On Wiman’s theorem for graphs

Discrete Mathematics 338 (2015) 1793–1800

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On Wiman’s theorem for graphs Alexander Mednykh a,b,∗ , Ilya Mednykh a,c a

Sobolev Institute of Mathematics, Novosibirsk State University, 630090, Novosibirsk, Russia

b

Univerzita Mateja Bela, 97401, Banská Bystrica, Slovakia

c

Siberian Federal University, 660041, Krasnoyarsk, Russia

article

info

Article history: Received 20 November 2013 Accepted 2 March 2015 Available online 23 March 2015 keywords: Riemann surface Graph Fundamental group Automorphism group Harmonic morphism Branched covering

abstract The aim of the paper is to find discrete versions of the Wiman theorem which states that the maximum possible order of an automorphism of a Riemann surface of genus g ≥ 2 is 4g + 2. The role of a Riemann surface in this paper is played by a finite connected graph. The genus of a graph is defined as the rank of its homology group. Let ZN be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2. We prove that N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g . In this case, the signature of the orbifold X /ZN is (0; 2, g + 1), that is X /ZN is a tree with two branch points of order 2 and g + 1 respectively. Moreover, if N < 2g + 2, then N ≤ 2g . The upper bound N = 2g is attained for any g ≥ 2. The latter takes a place when the signature of the orbifold X /ZN is (0; 2, 2g ). © 2015 Elsevier B.V. All rights reserved.

0. Introduction Klein’s quartic curve, x3 y + y3 z + z 3 x = 0, admits the group PSL2 (7) as its full group of conformal automorphisms. It is characterized as the curve of smallest genus realizing the upper bound 84(g − 1) on the order of a group of conformal automorphisms of a curve of genus g > 1, given by Hurwitz [9] in 1893. Around the same time, Wiman [15] characterized the curves w 2 = z 2g +1 − 1 and w 2 = z (z 2g − 1), g > 1, as the unique curves of genus g admitting cyclic automorphism groups of the largest and the second largest possible order (4g + 2 and 4g, respectively). The modern proof of these and similar results is contained in the paper by K. Nakagawa [13]. Over the last decade, counterparts of many theorems from the classical theory of Riemann surfaces were derived in the discrete case [1,3,5,12]. In these theorems, the finite connected graphs play the role of algebraic curves and the conformal automorphisms are replaced by harmonic ones. We say that a finite group acts harmonically on a graph if it acts freely on the set of directed edges. Following [1] we define the genus of a graph as the rank of its homology group. Then the upper bound on the order of a group acting harmonically on a graph of genus g > 1 is 6(g − 1). This result was obtained by S. Corry [5]. The aim of the present paper is to find a discrete version of the Wiman theorem. Let ZN be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2. We prove that N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g. Moreover, if N < 2g + 2, then N ≤ 2g. The upper bound N = 2g is attained for any g ≥ 2. We describe also the signature of the quotient graphs X /ZN arising in these cases. See Theorems 3 and 4 for explicit statements of the results.

∗

Corresponding author at: Sobolev Institute of Mathematics, Novosibirsk State University, 630090, Novosibirsk, Russia. E-mail addresses: [email protected] (A. Mednykh), [email protected] (I. Mednykh).

http://dx.doi.org/10.1016/j.disc.2015.03.003 0012-365X/© 2015 Elsevier B.V. All rights reserved.

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The basic tools to establish main results of the paper are the Riemann–Hurwitz theorem for graphs proved in [1] and the discrete version of the Harvey theorem (Theorem 2) proved in Section 2. In turn, the proof of Theorem 2 is based on the theory of harmonic morphisms [1] and the Bass–Serre uniformization theory [2] for graphs of groups. The necessary preliminary results are given in Section 1. 1. Basic definitions and preliminary results 1.1. Graphs In this paper, by a graph X we mean a connected multigraph. Denote by V (X ) the set of vertices of X and by E (X ) the set of directed edges of X . Following J.-P. Serre [14] we introduce two maps ∂0 , ∂1 : E (X ) → V (X ) (endpoints) and a fixed point free involution e → e¯ of E (X ) (reversal of orientation) such that ∂i e¯ = ∂1−i e. We will often identify X with V (X ), writing a ∈ X to mean a is a vertex, but keeping the notation E (X ) for edges. We put St(a) = StX (a) = ∂0−1 (a) = {e ∈ E (X )|∂0 e = a}, the star of a and call deg(a) = |(St (a))| the degree (or valency) of a. A morphism of graphs ϕ : X → Y carries vertices to vertices, edges to edges, and, for e ∈ E (X ), ϕ(∂i e) = ∂i ϕ(e), (i = 0, 1) and ϕ(¯e) = ϕ( ¯ e). For a ∈ X we then have the local map

