aa r X i v : . [ m a t h . A T ] M a y ON THE RIEMANN-HURWITZ FORMULA FOR GRAPHCOVERINGS
A.D. MEDNYKH
Abstract.
The aim of this paper is to present a few versions of theRiemann-Hurwitz formula for a regular branched covering of graphs. Bya graph, we mean a finite connected multigraph. The genus of a graphis defined as the rank of the first homology group. We consider a finitegroup acting on a graph, possibly with fixed and invertible edges, and therespective factor graph. Then, the obtained Riemann-Hurwitz formularelates genus of the graph with genus of the factor graph and orders ofthe vertex and edge stabilisers.
Mathematics Subject Classifications (2010) : 57M12, 57M60
Keywords : Graph, Branched covering, Invertible edges, Graph withsemi-edges. Introduction
Recall the classical Riemann-Hurwitz formula. Given surjective holomorphicmap ϕ : S → S ′ between Riemann surfaces of genera g and g ′ , respectively, one has2 g − ϕ )(2 g ′ −
2) + X x ∈ S ( r ϕ ( x ) − , where r ϕ ( x ) denotes the ramification index of ϕ at x. Let G be a finite group ofconformal automorphisms acting on S and ϕ : S → S ′ = S/G is the canonical mapinduced by the group action. Then the above formula can be rewritten in the form2 g − | G | (2 g ′ −
2) + X x ∈ S ( | G x | − , where G x stands for the stabiliser of x in G and | G x | is the order of the stabiliser.Remark that S has only finite number of points with non-trivial stabiliser.The latter formula has a natural discrete analogue. Let G be a finite group actingon the set of directed edges of a graph X of genus g freely and without invertibleedges. Denote by g ′ genus of the factor graph X ′ = X/G.
Then by [2] and [5] wehave g − | G | ( g ′ −
1) + X x ∈ V ( X ) ( | G x | − , where V ( X ) is the set of vertices of X. The aim of this paper is to extend the above mentioned result to group actionswith fixed and invertible edges. The main difficulty in this case is the correctdefinition of a factor graph
X/G, when G acts on a graph X with invertible edges.There are at least three different ways to define the graph X/G.
The first way isto consider
X/G = (
X/G ) loop as a graph with loops obtained as images of the The work is supported by the Russian Foundation for Basic Research (grant 15–01–07906). invertible edges of X under the canonical projection X → X/G.
The second way isto consider the images of invertible edges as semi-edges (or tails) of the factor graph
X/G = (
X/G ) tail , and the third one is to create the factor graph X/G = (
X/G ) free by deleting loops (or semi-edges) that are the images of invertible edges. All thethree ways are well known in the literature ([7], [3], [2]); they are effectively usedin various questions of the graph theory. Figure 1.
Different ways to define a factor graph
X/G. Preliminary results and definitions
In this section we introduce the notion of a graph with semi-edges that is aslightly more general then the standard notion of a graph. This gives us a way todefine the action of group on the graph with multiple edges and loops. Also, we areinteresting in the group actions with fixed edges, as well as with invertible edges.The factor space of such an action, in general, is not necessary a graph. But, it canbe recognised as a graph with semi-edges.Following [7] we define a graph with semi-edges as an ordered quadruple X =( D, V ; I, λ ) where D = D ( X ) is a set of darts , V = V ( X ) is a nonempty set of vertices , which is required to be disjoint from D, I is a mapping of D onto V, called the incidence function , and λ is an involutory permutation of D, called the dart-reversing involution. For convenience or if λ is not explicitly specified wesometimes write ¯ x instead of λx. Intuitively, the mapping I assigns to each dartits initial vertex , and the permutation λ interchanges a dart and its reverse. The terminal vertex of a dart x is the initial vertex of λx. The 2-orbits of λ are called edges . The 1-orbits of λ are called semi-edges or tails . An edge is called a loop if λx = x and Iλx = Ix.
