Group Actions on Graphs, Maps and SurfacesDoc. RNDr. Roman Nedela, CSc. Group Actions on Graphs,...

Ministry of Education of Slovak RepublicScientific Board of P. J. Šafárik University

Doc. RNDr. Roman Nedela, CSc.

Group Actions on Graphs, Maps andSurfaces

Dissertationfor the Degree of Doctor of Physical-Mathematical Sciences

Branch: 11-11-9 Discrete Mathematics

Košice, May 2005

Contents

1 Introduction 5

2 Half-arc-transitive actions of groups on graphs of valency four 72.1 Graphs and groups of automorphisms . . . . . . . . . . . . . . . 72.2 Half-arc-transitive action, G-orientation. . . . . . . . . . . . . . . 72.3 Orbital graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 A construction of half-arc-transitive graphs of valency 4 . . . . . 102.5 Alternating cycles, classification of tightly attached graphs . . . . 102.6 Regular maps and half-arc-transitive graphs of valency four . . . 112.7 Pl and Al operators on graphs of valency 4 . . . . . . . . . . . . 132.8 Graphs of valency 4 and girth 4 . . . . . . . . . . . . . . . . . . . 142.9 Classification of point stabilizers . . . . . . . . . . . . . . . . . . 152.10 Relations to group actions of other sorts . . . . . . . . . . . . . . 17

3 Maps, Regular Maps and Hypermaps 213.1 Topological and combinatorial maps, permutation representation

of maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Generalization to hypermaps, Walsh map of a hypermap . . . . . 283.3 Maps, hypermaps and groups . . . . . . . . . . . . . . . . . . . . 303.4 Regular maps of large planar width and residual finiteness of

triangle groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Maps, hypermaps and Riemann surfaces . . . . . . . . . . . . . . 403.6 Enumeration of maps of given genus . . . . . . . . . . . . . . . . 433.7 Regular hypermaps on a fixed surface . . . . . . . . . . . . . . . 483.8 Operations on maps and hypermaps, external symmetries of hy-

permaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.9 Lifting automorphisms of maps . . . . . . . . . . . . . . . . . . . 543.10 Regular embeddings of graphs . . . . . . . . . . . . . . . . . . . . 57

4 Minimal triangulations of given edge width 75

5 Publication record and citation index 795.1 Publication record of the author . . . . . . . . . . . . . . . . . . 795.2 Citation index of the author . . . . . . . . . . . . . . . . . . . . . 835.3 Survey of publications and citations . . . . . . . . . . . . . . . . 995.4 List of invited visits and selected talks in conferences . . . . . . . 995.5 Other activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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R. Nedela: Group Actions on Surfaces

6 Appendix: Reprints of papers 103On the point stabilizers of transitive groups with non-self-paired sub-

orbits of length 2 (with D. Marušič) . . . . . . . . . . . . . . . . 103Exponents of orientable maps (with M. Škoviera) . . . . . . . . . . . . 103Classification of regular maps of prime negative Euler characteristic

(with A. Breda and J. Širáň) . . . . . . . . . . . . . . . . . . . . 103K-minimal triangulations of surfaces (with A. Malnič) . . . . . . . . . 103

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Chapter 1

Introduction

The presented thesis deals with three topics: half-arc-transitive actions of groupson graphs, regular maps and hypermaps and triangulations of given planaredge-width. These topics are unified by the interaction of graphs, groups andsurfaces. In the first two, the role of group theoretical methods and aspects isstressed, although we briefly mention other related mathematical objects suchas Riemann surfaces or algebraic curves, for instance.

Chapter 2 can be considered as a survey on half-arc-transitive group actionson 4-valent graphs, or equivalently, on transitive permutation groups with non-self-paired suborbits of length two. Most of the related work of the author wasdone in a fruitful collaboration with Dragan Marušič (Univ. Ljubljana) duringyears 1994-2000. The strongest result we have achieved is a classification ofpoint stabilisers of such actions (see the attached reprint for details).

Chapter 3 deals with maps and hypermaps posessing maximum number ofsymmetries, called regular maps and regular hypermaps. First, parts of thegeneral theory of maps and hypermaps is built. Then an exhaustive surveyfollows. In each section one of the fundamental problems is discussed, the readeris provided by the most recent information on the current stage of art of theresearch in the field. Of course, any such a survey is subjective, some aspectsare more stressed (namely the ones in which the author actively contributed),some others are suppressed. Regular maps and hypermaps were (and still are)in the center of my research activities. Most of my results in the field weredone in collaboration with other mathematicians, to name just few of them Imention Antonio Breda, Shao-Fei Du, Jin-Ho Kwak, Gareth Jones, AleksanderMalnič, Alexander Mednykh, Martin Škoviera and Jozef Širáň. Among manyinteresting results I have chosen (for the Appendix) a recent paper by Breda,Nedela and Širáň classifying regular maps on surfaces of prime negative Eulercharacteristic. As a consequence, an infinite family of non-orientable surfacesadmitting no regular maps was found. This way we have solved a problem ofConder and Everitt (1995). Another selected paper by Škoviera and myself isdevoted to a systematic study of exponents of maps (integer invariants relatedwith certain map operations). Exponents of maps play an important role inclassification of regular embeddings of graphs.

Chapter 4 deals with minimal triangulations of given edge-width. In collab-oration with Aleksander Malnič we have proved (independently on Robertson-Seymour theory) that the class of triangulations on given surface and of bounded

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edge-width has a finite basis with respect to the vertex-splitting operation. Ourresult generalises the well-known statement of Steinitz on polyhedral triangula-tions, see the attached paper for details.

Chapter 5 is obligatory. It contains some information on research activitiesof the author.

In the Appendix the above-mentioned papers of the author related withChapters 2,3 and 4, respectively, are included.