ϕa : StX (a) → StY (ϕ(a)). A map ϕ is locally bijective if ϕa is bijective for all a ∈ X . We call ϕ a covering if ϕ is surjective and locally bijective. A bijective morphism is called an isomorphism, and an isomorphism ϕ : X → X is called an automorphism. 1.2. Harmonic morphisms and harmonic actions Let X , Y be graphs. Let ϕ : X → Y be a morphism of graphs. We now come to one of the key definitions in this paper. A morphism ϕ : X → Y is said to be harmonic (alternatively it is called branched covering, quasi-covering or vertically holomorphic map) if, for all x ∈ V (X ), y ∈ V (Y ) such that y = ϕ(x), the quantity

|e ∈ E (X ) : x = ∂0 e, ϕ(e) = e′ | is the same for all edges e′ ∈ E (Y ) such that y = ∂0 e′ . One can check directly from the definition that the composition of two harmonic morphisms is again harmonic. Therefore the class of all graphs, together with the harmonic morphisms between them, forms a category. It is important to say that the definition of a harmonic morphism given in [1] is slightly more general. We note also that an arbitrary covering of graphs is a harmonic morphism. Let ϕ : X → Y be harmonic and let x ∈ V (X ). We define the multiplicity of ϕ at x by mϕ (x) = |e ∈ E (X ) : x = ∂0 e, ϕ(e) = e′ |

(1)

for any edge e ∈ E (X ) such that ϕ(x) = ∂0 e . By the definition of a harmonic morphism, mϕ (x) is independent of the choice of e′ . If deg(x) denotes the degree of a vertex x, we have the following basic formula relating the degrees and multiplicity: ′

′

deg(x) = deg(ϕ(x))mϕ (x).

(2)

We define the degree of a harmonic morphism ϕ : X → Y by the formula deg(ϕ) := |e ∈ E (X ) : ϕ(e) = e′ |

(3)

for any edge e ∈ E (Y ). By [1, Lemma 2.2] the right-hand side of (3) does not depend on the choice of e and therefore deg(ϕ) is well defined. Let G < Aut (X ) be a group of automorphisms of a graph X . An edge e ∈ E (X ) is called invertible if there is an automorphism in group G sending e to e¯ . We say that the group G acts harmonically on a graph X if G acts freely on the set E (X ) of directed edges of X . If, additionally, G acts without invertible edges we say that G acts purely harmonically on X . In the latter case the quotient graph X /G is well defined. The vertices and the edges of X /G are formed by orbits Gu, u ∈ V (X ) and Ge, e ∈ E (X ) respectively. The image of an edge e with endpoints {u, v} under the canonical map X → X /G is the edge Ge with endpoints {Gu, Gv}. The following observation made by Scott Corry [6] plays a crucial role. ′

′

Observation 1. Suppose that a group G acts purely harmonically on a graph X . Then the quotient graph X /G is well defined and the canonical projection X → X /G is a harmonic morphism. Let G be a finite group acting purely harmonically on a graph X . For every v˜ ∈ V (X ) denote by Gv˜ the stabilizer of v˜ in the group G and by |Gv˜ | the order of the stabilizer. Then to each vertex v ∈ V (X /G) we prescribe the number mv = |Gv˜ |, where v˜ ∈ ϕ −1 (v). Since G acts transitively on each fiber of ϕ , the numbers mv are defined correctly.