We identify the set of edges E ( X ) of X with the following set of unordered pairsof darts: E ( X ) = {{ x, ¯ x } : x ∈ D ( X ) , x = ¯ x } . We will refer to the vertices Ix and I ¯ x as endpoints of the edge { x, ¯ x } . In a similarway, the set of semi-edges T ( X ) of X is identified with the set T ( X ) = {{ x } : x ∈ D ( X ) , x = ¯ x } . A directed edge of X is an ordered pair ( x, ¯ x ) , where x ∈ D ( X ) and x = ¯ x. Wenote that all edges { x, ¯ x } ∈ E ( X ) , including loops, are provided by two directededges ( x, ¯ x ) and (¯ x, x ) . N THE RIEMANN-HURWITZ FORMULA FOR GRAPH COVERINGS 3 A morphism of graphs f : X = ( D, V ; I, λ ) → X ′ = ( D ′ , V ′ ; I ′ , λ ′ ) is a function f : D ∪ V → D ′ ∪ V ′ such that f D ⊆ D ′ , f V ⊆ V ′ , f I = I ′ f and f λ = λ ′ f. Thus,a morphism is an incidence-preserving mapping which takes vertices to vertices andedges to edges or semi-edges. Note that the image of an edge can be an edge, a loopor a semi-edge, the image of a loop can be a loop or a semi-edge, and the image ofa semi-edge can be just a semi-edge.A bijective morphism f : X → X ′ is called an isomorphism, and an isomorphismof X onto itself is called an automorphism. The group Aut( X ) of automorphismsof X is a subgroup of S D ( X ) leaving invariant each of the sets V ( X ) , E ( X ) , T ( X )and preserving incidence.We say that a group G acts on X if G is a subgroup of Aut( X ) . We will refer to X as a graph if it has the empty set of semi-edges T ( X ) = ∅ . Let X be a finite connected graph. We define the genus of X to be the number g ( X ) = 1 − | V ( X ) | + | E ( X ) | . We note that g ( X ) coincides with the Betti number of X that is the rank of thefirst homology group H ( X, Z ) . Let G be a finite group acting on the graph X. Anedge e = { x, ¯ x } ∈ E ( X ) is said to be invertible by G if there is an element g ∈ G such that g sends x to ¯ x and ¯ x to x. An edge e = { x, ¯ x } ∈ E ( X ) is said to be fixed by G if there is a non-trivial element g ∈ G that fixes x and ¯ x. We say that G actson X without invertible edges if X has no edges invertible by G. Also, G acts on X without fixed edges if X has no edges fixed by G. In this paper we will use two kinds of edge stabiliser. The first one, G e , consistsof all elements of G which fix the edge e, that is fix both x and ¯ x. The second, G { e } , is the setwise stabiliser of the set e = { x, ¯ x } in G. We note that | G { e } | = 2 | G e | ifthe edge e is invertible and | G { e } | = | G e | otherwise.3. Groups acting on a graph without invertible edges
Our first result is the following theorem for groups acting on a graph withoutinvertible edges.
Theorem 1.
Let X be a graph of genus g and G is a finite group acting on X without invertible edges. Denote by g ( X/G ) genus of the factor graph X/G.
Then g − | G | ( g ( X/G ) −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | − , where V ( X ) is the set of vertices, E ( X ) is the set of edges of X, G x stands for thestabiliser of x ∈ V ( X ) ∪ E ( X ) in G and | G x | is the order of a stabiliser. Proof: Since G acts on X without invertible edges, the factor graph X/G is welldefined. The vertices of
X/G are orbits
Gv, v ∈ V ( X ) , while the edges are orbits Ge, e ∈ E ( X ) . Vertices Gv and Gv are incident to an edge Ge in X/G if and only if v and v are incident to the edge e in X. Prescribe to every ˜ x ∈ V ( X/G ) ∪ E ( X/G )a group G ˜ x isomorphic to G x , where x is one of the preimages ˜ x under the canonicalmap ϕ : X → X/G.