In the end I would like to thank to all my friends and collaborators givingme an opportunity to share a beauty of mathematics, supporting me duringcrises and coming with new ideas giving me an additional boost and inspirationto work on research projects. Between all my collaborators the one, who havemostly influenced my development as a mathematician has a special position,this is Martin Škoviera. Thanks Martin!

In Banská Bystrica,March 31, 2005

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Chapter 2

Half-arc-transitive actionsof groups on graphs ofvalency four

2.1 Graphs and groups of automorphisms

Throughout this section by a graph we mean an ordered pair (V,E), whereV is a finite nonempty set and E is a symmetric irreflexive relation on V ,whose transitive closure is the universal relation. Graphs are thus simple andconnected, unless specified otherwise. By a directed graph X we mean an orderedpair (V, A), where V is a finite nonempty set and A, the set of arcs, is anasymmetric relation on V . A directed graph is balanced if for every vertex thenumber of incoming arcs is equal to the number of outgoing arcs. For a graphX, we let V (X), E(X), A(X) and Aut (X) denote the respective sets of vertices,edges and arcs, and the automorphism group of X. Given (undirected) graphX the set of arcs of X is said to be A = {(x, y), (y, x)|[x, y] ∈ E(X)}.

An automorphism of a graph X is a permutation ψ of V such that [ψ(x), ψ(y)]∈ E(X) for every edge [x, y] ∈ E(X). Similarly, an automorphism of an orientedgraph X is a permutation ψ of V such that [ψ(x), ψ(y)] ∈ A(X) for every arc[x, y] ∈ A(X). A group of automorphisms G ≤ Aut (X) of a graph has aninduced action on edges of X and on arcs of X. A graph X is said to be vertex-transitive, edge-transitive and arc-transitive, respectively, if its automorphismgroup Aut (X) acts vertex-transitively, edge-transitively and arc-transitively.

Furthermore, all groups are assumed to be finite. For graph-theoretic andgroup-theoretic terms not defined here we refer the reader to [5, 12, 32, 33, 65].

2.2 Half-arc-transitive action, G-orientation.

An edge-transitive group G ≤ Aut (X) of automorphisms of a graph X is eithervertex-transitive, or X is bipartite. Complete bipartite graphs Km,n with m 6= nare obvious representatives of bipartite edge- but not vertex-transitive graphs.

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If G is vertex- and edge-transitive on X then the induced action on the set ofarcs has at most two orbits. Thus such actions split into two families: arc-transitive group actions acting with one orbit on A(X), and half-arc-transitivegroup actions giving two orbits on A(X). Half-arc-transitive group actions ongraphs present the main objective of investigation in this chapter.

Let G act half-transitively on X = (V, E) and let take an orbit O ⊂ A(X)of the action. It follows that O intersects each pair of oppositely directed arcs(x, y), (y, x) in exactly one arc. Then

−→X = (V,O) is an associated directed

graph. Note that X can be obtained from−→X = (V, O) by forgetting the ori-

entation arcs of−→X = (V, O). If Ō denotes the complementary orbit to O then←−

X = (V, Ō) can be obtained by reversing the orientation of each arc of−→X .

Sometimes we need to express that given directed graph−→X arises from a graph

X and a half-arc-transitive group G ≤ Aut (X). In such a case we call any of−→X ,

←−X a G-orientation of X.There are graphs admitting two half-arc-transitive group actions such that

the respective G-orientations are essentially different (see Figure 2.1).

1

2 2 2

3 3 33

4 4 4 4

5

5 5

5

6

6 6

6

7

7 7

71

1 1

1

2

1 1

1

Figure 2.1: Two essentially different G-orientations of C4 × C4.

Note that if X is half-arc-transitive then all groups G ≤ Aut (X) actinghalf-arc-transitively on X induce the same G-orientation of X.

The following fundamental statement comes from Tutte [57, page 59].

Theorem 2.1 Let X = (V, E) be a graph and G ≤ Aut (X) acts half-arc-transitively. Then X is k-valent for some even integer k.

Proof. Take a G-orientation−→X of X. By its definition G acts transitively

on vertices and on arcs of−→X . Let v be a fixed vertex and let m = indeg(v)

and n = outdeg(v) be the numbers of incoming and outcoming arcs incident v,respectively. Since G is transitive on vertices, these numbers do not depend onthe choice of v. Since m|V | = ∑v∈V indeg(v) =

∑v∈V outdeg(v) = n|V | we

have m = n. Hence the valency of each vertex is k = 2m = 2n. ¤

Thus a G-orientation of a graph admitting a half-arc-transitive action ofG is always balanced. Since connected graphs of valency 2 are just cycles thesmallest non-trivial case to consider is valency four.

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Note that the above Theorem 2.1 does not generalize to infinite locally finitegraphs. Given directed graph

−→X by Pl(

−→X ) we denote the partial line graph of−→

X with the vertex set A = A(X) and arc-set defined as follows: ((x, y), (z, w))is an arc of Pl(

−→X ) if and only if y = z and x 6= w. If a group G acting half-arc-

transitively is fixed then we apply the partial line graph operator on (undirected)graph X via its G-orientation. The resulting graph Pl(X) is undirected. It arisesby forgetting the direction of arcs of Pl(

−→X ).

Theorem 2.2 For every odd integer k ≥ 5 there is an infinite half-arc-transitivegraph X of valency k. If X is a cubic graph admitting a half-arc-transitiveaction of a group then X is a cubic tree. In particular, there is no cubic half-arc-transitive graph.