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The following version of the Riemann–Hurwitz formula can be found in [1,5,6]. Proposition 1. Let G be a finite group acting purely harmonically on a graph X of genus g. Denote by γ the genus of the quotient graph X /G. Then



 

g − 1 = |G| γ − 1 +

v∈V (X /G)

1−

1 mv



,

where the numbers mv are the same as above. 1.3. Graph quotients, orbifolds and signatures If a group G acts harmonically, but not purely harmonically on a graph X , then X has invertible edges. The main difficulty in this case is the correct definition of a quotient graph X /G. There are at least three different ways to do this. The first way is to consider X /G = (X /G)loop as a graph with loops obtained as images of the invertible edges of X under the canonical projection X → X /G. The second one is to consider the images of invertible edges as semi-edges (or tails) of the graph X /G = (X /G)tail , and the third way is to create the quotient graph X /G = (X /G)free by deleting loops (or semi-edges) that are the images of invertible edges. All of the three ways are well known in the literature [1,3,11]; they are effectively used in various questions of the graph theory. We prefer to define a quotient graph as X /G = (X /G)tail , since it preserves more information about the action of group G on X . So, in what follows, X /G is a graph with semi-edges. Let X ′ be the barycentric subdivision of the graph X . We consider X ′ as a bipartite graph with the white vertices at the middle of edges of X and the black ones for the vertices of X . For convenience, we identify the set of edges of X ′ with the set of semi-edges of X . So, the semi-edge of X is just a bicolored edge of X ′ with white and black vertices. Also, there is a natural one-to-one correspondence between semi-edges and directed edges of the graph X . The group G naturally acts on the bipartite graph X ′ preserving vertex colors. In particular, this means that G acts on X ′ without edge inversions. That is, the quotient X ′ /G is a well defined bipartite graph. Note that the image of any invertible edge of X ′ in X ′ /G is now a bicolored edge with a white vertex of valency one. Consider the quotient X /G as the graph obtained from the bipartite graph X ′ /G by smoothing all 2-valent white vertices. The images of invertible edges are still bicolored edges with white vertices of valency one. We will refer to them as semiedges of graph X /G. Also, we consider each white vertex of valency one as a free end of the respective semi-edge. For the basic facts from the theory of graphs with semi-edges see the papers [11,3,4]. One can consider the quotient graph X /G as a one-dimensional orbifold. Now we introduce the signature of X /G. We will do this in two steps. First of all, let G be a group acting purely harmonically on a graph X . For every v˜ ∈ V (X ) denote by Gv˜ the stabilizer of v˜ in the group G and by |Gv˜ | the order of the stabilizer. Then to each vertex v ∈ V (X /G) we prescribe the number mv = |Gv˜ |, where v˜ ∈ ϕ −1 (v). Since G acts transitively on each fiber of ϕ , the numbers mv are defined correctly. If mv ≥ 2, we will call v a branch point of order mv . Let {m1 , m2 , . . . , mr } be the largest subset of {mv , v ∈ V (X /G)} satisfying 2 ≤ m1 ≤ m2 ≤ · · · ≤ mr and let γ be the genus of graph X /G. Then we define the signature of X /G as (γ ; m1 , m2 , . . . , mr ). Also, will refer to X /G as an orbifold of signature (γ ; m1 , m2 , . . . , mr ). Now, let G be a group acting harmonically, but not purely harmonically on a graph X . Then the signature of X /G is defined as the signature of orbifold X ′ /G. We note that X /G is obtained from X ′ /G by smoothing 2-valent vertices v with trivial stabilizers Gv . So, each vertex v with mv ≥ 2 is either a black vertex of X ′ /G or its white vertex of valency one. That gives two important observations. Let X /G be an orbifold of signature (γ ; m1 , m2 , . . . , mr ). Observation 2. The both quotients X /G and X ′ /G are graphs of genus γ . Observation 3. The branch points of the orbifold X ′ /G are either vertices of graph X /G or free ends of its semi-edges. Any free end is a branch point of order two. 1.4. Graphs of groups Following [2] we define a graph of groups to be a pair A = (A, A), where A is a connected graph, and A assigns groups Aa (a ∈ A), Ae = Ae¯ (e ∈ E (A)), and monomorphisms αe : Ae → Aa , where a = ∂0 e. In this paper we restrict ourself to a class of graphs of groups having trivial groups Ae = (1) for all edges e ∈ E (A) and finite groups Aa for all vertices a ∈ A. It will be enough for application to the theory of harmonic morphisms between graphs. In this case, αe Ae = (1) and

Aa/e = Aa /αe Ae = Aa are isomorphic copies of the group Aa prescribed to each pair (a, e) with a = ∂0 e. There are two equivalent definitions of the notion of the fundamental group of a graph of groups: the first is a direct algebraic definition via an explicit group presentation, and the second one using the language of groupoids. The algebraic