Since G acts transitively of fibres of ϕ the group G ˜ x is welldefined. One can consider the graph X/G with prescribed groups G v , v ∈ V ( X/G ) A.D. MEDNYKH and G e , e ∈ E ( X/G ) as a graph of groups in sense of the Bass-Serre theory [1]. Wenote that the fibre ϕ − (˜ x ) of ˜ x consists of | G || G ˜ x | elements. Hence,(1) | V ( X ) | = X v ∈ V ( X ) X ˜ v ∈ V ( X/G ) | G || G ˜ v | and(2) | E ( X ) | = X e ∈ E ( X ) X ˜ e ∈ E ( X/G ) | G || G ˜ e | . By definition of genus from (1) and (2) we obtain g − | E ( X ) | − | V ( X ) | = X ¯ e ∈ E ( X/G ) | G || G ˜ e | − X ˜ v ∈ V ( X/G ) | G || G ˜ v | = | G | ( X ˜ e ∈ E ( X/G ) − X ˜ v ∈ V ( X/G ) X ˜ e ∈ E ( X/G ) | G || G ˜ e | (1 − | G ¯ e | ) − X ˜ v ∈ V ( X/G ) | G || G ˜ v | (1 − | G ˜ v | )= | G | ( g ( X/G ) −
1) + X e ∈ E ( X ) (1 − | G e | ) − X v ∈ V ( X ) (1 − | G v | )= | G | ( g ( X/G ) −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | − . (cid:3) Groups acting on a graph with invertible edges
To describe the group action with edge revising we introduce a few definitions.First of all, we define the barycentric subdivision X ′ of a graph X as a graphresulting from the subdivision of all edges in X. The subdivision of some edge e with endpoints { u, v } yields a graph containing one new ( white ) vertex w, andwith an edge set replacing e by two new edges, { u, w } and { w, v } . As a result, weconsider the graph X ′ as a bipartite graph with black and white vertices that arein one to one correspondence with vertices and edges of the graph X, respectively.The reverse operation, smoothing a vertex w with regards to the pair of edges ( e, f )incident on w, removes both edges containing w and replaces ( e, f ) with a new edgethat connects the other endpoints of the pair. Here we emphasise that only 2-valentvertices can be smoothed.Let now G be a finite group acting on a graph X, possibly with invertible edges.In this case, there are at least three different ways to define the factor graph X/G.
The factor graph with loops.
Define the image of an edge e with endpoints { u, v } under the canonical map X → X/G to an edge Ge with endpoints Gu and Gv. If e is an invertible edge then the image of e is a loop Ge with the only one endpoint Gu = Gv.
We denote the obtained graph
X/G by (
X/G ) loop and its genus by g ( X/G ) loop . N THE RIEMANN-HURWITZ FORMULA FOR GRAPH COVERINGS 5
The factor graph with semi-edges.
Let X ′ be the barycentric subdivision of thegraph X. For geometric evidence, we identify the set E ( X ′ ) of edges of X ′ with theset D ( X ) of semi-edges of X. Then, the group G naturally acts on the bipartitegraph X ′ preserving the vertex colour. In particular, this means that G acts on X ′ without invertible edges. That is the quotient X ′ /G is the well defined bipartitegraph. Note that the image of any invertible edge of X in X ′ /G is now a bicolorededge with a white vertex of valency one.Consider ( X/G ) tail as a graph obtained from the bipartite graph X ′ /G bysmoothing all 2-valent white vertices. The images of invertible edges are still bicol-ored edges with white vertices of valency one. We will refer to them as semi-edges or the tails of graph ( X/G ) tail . For the basic facts of theory of graphs with semi-edges see Section 2 and the papers ([7], [3], [4]). Denote genus of (
X/G ) tail by g ( X/G ) tail . The factor graph without semi-edges.
Denote by (
X/G ) free the graph obtainedfrom ( X/G ) tail by removing all tails and replacing them by their black endpoint.Equivalently, ( X/G ) free can be obtained from ( X/G ) loop by removing all loopsarising as the images of invertible edges. This kind of factor graphs was introducedby M. Baker and S. Norine in [2] for group G generated by an involution. See also[6] for more detailed definitions. We write g ( X/G ) free for genus of ( X/G ) free . If the group G acts on a graph X with invertible edges then g ( X/G ) free = g ( X/G ) tail = g ( X/G ) loop . If the action of G has no invertible edges then all the three genera coincide with g ( X/G ) . Our next result is the following theorem.
Theorem 2.