Proof. Let k = 2m + 1. Take a k-valent infinite tree T with k ≥ 5. Onecan define inductively an unbalanced orientation of T thus forming

−→T such that

indeg(v) = m and outdeg(v) = m + 1 for every vertex v in−→T . There is an

associated group G ≤ Aut (T ) which induces the above orientation of T . SetX = Pl(

−→T ) = Pl(T ). Then G acts half-arc-transitively on X. Consider the

subgraph Y of X induced by the vertices that come from 2m+1 arcs incident to afixed vertex of T . By the definition of X, Y ∼= Km,m+1 and the subgraphs of thissort corresponding to vertices of the original graph form blocks of imprimitivityin the action of Aut (X). In particular, X cannot be arc-transitive since Km,m+1is not.

Let us assume that k = 3 and X is cubic. Let G be a group acting half-arc-transitively on X. By vertex-transitivity indeg(v) (outdeg(v) is the same forany vertex v of

−→X . Without loss of generality we assume that indeg(v) = 1 and

outdeg(v) = 2 for each vertex v. Let C be a cycle of X. Then G induces anorientation of arcs of C. The conditions indeg(v) = 1 and outdeg(v) = 2 forceC to have transitive orientation, i. e.

−→C ⊆ −→X is a directed cycle. Let x be an

arc incident to a vertex in−→C but not belonging to

−→C . By transitivity of the

action on arcs there is a directed cycle−→C ′ passing through x. Hence C ∩ C ′ is

nonempty, a contradiction. Thus X is a cubic tree. ¤

2.3 Orbital graphs

Let G be a transitive permutation group acting on a set V and let v ∈ V . Thereis a 1-1 correspondence between the set of orbits of the stabilizer Gv on V , thatis, the set of suborbits of G, and the set of orbitals of G, that is, the set of orbitsin the natural action of G on V ×V , with the trivial suborbit {v} correspondingto the diagonal {(v, v) : v ∈ V }. The paired orbital ∆t of an orbital ∆ is theorbital {(v, w) : (w, v) ∈ ∆}. If ∆t = ∆ we say that ∆ is a self-paired orbital. Aself-paired suborbit of G is a suborbit which corresponds to a self-paired orbital.The orbital graph X(G,V ;∆) of (G,V ) relative to ∆, is the graph with vertexset V and arc set ∆. The suborbit ∆ is said to be connected if the underlyingundirected graph X0(G, V ; ∆) of X(G,V ;∆) is connected. Of course, if ∆ = ∆t

is a self-paired orbital then X(G,V ;∆) can be viewed as an undirected graphwhich admits a vertex- and arc-transitive action of G. On the other hand, if∆ 6= ∆t is a non-self-paired orbital then ∆ ∩ ∆t = ∅ and X0(G,V ;∆) admits

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a 12 -arc-transitive action of G. Conversely, given an edge [u, v] of a graph X ofvalency 2d admitting a 12 -arc-transitive action of some group G ≤ Aut (X), thetwo arcs (u, v) and (v, u) give rise (via the action of G) to two oriented graphs,namely the orbital graphs of G relative to two paired orbitals, where the lengthof the corresponding two suborbits is d.

The above discussion is summarized in the following theorem.

Theorem 2.3 Let X = (V, E) be a 2d-valent graph and G ≤ Aut (X) acts half-arc-transitively on X. Then there exists a corresponding orbital ∆ in the actionof G on V ×V with a non-self-paired suborbit of length d such that the respectiveorbital graph XO(G, V ;∆) ∼= X

Vice-versa, each transitive action of a permutation group G with a non-self-paired connected suborbit of length d gives rise an orbital graph XO(G,V ; ∆)which admits a half-arc-transitive action of G.

Since a transitive action of a group G is equivalent to the action of G on leftcosets of Gv by left multiplication a half-arc-transitive graph is determined byspecifying G, H = Gv ≤ G and d elements a1, . . . , ad such that {(H, aiH)|i =1, . . . d} is a non-self-paired suborbit of length d.

In the specific instance of this situation, when an orbital ∆ has a non-self-paired connected suborbit of length 2 in the action of a group G on the set of leftcosets of its subgroup H, we say that the associated 4-valent graph X0(G,H, ∆)is (G, 12 ,H)-arc-transitive, or (G,

12 )-arc-transitive in short. Of course, when G =

Aut (X) then X is a 12 -arc-transitive graph. Hence to study half-arc-transitiveactions of groups on graphs of valency four is the same as to study transitivepermutation groups with a connected non-self-paired suborbit of length 2.

2.4 A construction of half-arc-transitive graphsof valency 4

Let r ≥ 3 be an odd integer, t ≥ 3 be an integer and let s ∈ Z∗r satisfyst ∈ {1,−1} modulo r. Set V = {vji ; i ∈ Zt, j ∈ Zr} and E = {vji vj±s

i

i+1 |i ∈Zt, j ∈ Zr}. Denote X(s; t, r) = (V, E). For instance, X(2; 3, 9) is the smallesthalf-arc-transitive graph called the Holt’s graph, see [20], while X(2; 6, 9) isthe smallest graph in Bouwer’s family [3]. The permutations ρ, σ and τ givenby the rules vji ρ = v

j+1i , v

ji σ = v

sji+1 and v

ji τ = v

−ji generate a subgroup G =

G(s, t, r) of automorphisms of X(s; t, r) acting half-arc-transitively on X(s; t, r).Note that 〈ρ, σ〉 acting transitively on vertices is metacyclic, so X(s; t, r) is ametacirculant. The half-arc-transitivity of graphs X(s; t, r) was considered in[1] and proved in two particular cases: t = 3 and r ≥ 9, or r prime and ta composite integer. Furthermore, in [49] is shown that X(s; 4, r) is half-arc-transitive provided s2 6= ±1.