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definition is easier to state. First, choose a spanning tree T in A. The fundamental group of A with respect to T , denoted π1 (A, T ), is defined as the quotient of the free product

[(∗a∈A Aa ) ∗ F (E (A))]/R, where F (E (A)) denotes the free group with free basis E (A) and R is the following set of relations: (i) e¯ = e−1 for every e in E. (ii) e = 1 for every edge e of the spanning tree T . There is also a notion of the fundamental group of A with respect to a base-vertex a in A, denoted by π1 (A, a), which is defined using the formalism of groupoids (see [2,7] for details). It turns out that for every choice of a base-vertex a and every spanning tree T in A the groups π1 (A, T ) and π1 (A, a) are naturally isomorphic. We note also [2, Section 1.22] that for given a, b ∈ A the groups π1 (A, a) and π1 (A, b) are conjugate in the fundamental groupoid of A. In what follows we will use notation π1 (A) ignoring the way the fundamental group was constructed. It follows from the above definition that if A is a graph of genus g, then F (E (A))/R = Fg is the free group of rank g. Then

π1 (A) = (∗a∈A Aa ) ∗ Fg . To every graph of groups A, with a specified choice of a base-vertex a ∈ A, one can associate a Bass–Serre universal covering tree  A = ( A, a), which is a tree admitting a natural group action of the fundamental group π1 (A) = π1 (A, a) without edge-inversions. Moreover, the quotient graph of groups  A/π1 (A) for the action of π1 (A) on  A can be chosen to be naturally isomorphic to A. 1.5. Coverings of graphs of groups and harmonic morphisms We start with the following definition which relates the notion of covering in the category of graphs of groups with the notion of harmonic morphisms in the category of graphs. Definition 1. Let A = (A, A) and A′ = (A′ , A′ ) be graphs of groups with trivial edge groups. By a covering of graphs of groups

8 = (ϕ, Φ ) : A → A′ we mean (i) a harmonic morphism ϕ : A → A′ ; (ii) a set Φ of injective homomorphisms ϕa : Aa → A′ϕ(a) , a ∈ A such that mϕ (a)|Aa | = |A′ϕ(a) |, where mϕ (a) is the multiplicity of ϕ at the point a. The above definition is consistent with that given in [7] which, in turn, is equivalent to the definition proposed by Bass [2]. We produce one basic example to illustrate the notion of covering in the graph of groups category. Example 1. Let G be a group of automorphisms of a finite connected graph A. Suppose that G acts on the set of directed edges of A freely and without edge inversions. Consider the canonical map ϕ : A → B = A/G. Denote by StG (a) the stabilizer of a vertex a in group G. Then ϕ is harmonic morphism with mϕ (a) = |StG (a)|, a ∈ A. Denote by A the graph group obtained from A by prescribing a trivial group to each vertex and each edge of A. To define B we prescribe to each vertex b = ϕ(a) of B a group Bϕ(a) isomorphic to StG (a) and assign a trivial group to each edge of B. Since G acts transitively on each fiber of ϕ , the group Bϕ(a) is well defined. Consider the set Φ of trivial embeddings ϕa : Aa → Bϕ(a) , a ∈ A. We have mϕ (a)|Aa | = |Bϕ(a) |. Then

8 = (ϕ, Φ ) : A → B = A/G is the covering of graphs of groups. 2. Cyclic group acting harmonically on a graph We start with the following well-known theorem by W.J. Harvey [8]. Theorem 1. The cyclic group ZN acts on a closed Riemann surface X as a group of conformal automorphisms with a quotient space X /ZN of the signature (γ ; m1 , . . . , mr ) if and only if the following is satisfied:

(1°) (2°) (3°) (4°)

lcm(m1 , . . . , mˆ i , . . . , mr ) = M, where mˆ i denotes the omission of mi and M = lcm(m1 , . . . , mr ) M divides N, and for γ = 0, M = N r ̸= 1 and if γ = 0, r ≥ 3 If M is even, then the number of periods mi divisible by the maximum power of 2 dividing M is even.