Let X be a graph of genus g and G is a finite group acting on X, possi-bly with invertible edges. Denote by g ( X/G ) tail genus of the factor graph ( X/G ) tail . Then g − | G | ( g ( X/G ) tail −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | −
1) + X e ∈ E inv ( X ) | G e | , where V ( X ) is the set of vertices, E ( X ) is the set of edges of X, G x is the stabiliserof x ∈ V ( X ) ∪ E ( X ) in G, and E inv ( X ) is the set of invertible edges of X. Proof: Consider the set D ( X ) of semi-edges of X . Let e = ( x, ¯ x ), where x ∈ D ( X ) , be a directed edge of X . Denote by X ′ the barycentric subdivision of graph X . Without loss of generality, one can assume that the edges of X ′ are the elementsof D ( X ). Denote by ∂ / e the vertex of X ′ subdividing the edge e in two edges x = h e and ¯ x = h ¯ e of X ′ .Since G acts on X ′ without invertible edges, by Theorem 1 we have(3) g − | G | ( g ( X ′ /G ) −
1) + X v ∈ V ( X ′ ) ( | G v | − − X e ∈ E ( X ′ ) ( | G e | − . Here, the set of vertices V ( X ′ ) of the bipartite graph X ′ is the union V ( X ′ ) = B ( X ′ ) ∪ W ( X ′ ) of the sets of black and white vertices, where B ( X ′ ) = V ( X ) and W ( X ′ ) = { ∂ / e : e ∈ E ( X ) } . The stabiliser of a point w = ∂ / e in G consists of the elements of G thatpermute the endpoints of e leaving e invariant or the ones that fix e. Hence, G ∂ / e = A.D. MEDNYKH G { e } , where G { e } is the setwise stabiliser of the set e = { h e , h ¯ e } in G. As a result,we obtain(4) X v ∈ V ( X ′ ) ( | G v | −
1) = X v ∈ V ( X ) ( | G v | −
1) + X e ∈ E ( X ) ( | G ∂ / e | − X v ∈ V ( X ) ( | G v | −
1) + X e ∈ E ( X ) ( | G { e } | − . For each e ∈ E ( X ) we have G e = G h e = G h ¯ e . Hence(5) X e ∈ E ( X ′ ) ( | G e | −
1) = X e ∈ E ( X ) ( | G h e | −
1) + X e ∈ E ( X ) ( | G h ¯ e | − X e ∈ E ( X ) ( | G e | − . Subsituiting equations (4) and (5) into (3) we obtain g − | G | ( g ( X ′ /G ) −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | − X e ∈ E ( X ) ( | G { e } | − | G e | ) . (6)Denote by E inv ( X ) the set of invertible edges of X. Then | G { e } | = 2 | G e | if e ∈ E inv ( X ) and | G { e } | = | G e | otherwise. The smoothing of a white vertex in graph X ′ /G decreases the number of vertices and the number of edges of the graph by one.So, it does not affect the genus g ( X ′ /G ) = 1 − | V ( X ′ /G ) | + | E ( X ′ /G ) | . Hence, bydefinition of (
X/G ) tail we have g ( X/G ) tail = g ( X ′ /G ) . Then (6) can be rewrittenin the form g − | G | ( g ( X/G ) tail −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | −
1) + X e ∈ E inv ( X ) | G e | . (cid:3) Since g ( X/G ) free = g ( X/G ) tail , as an immediate consequence of Theorem 2 wehave the following statement. Theorem 3.
Let X be a graph of genus g and G is a finite group acting on X, possibly with invertible edges. Denote by g ( X/G ) free genus of the factor graph ( X/G ) free . Then g − | G | ( g ( X/G ) free −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G e | −
1) + X e ∈ E inv ( X ) | G e | , where V ( X ) is the set of vertices, E ( X ) is the set of edges of X, G x is the stabiliserof x ∈ V ( X ) ∪ E ( X ) in G, and E inv ( X ) is the set of invertible edges of X. Following [6] we say that the group G acts harmonically on a graph X if G actsfreely on the set of darts D ( X ) of X or, equivalently, on the set of directed edgesof X . In this case we have | G e | = 1 for each e ∈ E ( X ) . We have the followingcorollary from Theorem 3.