2.5 Alternating cycles, classification of tightlyattached graphs

Let consider a 4-valent graph with a fixed G ≤ Aut (X) acting half-arc-transi-tively on X. A walk W in an oriented graph is called alternating if every other

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vertex is a tail and every other vertex is a head of two consecutive arcs in W .Since the G-orientation of X is balanced, each arc lies in a unique alternatingcycle called a G-alternating cycle. The set of G-alternating cycles decomposethe set of edges. They are of even length 2r, the parameter r ≥ 2 is calledG-radius of X. The following result was proved by Marušič [32].

Theorem 2.4 (Marušič [32]) Let X be a (G, 12 )-transitive graph of valency 4for some G ≤ Aut (X). Then there exists an integer r ≥ 2 such that

all G-alternating cycles have length 2r for some r and the set of G-alternating cycles form a decomposition of the set of edges of X;

either X has precisely two G-alternating cycles, both spanning V (X),which occurs if and only if X ∼= Cay(Z2r; {1,−1, s,−s} for some odds ∈ Z∗2r \ {1,−1} such that s2 = ±1; in this case X is arc-transitive;or X has at least three G-alternating cycles, which are all induced cycles.

Marušič has observed in [32] that any two adjacent G-alternating intersect inthe same number of vertices aG(X) and that this number divides the G-radius.If aG(X) = rG(X), aG(X) = 1 or aG(X) = 2 the graph is called, respectively,tightly G-attached, loosely G-attached or antipodally G-attached. Marušič andPraeger [44] proved that a (G, 12 )-transitive graph of valency four is a cover of(G, 12 )-transitive graph Y of valency 4 which is loosely, antipodally or tightlyG-attached. Hence understanding the above three classes of half-arc transitivegraphs of valency four is of crucial importance. A partial classification in thisdirection is proved in [32].

Theorem 2.5 (Marušič [32]) Let X be a tightly attached half-transitive graphof valency four with an odd radius r. Then X ∼= X(s; t, r), where t ≥ 3 is aninteger, s ∈ Z∗r and st ∈ {1,−1}, and moreover none of the following threeconditions is satisfied:

s2 = ±1;(s; t, r) = (2, 3, 7), (2; 6, 7);

(s; t, r) = (s; 6, 7k), where k ≥ 3, (7, k) = 1, s6 = 1 and the equationx2 + x − 2 = 0 has a unique solution q ∈ {s,−s, 1/s,−1/s} such that7(q − 1) = 0 and q ≡ 5 (mod 7).

In [44] the classification of (G, 12 )-transitive graphs of even radius is com-pleted. Further results related on G-alternating and related invariants can befound in [40] and [44].

2.6 Regular maps and half-arc-transitive graphsof valency four

A map is a 2-cell decomposition of a surface. In this section we shall assumethat the underlying surface of a map is orientable and that the degree of everyvertex and size of every face is at least three. A map M is (orientably) regular ifthe (orientation preserving) map automorphism group acts arc-transitively on

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M. Given map M we can form a new map Med(M), called medial map ofM, which vertices are centers of edges of M and two vertices are joined by anedge if the respective edges of M are consecutive in the boundary walk of a faceof M. It is assumed that the edges of Med(M) are placed inside faces of Mand that this is done in a way that edges of Med(M) do not cross. Since everyedge lies in exactly four angles, the underlying graph X of Med(M) is 4-valent.The faces of Med(M) split into two disjoint classes: vertex-faces containing thevertices of the original map M and face-faces containing the centers of faces ofM. Since the surface is orientable one can orient the arcs of the medial graphsuch that the boundary walks of vertex- and face-faces form transitive closedwalks inducing the same, say counter-clockwise, orientation of vertex-faces andthe reverse orientation of face-faces.

Figure 2.2: Cube and its medial

If M is regular with G = Aut (M) then the medial graph X of M is(G, 12 , Z2)-transitive of valency four. The G-orientation is just the one describedabove. Vice-versa, given (G, 12 , Z2)-transitive 4-valent graph X, there is a reg-ular map M such that X is the medial graph of M and G = Aut (M). Hencewith any (G, 12 , Z2)-transitive 4-valent graph there is an associated orientablesurface. The above described relationship is studied in detail by Marušič andNedela in [36]. A map is called chiral if it does not admit a reflection (an orien-tation reversing automorphism). The results proved in [36] are applied to showthat apart from few small exceptions Coxeter chiral maps on torus of type {3, 6}(see [11, Chapter 7]) have half-arc-transitive medials. Note that every such amap M is (up to duality) determined by a couple of non-negative integers b,c; thus we write M = Mb,c = {3, 6}b,c or M = {6, 3}b,c in the notation ofCoxeter-Moser [11]). The result follows.

Theorem 2.6 A half-arc-transitive graph of valency 4 is toroidal if and only ifit is isomorphic to the medial graph of the toroidal regular map Mb,c, where b,c satisfy bc(b− c) 6= 0 and b + c > 3.

The correspondence between regular maps and (G, 12 , Z2)-transitive graphsof valency four is a particular instance of a more general relation between regular

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hypermaps and (G, 12 )-transitive graphs with cyclic G-stabilizers. This general-ization is studied in [4].

2.7 Pl and Al operators on graphs of valency 4

Given oriented graph X the line graph L(X) is a (oriented) graph whose verticesare arcs of X, and xy is an arc in L(X) if xy form a 2-arc in X. If X is balancedof valency four then L(X) is balanced of valency four as well. Moreover, theset of alternating 4-cycles decomposes the set arcs of X. Assume a balanced4-valent oriented graph Y share the following two properties:

(i) the set of its alternating 4-cycles decomposes the set of arcs of Y ,(ii) two alternating 4-cycles meet in at most one vertex.Then there is a unique balanced oriented 4-valent graph X such that Y =

L(X). It follows that there is a reverse operator Al defined on balanced 4-valent oriented graphs satisfying (i) and (ii) and we can write X = Al(Y ) =Al(L(X)). In particular, the above defined operators apply on 4-valent orientedgraphs, with the orientation induced by a half-arc-transitive action of a groupof automorphisms.