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The above theorem is widely used in the theory of finite groups acting on Riemann surfaces and in the Teichmüller spaces. To investigate cyclic actions on graphs we need the following discrete version of the Harvey theorem. Theorem 2. The cyclic group ZN acts harmonically on a graph X of genus g ≥ 2 with a quotient graph X /ZN of the signature (γ ; m1 , . . . , mr ) if and only if the following is satisfied:

(1°) M divides N, and for γ = 0, M = N, where M = lcm(m1 , . . . , mr ) (2°) if γ = 1, r ≥ 1 (3°) if γ = 0, r ≥ 2 with m2 > 2 if r = 2. The proof of the Harvey theorem for graphs is based on the following algebraic lemma. Lemma 1. Let the group Γ = Z∗γ ∗ Zm1 ∗ · · · ∗ Zmr be a free product of cyclic groups, ZN is the cyclic group of order N and M = lcm(m1 , . . . , mr ). Then there exists an order preserving homomorphism from Γ onto ZN if and only if M divides N, and for γ = 0, M = N. m

r = 1 Proof. Let the group Γ be generated by the elements a1 , . . . , aγ , x1 , . . . , xr satisfying the relations x1 1 = 1, . . . , xm r and ZN = ⟨z : z N = 1⟩. Suppose that ϕ : Γ → ZN defined by

ϕ(ai ) = z ki ,

i = 1, . . . , γ ,

ϕ(xj ) = z lj ,

j = 1, . . . , r ,

is an order preserving epimorphism. Then all mj are divisors of N and lj = Nsj /mj for some integers sj relatively prime with mj . In this case, the subgroup S of ZN generated by the elements z Nsj /mj , j = 1, . . . , r, is also generated by z N /mj , j = 1, . . . , r. The order of subgroup S is M = lcm(m1 , . . . , mr ). Hence M divides N. If γ = 0, then S = ZN and so M = N. Conversely, let M divide N, and for γ = 0, M = N. To create an order preserving epimorphism ϕ : Γ → ZN we set

ϕ(ai ) = z ki ,

i = 1, . . . , γ ,

ϕ(xj ) = z N /mj ,

j = 1, . . . , r ,

where ki , i = 1, . . . , γ are arbitrary integers. If γ ≥ 1, we choose k1 = 1. Then the group generated by ϕ(ai ), i = 1, . . . , γ , ϕ(xj ), j = 1, . . . , r, contains z and so the homomorphism ϕ : Γ → ZN is ‘‘onto’’. If γ = 0, then the order of the group generated by ϕ(xj ), j = 1, . . . , r, is M = lcm(m1 , . . . , mr ). Since M = N, the homomorphism ϕ is again ‘‘onto’’.  Now we provide the proof of Theorem 2. Proof. Let the cyclic group ZN act harmonically on a graph X of genus g ≥ 2 with a quotient space X /ZN of the signature (γ ; m1 , . . . , mr ). By replacing X with X ′ and X /ZN with X ′ /ZN we can assume that ZN acts purely harmonically on X . Recall that X and X ′ are graphs of the same genus, while X /ZN and X ′ /ZN are orbifolds of the same signature (γ ; m1 , . . . , mr ). Let Y = X /ZN be the quotient graph. Then the branch set of the covering ϕ : X → Y = X /ZN consists of r vertices of Y with orders m1 , m2 , . . . , mr . Let X be a graph of groups with a trivial group assigned to each vertex and each edge of X . Consider a graph of groups Y obtained by prescribing the respective group Zmi , i = 1, . . . , r, to each of r points of the branch set and trivial groups to all other vertices and edges of Y . Then the morphism ϕ : X → Y can be naturally extended to a covering 8 : X → Y of graph of groups. Denote by H = π1 (X) and Γ = π1 (Y) the fundamental groups and by  X and  Y the universal covering trees of graphs of groups X and Y respectively. By the Bass uniformization theorem [2, Proposition 2.4] there exists a lift of 8 to : an isomorphism 8 X → Y between the covering trees equivariant under the action of H and Γ on  X and  Y respectively. ∼ ∼      We note that X = X/H and Y = Y/Γ . Identifying X and Y via isomorphism 8 we replace the covering 8 : X → Y by the covering  X/H →  X/Γ induced by a group inclusion H ▹ Γ with Γ /H ∼ = ZN . We note [2, p. 7] that H = Fg is a free group of rank g ≥ 2 and