N THE RIEMANN-HURWITZ FORMULA FOR GRAPH COVERINGS 7
Corollary 1.
Let X be a graph of genus g and G is a finite group acting on X harmonically, possibly with invertible edges. Denote by g ( X/G ) free genus of thefactor graph ( X/G ) free . Then g − | G | ( g ( X/G ) free −
1) + X v ∈ V ( X ) ( | G v | −
1) + | E inv ( X ) | , where V ( X ) is the set of vertices, E ( X ) is the set of edges of X, G v is the stabiliserof v ∈ V ( X ) in G, and E inv ( X ) is the set of invertible edges of X. Remark .
The Riemann-Hurwitz formula given in Corollary 1, up to notation,coincides with formula (2 .
16) from [2]. Here, the set E inv ( X ) of invertible edges isexactly the set of vertical edges in terminology of [2].One more consequence of Theorem 2 is the following result. Theorem 4.
Let X be a graph of genus g and G is a finite group acting on X, possibly with invertible edges. Denote by g ( X/G ) loop genus of the factor graph ( X/G ) loop . Then g − | G | ( g ( X/G ) loop −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G { e } | − , where V ( X ) is the set of vertices, E ( X ) is the set of edges of X, G v stands for thestabiliser of v ∈ V ( X ) , G { e } stands for the stabiliser of the set consisting of twosemi-edges of e ∈ E ( X ) , and | G x | is the order of the stabiliser. Proof: Let X ′ be the barycentric subdivision of X . Let ϕ : X ′ → X ′ /G bethe canonical projection and ϕ : X → ( X/G ) tail be a map obtained from ϕ ′ bysmoothing white 2-valent vertices of X ′ and X ′ /G . Denote by T ( X/G ) the setof tails of (
X/G ) tail . The graph (
X/G ) loop can be obtained from ( X/G ) tail byreplacing every tail of graph ( X/G ) tail with a loop. Hence,(7) g ( X/G ) loop = g ( X/G ) tail + | T ( X/G ) | . Recall that each element of T ( X/G ) is the image of an invertible edge e ∈ E ( X )under ϕ. Hence, for any ˜ x ∈ T ( X/G ) the fiber ϕ − (˜ x ) consists of | G || G { e } | invertibleedges of X . Since G acts transitively on the fiber ϕ − (˜ x ), the number | G || G { e } | doesnot depend on the choice of e in the fiber. Therefore(8) | T ( X/G ) | = X ˜ x ∈ T ( X/G ) X e ∈ E inv ( X ) | G { e } || G | = 1 | G | X e ∈ E inv ( X ) (cid:12)(cid:12) G { e } (cid:12)(cid:12) . By Theorem 2 we have(9) g − | G | ( g ( X/G ) tail − X v ∈ V ( X ) ( | G v |− − X e ∈ E ( X ) ( | G e |− X e ∈ E inv ( X ) | G e | . Since | G { e } | = 2 | G e | if e ∈ E inv ( X ) and | G { e } | = | G e | otherwise, from (7) and (8)we get(10) | G | ( g ( X/G ) tail −
1) = | G | ( g ( X/G ) loop − − X e ∈ E inv ( X ) | G e | . A.D. MEDNYKH
Also(11) − X e ∈ E ( X ) ( | G e | −
1) = − X e ∈ E ( X ) ( | G { e } | −
1) + X e ∈ E ( X ) ( | G { e } | − | G e | )= − X e ∈ E ( X ) ( | G { e } | −
1) + X e ∈ E inv ( X ) | G e | . Substituting (10) and (11) into (9) we finally obtain g − | G | ( g ( X/G ) loop −
1) + X v ∈ V ( X ) ( | G v | − − X e ∈ E ( X ) ( | G { e } | − . (cid:3) Acknowledgments
The author is thankful to Gareth Jones, Tom Tucker, Tomaˇz Pisanski and RomanNedela for fruitful discussions of the results of the paper.
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Lifting graph automorphisms by voltage assignments //Eu-ropean Journal of Combinatorics. no. 7 (2000), 927–947. Alexander Dmitrievich MednykhSobolev Institute of MathematicsNovosibirsk State University630090, Novosibirsk, Russia
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