Given (G, 12 )-transitive graph of valency four the partial line graph Pl(X)of X is the underlying undirected graph of a G-orientation of X. Marušičand Nedela [38] have studied the Pl and Al operators in the context of half-arc-transitive graphs in detail. In particular, it is proved there that G acts on Pl(X)as an orientation preserving group of automorphisms. The action is transitive onvertices. Since the number of vertices in Pl(X) is doubled, the size of the vertex-stabilizer is halved. Using this observation one can easily deduce that the orderof the stabilizer of G-action is a power of two, say 2h. The constant h is calledG-height of X. If h > 1 then Pl(X) is (G, 12 )-transitive. For h = 1 the partialline graph of X is a Cayley graph Cay(G; {a, a−1, b, b−1}), where the generatorsG = 〈a, b〉 are non-involutory. Thus Cayley graphs Cay(G; {a, a−1, b, b−1}) canbe considered to be the 4-valent graphs of height 0. It follows that every 4-valent half-arc-transitive graph with the automorphism group G of height h canbe constructed from a 4-valent Cayley graph Cay(G; {a, a−1, b, b−1}) applyingthe Al operator h times. The following theorem summarizes the properties ofPl and Al operators on 4-valent (G, 12 )-transitive graphs. More results with adetailed explanation can be found in [38].

Theorem 2.7 Let X, Y be graphs of valency four. Then

(i) If X is (G, 12 , H)-arc-transitive for some H ≤ G ≤ Aut (X) and |H| > 2,then Pl(X) is (G, 12 ,K)-arc-transitive with G-radius 2 for some K < H ofindex 2 in H. Conversely, if Y is (G, 12 ,K)-arc-transitive with G-radius2 and G-attachment number 1 for some K ≤ G ≤ Aut (Y ) such that|K| ≥ 2, then X = Al(Y ) is (G, 12 , H)-arc-transitive for some H ≤ Gsuch that [H : K] = 2 and thus |H| > 2.

(ii) If X is (G, 12 , Z2)-arc-transitive for some non-abelian subgroup G ≤ Aut (X),then there exist non-involutory generators generators a and b of G suchthat (ab−1)2 = 1 and Pl(X) ∼= Cay(G; {a±1, b±1}). Conversely, if Y issuch a Cayley graph, then Al(Y ) is (G, 12 , Z2)-transitive.

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Using Theorem 2.7 the following characterization of actions of small heightis proved in [38].

Corollary 2.8 Let X be a (G, 12 )-transitive graph of height h ≤ 3. Then thereare non-involutory generators a, b such that X ∼= Alh(Cay(G; a, b)) and a, bsatisfy the following relations:

(i) (ab−1)2 = 1 if h = 1,

(ii) (ab−1)2 = 1 and (a2b−2)2 = 1 if h = 2,

(iii) (ab−1)2 = 1, (a2b−2)2 = 1 and (a3b−3)2 = 1

or (ab−1)2 = 1, (a2b−2)2 = 1 and a3b−3a3b−1a−1b−1 = 1 if h = 3.

Remark. To stress the orientation of edges used to apply the Al operator wewrite Alh(Cay(G; a, b) for the respective undirected graph.

Example 2.9 Corollary 2.8(iii) is used by Conder and Marušič in [10] to con-struct a first example of a half-arc-transitive graph of valency 4 with a non-abelian stabilizer. The group G = 〈a, b〉 is a subgroup of the symmetric groupS32 given by

a = (1, 2, 3, 4, 5, 6, 7, 8)(9, 10, 11, 12, 13, 14, 15, 16)(17, 18, 19, 20, 21, 22, 23, 24)(25, 26, 27, 28, 29, 30, 31, 32),andb = (1, 2, 11, 18, 21, 28, 27, 22, 5, 14, 15, 16, 9, 10, 3, 26, 29, 20, 19, 30, 13, 6, 7, 8)(4, 23, 32, 17, 12, 31, 24, 25).The proof that there are no more graph automorphisms is computer assisted.

2.8 Graphs of valency 4 and girth 4

In the previous section we have seen that 4-valent graphs of girth 4 form animportant subset of 4-valent graphs admitting half-arc-transitive group actionof a group G. An important question arises: Under what condition in sucha graph all the 4-cycles are necessarily G-alternating? A 4-cycle C is calledG-directed if the action of G induces a transitive orientation of C.

Exceptional graphs. Let r, t be integers, r odd and s ∈ Z∗r satisfyingst ∈ {1,−1}. Set X(t, r) = X(1; t, r). For r and t even let Y (t, r) denotethe graph with vertex set {vji : i ∈ Zt, j ∈ Zr} and edges of the form vji vji+1,vji v

j+(−1)i+1i+1 , i ∈ Zt, j ∈ Zr. Moreover, let Z(t; r) denote the graph with the

vertex set {vji : i ∈ Zt, j ∈ Zr} and edges of the form vji vji+1, vji vj+(−1)i+1

i+1 fori ∈ Zt \ {−1}, j ∈ Zr and vj−1vj+r/20 , vj−1vj+1+r/20 , j ∈ Zr. Finally, denote byY = C2r ⊗ C2r, r ≥ 3 the halved quotient of the cartesian product C2r × C2rarising by identifying the (antipodal) couples of vertices (i, j), (i+r, j+r) for alli, j ∈ Z2r. Let us note that alternatively Y can be constructed as a medial graphof the quadrangular regular embedding of X = Cr×Cr into the torus. While Xis the underlying graph of the Coxeter map {4, 4}r,0, Y is the underlying graphof {4, 4}r,r.

We say that a graph of valency 4 admitting a half-arc-transitive group actionis exceptional if it belongs to one of the following families of (arc-transitive)graphs.