Γ = Z∗γ ∗ Zm1 ∗ · · · ∗ Zmr . Let θ be the homomorphism of Γ onto ZN with kernel H. Since H is torsion free, θ is an order preserving homomorphism. Lemma 1 implies that M divides N, and for γ = 0, M = N. So, statement 1° of the theorem is satisfied. To prove 2° and 3° we have to use the Riemann–Hurwitz formula from Proposition 1. In our case it has the form

 g −1=N

γ −1+

r   i =1

1−

1 mi



,

(4)

where 2 ≤ m1 ≤ m2 ≤ · · · ≤ mr . Since g ≥ 2 the conditions 2° and 3° immediately follow from (4). Now, we suppose that conditions 1°, 2° and 3° are satisfied. Our aim is to prove that there exists a graph X of genus g ≥ 2 and the group ZN acting harmonically on X with the quotient graph X /ZN of signature (γ ; m1 , . . . , mr ). From 1° by Lemma 1 we conclude that there exists an order preserving epimorphism θ : Γ → ZN with a torsion free kernel H = ker (θ ). Considering the groups Γ and H as the genus zero Fuchsian groups with parabolic elements by [10, Chapter III.7] we still

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Fig. 1. Graph of groups Y with π1 (Y) = Γ .

have the Riemann–Hurwitz formula (4). In particular, H = Fg is a free group of rank g. By 2° and 3° the right hand side of Eq. (4) is positive. Hence, g ≥ 2. Denote by Y the graph of groups shown in Fig. 1. We note that π1 (Y) = Γ . By the Bass–Serre theory graph of groups Y has the universal covering tree  Y such that  Y/Γ ∼ Y/H. Since H is torsion free, it acts without fixed points on the = Y. Let X =   graph Y. Then X is a graph of groups with the trivial group assigned to each vertex and each edge. Moreover, the fundamental group of X is H = Fg . So, one can identify X with an ordinary graph X of genus g. Then, the morphism X =  Y/H → Y =  Y/Γ induced by a group inclusion H ▹ Γ with the covering group Γ /H ∼ = ZN can be considered as a covering X → X /ZN with the quotient graph X /ZN of signature (γ ; m1 , . . . , mr ). (See Example 1 for details.)  3. Main results 3.1. The statements of main results Now we are going to present the main results of the paper. Theorem 3. Let X be a graph of genus g ≥ 2 and ZN is a cyclic group acting harmonically on X . Then N ≤ 2g + 2. The upper bound N = 2g + 2 is attained for any even g. In this case, the signature of orbifold X /ZN is (0; 2, g + 1), that is, X /ZN is a tree with two branch points of order 2 and g + 1, respectively. Theorem 4. Let X be a graph of genus g ≥ 2 and ZN is a cyclic group acting harmonically on X . Let N < 2g + 2 then N ≤ 2g. The value N = 2g is attained only in the following cases: (i) N = 2g and X /ZN is an orbifold of the signature (0; 2, 2g ), g ≥ 2; (ii) N = 12 and X /ZN is an orbifold of the signature (0; 3, 4), g = 6. The next possible value N = 2g − 1 is attained only in two cases: (iii) N = 3 and X /ZN is an orbifold of the signature (0; 3, 3), g = 2; (iv) N = 15 and X /ZN is an orbifold of the signature (0; 3, 5), g = 8. The proof of the above theorems is based on the following lemma. Lemma 2. Let ZN be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2 with a quotient graph X /ZN of the signature (γ ; m1 , . . . , mr ). If N > 2g − 2, then γ = 0, r = 2 and (m1 , m2 ) = (3, 5), (3, 4), (3, 3) or (2, m) for m ≥ 3. Proof. (i) Let γ ≥ 2. Then by the Riemann–Hurwitz formula (Proposition 1) we have g − 1 ≥ N (γ − 1) ≥ N . This contradicts the hypothesis. (ii) Let γ = 1. Then it follows from Theorem 2 that r ≥ 1. By Proposition 1 we obtain g −1=N

r   i=1

1−

1 mi

 ≥

N 2

.

That is N ≤ 2g − 2. This also contradicts the hypothesis.

