0 ) + 2. By repeated use of Theorem 1.5, we can obtain a fat graph structure on G which provides gemax (G).

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References [1] Anderson, J. W., Hyperbolic Geometry , Springer undergraduate mathematics series. [2] Anderson, J. W., Parlier, H., Pettet, A.; Relative shape of thick subset of moduli space, Amer. J. Math. 138 (2016), no. 2, 473-498. [3] Aougab, T., Huanh, S. Minimally intersecting filling pairs on surfaces., Algebr. Geom. Topol. 15(2015), no 2, 903-932. [4] Balacheff, F., Parlier, H.; Short loop decomposition of surfaces and the geometry of Jacobians., Geom. Funct. Anal. 22 (2012), no. 1, 37-73. [5] Bridson, Martin R., Haefliger, Andre., Metric Spaces of Non-Positive Curvature, Berlin, Springer, 1999., Comprehensive Studies in Mathematics, 319. [6] Buser, P., Geometry and Spectra of compact Riemann Surfaces, Springer-Verlag, New York, 1983. [7] Cannon J. W., The combinatorial structure of co-compact discrete hyperbolic groups. Geometriae Dedicata. 16 (1984), no. 2, 123-148. [8] Duke, R. A., The genus, regional number, and Betti number of a graph. Canad. J. Math. 18 (1966), 817-822. [9] Farb, B. and Margalit, D.; A Primer on Mapping Class Groups, PMS-49, Princeton University Press. [10] N. H. Xuong, How to determine the maximum genus of a graph. J. Combinatorial Theory. B 26 (1979), 217-225. [11] N. H. Xuong, Upper embeddable graphs and related topics. J. Combinatorial Theory. B 26 (1979), no. 2, 226-232. [12] Parlier, H.; Kissing numbers for surfaces, J. Topol. 6 (2013), no. 3, 777-791. [13] Ramirez, A.; Open-closed string topology via fat graphs. 2006. arXiv:math/0606512v1. [14] Ringeisen, R. D., Survey of results on the maximum genus of a graph. J. Graph Theory. 3 (1979), 1-13. [15] Sanki, B., Filling of closed surfaces. To appear in J. Topol. Anal. [16] Sanki, B., Gadgil, S., Shortest length geodesics on closed hyperbolic surfaces, arXiv preprint 2016, arXiv:1503.01891. [17] Stahl, S., The embeddings of a graph-A survey. J. Graph Theory. 2 (1978), 275-298. [18] W. Thurston, A spine for Teichm¨ uller space, preprint(1986). Institute of Mathematical Sciences, CIT Campus, Tharamani, Chennai, 600113, India E-mail address: [email protected]