14


(i) the family F1 of circulants Cay(Z2r; {±1;±s}),where s ∈ Z∗2r satisfiess2 ∈ {1,−1} ;

(ii) the family F2 of graphs X(t; r),where t ≥ 3 is an integer and r ≥ 3 is anodd integer;

(iii) the family F3 of graphs Y (t; r) ,where t ≥ 4 and r ≥ 4 are even integers;(iv) the family F4 of graphs Z(t; r) ,where t ≥ 4 and r ≥ 4 are even integers;(v) the family F5 of lexicographic products Ct[2K1],where t ≥ 3 is an integer;(vi) the family F6 of Cartesian products C2r × C2r, r ≥ 2;(vii) the family F7 of graphs C2r ⊗ C2r , r ≥ 3.

The following statement proved in [39] gives an answer to the above problem.

Theorem 2.10 Let X be a (G, 12 )-transitive graph of valency 4 and girth 4,where G ≤ Aut (X). Then either every 4-cycle of X is G-alternating or every 4-cycle of X is G-directed or X is one of the exceptional graphs above. Moreover, ifevery 4-cycle of X is G-directed then either the G-height is 1 or X is exceptional.

Corollary 2.11 ([39]) Let X be a half-arc-transitive graph of valency 4 andgirth 4. Then either

(i) every 4-cycle is alternating or

(ii) every 4-cycle is directed.

Moreover, in case (ii) X is of height 1.

2.9 Classification of point stabilizers

Information on the structure of point stabilizers of transitive permutation groupsis important in classification results. In this section we first briefly mentionknown results then we shall concentrate to vertex stabilisers of half-arc-transitivegroup actions on 4-valent connected graphs. Their structure was described byMarušič and Nedela in [37], see the attached paper.

Arc-transitive actions. A classical result of Tutte [55] establishes a bound3 · 24 on the size of a vertex stabilizer of an arc-transitive group acting on aconnected cubic graph. Later, it turned out (see Sims [50, 51] and Wong [66])that exactly five groups, namely Z3, S3, D12, S4 and S4 × Z2, can serve aspoint stabilizers of such actions. For valency > 3 the size of vertex stabilizers isunbounded. However, when we are restricted to primitive group actions thereare many particular results [22, 23, 45, 48, 51, 61] bounding the size of a stabilizerby a function of valency. Hence, there is an evidence that the following Sims’conjecture could be true:

Sims’ conjecture. Vertex stabilizers of primitive groups acting arc-transi-tively on graphs of valency k are bounded by a function of k.

Sims’ conjecture was finally verified by Cameron, Praeger, Seitz and Saxl[6] using the classification of simple non-abelian groups. A bound in terms of

15


valency in case of arbitrary transitive permutation groups and doubly primitivevertex stabilizer is established in [16, 17]. On the other hand, Weiss [63] conjec-tured that, if the vertex-stabilizer Gv acts primitively on the neighborhood of vin an arc-transitive action of G on a connected graph X then its size is boundedby a function of the valency of X, see the survey [47] for details. Recently, areduction of the problem to simple groups is done in [9].

Since the Sims’ conjecture was verified a natural problem of determiningpossible vertex-stabilizers of primitive group actions on graphs of fixed (small)valency arises. As was already mentioned this problem was solved for valency3. As concerns valency four, the upper bound 2436 was published in [23]. Inthe recent paper of Li, Lu and Marušič [27] the precise list of 10 possible groupswhich appear as vertex-stabilizers of primitive arc-transitive group actions on4-valent graphs is derived. It follows that a sharp upper bound for the stabilizeris 2432, achieved by the group S4 × S3.

Half-arc-transitive actions on 4-valent graphs. Generally, the struc-ture of vertex-stabilizers of half-arc-transitive group actions on connected graphsis not known. A half-arc-transitive action on connected graphs of valency four isnot primitive and the size of the respective vertex-stabilizer is not bounded. Aswas already mentioned the vertex-stabilizer is a 2-group. Natural candidates areelementary abelian 2-groups, and indeed, it is easy to see that the lexicographicproduct Ch+1[K̄2] admits a half-arc-transitive action of G-height h. To con-struct, (G, 12 ,H)-transitive 4-valent graphs with non-abelian stabilizers is lesstrivial. Actually, as a consequence of the following classification result provedby Marušič and Nedela [37] we obtain that the stabilizer H is almost abelianmeaning the nilpotency class of H is at most two.

Theorem 2.12 Let G act half-arc-transitively on a connected 4-valent graphwith a vertex stabilizer H. Then there are positive integers h, d with 23h ≤ d ≤ hand a set of generators 〈τ1, τ2, . . . , τh〉 of H satisfying the following relations:

(R1) τ2i = 1 for i = 1, 2, . . . , h,

(R2) (τiτj)2 = 1 if 0 < |i− j| < d,

(R3) (τiτj)2 = τ²(j−i,0)h−d+i τ

²(j−i,1)h−d+i+1 . . . τ

²(j−i,2d−2h+j−1)(h−d+i)+2d−2h+j−1

for all 1 ≤ i < j ≤ h,where j − i ≥ d and ²(r, s) ∈ {0, 1}.Furthermore, G = 〈a,H〉 = 〈a, b〉, for some a ∈ G \H, b = τ1a and conju-

gation by a sends τi onto τi+1 for i = 1, 2, . . . , h− 1.On the other hand, let (G,H) be a pair of abstract groups satisfying the

above conditions. Then the action of G on left cosets of H is faithful with aconnected non-self-paired suborbit of length two and point stabilizer H, exceptwhen H ∼= Zh2 is normal in G, or when h = 1, H ∼= Z2 and G is dihedral.