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(iii) Let γ = 0. Then by Theorem 2 we receive r ≥ 2. From the inequality N > 2g − 2 by the Riemann–Hurwitz formula we get

 g −1=N

−1 +

r  

1

1−



mi

i=1

 r ≥ N −1 + > (g − 1)(r − 2). 2

Since g − 1 > 0 we obtain 1 > r − 2. Hence 2 ≤ r < 3, or r = 2. Now we have

 g −1=N

      1 1 1 1 −1 + 1 − + 1− =N 1− − . m1

m2

m1

m2

Since g > 1 and N > 2g − 2 we obtain 0
 1−

1 m1

−

1 m2



<

N 2

.

That is, 0 < 1 − m1 − m1 < 12 , where 2 ≤ m1 ≤ m2 . The latter inequality has precisely the following integer solutions 1 2 (m1 , m2 ) = (3, 5), (3, 4), (3, 3) and (2, m) for m ≥ 3.  3.2. The upper bound for N in the case N > 2g − 2 Let us suppose that N > 2g − 2. Then by Lemma 2 the signature of quotient graph X /ZN is (0; m1 , m2 ) and the Riemann–Hurwitz formula is reduced to

 g −1=N

1−

1 m1

−

1 m2



,

where (m1 , m2 ) = (3, 5), (3, 4), (3, 3) or (2, m) for m ≥ 3. Consider all of these situations separately. Since γ = 0, by Theorem 2 we have N = M, where M = lcm(m1 , m2 ). 1°. (m1 , m2 ) = (3, 5). In this case, N = lcm(3, 5) = 15 and consequently, by the Riemann–Hurwitz formula g = 8. We also have N = 2g − 1. 2°. (m1 , m2 ) = (3, 4). Then N = lcm(3, 4) = 12 and g = 6. That is, N = 2g. 3°. (m1 , m2 ) = (3, 3). Now N = 3 and g = 2. In this case N = 2g − 1. 4°. (m1 , m2 ) = (2, m), m ≥ 3. Since N = lcm(2, m), we have N = m if m is even and N = 2m otherwise. The Riemann–Hurwitz formula gives g = m in the first case and g = m − 1 in the second. As a result we obtain: 2 4.1°. Let m = 2g be even, then N = 2g and the quotient graph X /ZN is an orbifold of the signature (0; 2, 2g ), g ≥ 2. 4.2°. Let m = g + 1 be odd, then N = 2g + 2, the quotient graph X /ZN is an orbifold of the signature (0; 2, g + 1), and g ≥ 2 is even. 3.3. Proof of Theorems 3 and 4 If N ≤ 2g − 2, then both the Theorems 3 and 4 are valid. So we can suppose that N > 2g − 2. By the previous section in this case the orbifold X /ZN has the signature (0; m1 , m2 ) for suitable values m1 and m2 . Also, there are only three possibilities for N : N = 2g + 2, N = 2g, and N = 2g − 1. In particular, we have the required upper bound N ≤ 2g + 2. That is, the first statement of Theorem 3 is proved. The equality N = 2g + 2 holds only in the case 4.2°. Then (m1 , m2 ) = (2, m), where m = g + 1 and g is even, and the quotient graph X /ZN is an orbifold of the signature (0; 2, g + 1). This is the second statement of Theorem 3. It follows from the previous section that the case N = 2g + 1 is never happened. Hence, we have to consider only two cases N = 2g and N = 2g − 1. The case N = 2g was completely described in points 2° and 4.1° above. These are exactly statements (ii) and (i) of Theorem 4. Finally, the case N = 2g − 1 can be realized only in points 1° and 3°. This proves statements (iv) and (iii) of Theorem 4. It is worth to mention that the upper bound N = 2g + 2 in Theorem 3 is attained for an action of the cyclic group Z2g +2 on the complete bipartite graph K2, g +1 for any even g ≥ 2. Acknowledgments The authors are thankful to the following grants for partial support of this investigation: the Russian Foundation for Basic Research (grant 15-01-07906), the Grant of the Russian Federation Government at Siberian Federal University (grant 14.Y26.31.0006), the Dmitry Zimin’s Foundation ‘‘Dynasty’’ and the Project ‘‘Mobility—Enhancing Research, Science and Education’’, Matej Bel University (ITMS code 26110230082) under the Operational Program Education cofinanced by the European Social Foundation. The authors are very grateful to anonymous referees for fruitful remarks and suggestions.

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