Theorem 2.13 Every group H with presentation of the form

H = 〈τ1, τ2, . . . , τh|(R1), (R2), (R3)〉

embeds into the vertex stabilizer Aut (X)v of a 4-valent vertex-transitive graphX, hence H ∼= H̄ ≤ Aut (X), and there is a ∈ Aut (X) \ H̄ such that X is(G, 12 ,H)-arc-transitive for G = 〈a, H̄〉.

16


Note that the above automorphisms have the following meaning: given G-orientation a, b takes a fixed arc x headed at a vertex v , respectively, onto thetwo arcs xa, xb outgoing from v. The element ai takes a fixed (h−1)-arc x ontoy = xa

i

. The stabilizer of y is of order two and the involution τi is the uniquenon-trivial element in the stabilizer of y.

Half-arc-transitive graphs with prescribed stabilizers. Most of theknown half-arc-transitive graphs are of height 1. In [30] 4-valent half-arc-transitive graphs with stabilizers Z2 × Z2 are constructed. Recently, Marušičgives [34] a construction of 4-valent half-arc-transitive graphs with arbitrar-ily large elementary abelian stabilizers. First example of a 4-valent half-arc-transitive graph with a non-abelian stabilizer (isomorphic to the dihedral groupD8) was constructed by Conder and Marušič [10], see Example 2.9. The follow-ing problem remains open.

Problem 2.1 Given group H with presentation of the form

H = 〈τ1, τ2, . . . , τh|(R1), (R2), (R3)〉

is there a 4-valent half-arc-transitive with vertex-stabilizer isomorphic to H?

2.10 Relations to group actions of other sorts

The same group admits, in general, many different actions. In the followingfew lines we present a brief (and incomplete) list of objects related to half-arc-transitive group actions. As already mentioned if X is a (G, 12 )-arc-transitive4-valent graph oh height h then Plh(X) is a Cayley graph of valency 4 whosegenerators satisfy the relation (ab−1)2 = 1. Hence Cayley graphs of valency4 based on a group G and (G, 12 )-arc-transitive graphs are closely related, formore details see for instance [14, 25, 26, 52]. Given (G, 12 ,H)-transitive graphX with H being cyclic, the construction of medial (hyper)map gives an associ-ated graph Y admitting a one-regular action of G with a cyclic stabiliser [36, 4].A relationship of half-arc-transitive group actions to semi-symmetric group ac-tions on bipartite graphs is investigated in [42, 43]. The latter graphs can beinterpreted as incidence graphs of some geometries. Half-arc-transitive groupactions arise naturally in an investigation of a homogeneous factorisations ofindex 2 which are not symmetric and for which each part is an orbit given byan action of some group [18]. Finally, half-arc-transitive group actions in theframe of association schemes and orbital graphs are studied in [19].

17

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99 (1967), 235–246.

20

Chapter 3

Maps, Regular Maps andHypermaps

3.1 Topological and combinatorial maps, per-mutation representation of maps

Topological maps

A map on a surface is a cellular decomposition of a closed surface into 0-cells called vertices, 1-cells called edges and 2-cells called faces. The verticesand edges of a map form its underlying graph. A map is said to be orientableif the supporting surface is orientable, and is oriented if one of two possibleorientations of the surface has been specified; otherwise, a map is nonoriented .

Typically, a map on a surface is constructed by embedding of a connectedgraph in the surface. Graphs considered in this chapter are finite, non-trivialand connected unless the opposite follows from the immediate context. Edges ofour graphs are of three kinds: links, loops and semiedges. Multiple adjacenciesare allowed. A link is incident with two vertices while a loop or a semiedgeis incident with a single vertex. A link or a loop gives rise to two oppositelydirected darts that are reverse to each other. A semiedge incident with a vertexu gives rise to a single dart initiating at u that is reverse to itself. From thetopological point of view, a semiedge is identical with a pendant edge exceptthat its pendant end-point is not listed as a vertex. Summing up, a graph seenas a topological space is just a finite 1-dimensional cell complex.

The following result relating numerical invariants of maps with the Eulercharacteristic of the supporting surface is well-known.

Theorem 3.1 (Euler formula) Let M be a map on a closed surface S of genusg with v vertices, e edges, s ≤ e semiedges and f faces. Then(1) v − e + s + f = 2− 2g, if S is orientable;(2) v − e + s + f = 2− g, if S is nonorientable.

21


Graphs

For the sake of technical convenience we shall usually replace topological graphsand maps by their combinatorial counterparts. Formally, a (combinatorial)graph is a quadruple G = (D,V ; I, L) where D = D(G) and V = V (G) aredisjoint non-empty finite sets, I : D → V is a surjective mapping, and L = LGis an involutory permutation on D. The elements of D and V are darts andvertices, respectively, I is the incidence function assigning to every dart itsinitial vertex , and L is the dart-reversing involution; the orbits of the group 〈L〉on D are edges of G. If a dart x is a fixed point of L, that is, L(x) = x, then thecorresponding edge is a semiedge. If IL(x) = I(x) but L(x) 6= x, then the edgeis a loop. The remaining edges are links. Let us remark that a similar definitionof graphs appears in Jones and Singerman [91] (see also [150, 138]).

The usual graph-theoretical concepts such as cycles, connectedness, etc.,easily translate to our present formalism.

Category of Maps and Ormaps

As far as maps on surfaces are concerned, there are two essentially differentapproaches to their combinatorial description. The first approach, based on arotation-involution pair acting on darts, involves the orientation of the support-ing surface and so is suitable only for maps on orientable surfaces [70, 91]. Thecorresponding combinatorial structure is called a combinatorial (or, sometimes,algebraic) oriented map. The other approach, using three involutions acting onmutually incident (vertex, edge, face)-triples called flags, is orientation insensi-tive and thus allows us to represent maps on non-orientable surfaces as well [92].The resulting combinatorial structure will be called a combinatorial nonorientedmap. Accordingly, we shall usually employ the same notation for a topologicalmap and for the corresponding combinatorial structure on it.

We start with necessary definitions concerning oriented maps. By a (combi-natorial) oriented map we henceforth mean a triple (D; R,L) where D = D(M)is a non-empty finite set of darts, and R and L are two permutations of D suchthat L is an involution and the group Mon(M) = 〈R, L〉 acts transitively onD. The group Mon(M) is called the oriented monodromy group of M. Thepermutation R is called the rotation of M. The orbits of the group 〈R〉 arethe vertices of M, and elements of an orbit v of 〈R〉 are the darts radiating (oremanating) from v, that is, v is their initial vertex. The cycle of R permutingthe darts emanating from v is the local rotation Rv at v. The permutation L isthe dart-reversing involution of M, and the orbits of 〈L〉 are the edges of M.The orbits of 〈RL〉 define the face-boundaries of M. The incidence betweenvertices, edges and faces is given by nontrivial set intersection. The vertices,darts and the incidence function define the underlying graph M, which is alwaysconnected due to the transitive action of the monodromy group.

An oriented map can be equivalently described as a pair (G; R) where G =(D, V ; I, L) is a connected graph and R is a permutation of the dart-set of Gcyclically permuting darts with the same initial vertex, that is, IR(x) = I(x)for every dart x of G.

Combinatorial nonoriented maps are built from three involutions acting ona non-empty finite set F of flags [92]. A (combinatorial) nonoriented map isa quadruple (F ; λ, ρ, τ) where λ, ρ and τ are fixed-point free involutory per-

22


Figure 3.1: The five Platonic solids

mutations of F = F (M) called the longitudinal , the rotary and the transversalinvolution, respectively, which satisfy the following conditions:

(i) λτ = τλ; and

(ii) the group 〈λ, ρ, τ〉 acts transitively on F .This group is the nonoriented monodromy group Mon(M) of M. We define thevertices of M to be the orbits of the subgroup 〈ρ, τ〉, the edges of M to bethe orbits of 〈λ, τ〉, and the face-boundaries to be the orbits of 〈ρ, λ〉 under theaction on F , the incidence being given by nontrivial set intersection. Note thateach orbit z of 〈λ, τ〉 has cardinality 2 or 4 according to whether z is a semiedgeor not.

Example 3.2 The underlying graph of the tetrahedron is the complete graphK4 = (D, V ; I, L) on 4 vertices. We may set V = {1, 2, 3, 4}, D = {12, 21, 13, 31,14, 41, 23, 32, 24, 42, 34, 43}, L(ij) = ji and I(ij) = i, for any ij ∈ D. Then therotation at vertices compatible with the counterclockwise global orientation in

23


Figure 3.2 is R = (12, 13, 14)(23, 21, 24)(31, 32, 34)(41, 43, 42). Vice-versa, hav-ing R we can identify the triangular faces of the map (D;R, L) via cycles ofthe permutation RL = (12, 24, 41)(21, 13, 32)(31, 14, 43)(23, 34, 42). Figure 3.3shows the same topological map (the tetrahedron) described by means of 3 invo-lutions acting on 24 flags as a map (F ;λ, ρ, τ) in the category of Maps.

2

3

2423

31

34

21

1

4 3243

13

12

14

42

41

Figure 3.2: The tetrahedron described as an oriented map (D; R,L)

ρλτ

Figure 3.3: The tetrahedron described as a map (F ;λ, ρ, τ)

The meaning of the condition on λ, ρ, τ requiring these involutions to befixed-points free becomes clear, if one decides to extend our theory onto mapson surfaces with boundary. This was done by Bryant and Singerman in [35].To do the generalisation we have to allow fixed points of λ,ρ,τ . The underlyingsurface of a map has a non-empty boundary if and only if at least one of λ,ρ,τ

24


fixes a flag. In fact, the category of nonoriented maps is not complete if we donot consider maps on surfaces with boundary, since a homomorphic image of amap on a closed surface can be a map on a surface with a non-empty boundary.As an example, consider an embedding of a cycle in the sphere and its quotienton the disk given by the reflection interchanging the two faces of the map andfixing the embedded graph point-wise.

Clearly, the even-word subgroup 〈ρτ, τλ〉 of Mon(M) has always index atmost two. If the index is two, then M is said to be orientable.

A

A

B

B

C

C

D

E

D

E

Figure 3.4: K6 in the Projective plane

With every oriented map (D; R, L) we associate the corresponding nonori-ented map M\ = (F \; λ\, ρ\, τ \) by setting F \ = D × {1,−1} and defining fora flag (x, j) ∈ D × {1,−1}:

λ\(x, j) = (L(x),−j), ρ\(x, j) = (Rj(x),−j), and τ \(x, j) = (x,−j).Conversely, from an orientable nonoriented map M = (F ; λ, ρ, τ) we can

construct a pair of oriented maps M′ = (D; R,L) and M′′ = (D; R−1, L) thatare the mirror image of each other. We take D to be the set F/τ of orbits of τon F . Let us denote by F+ ⊂ F one of the two orbits induced by the action ofthe even-word subgroup of Mon(M). For a dart {z, τ(z)} = [z], where z ∈ F+,we set R([z]) = [ρτ(z)] and L([z]) = [λτ(z)]. Instead of R we could have takenthe rotation R′([z]) = [τρ(z)], but since R′ = R−1 we get nothing but the mirrorimage – as expected.

Test of orientability

Let M = (F ; λ, ρ, τ) be an nonoriented map. We want to determine whetherthe respective supporting surface S is orientable, or not. The following simplealgorithm is well-known, see [70, 146]. Consider the associated 3-valent graph G,whose set of vertices is F , each vertex f ∈ F is incident with darts (f, λ), (f, ρ)and (f, τ) and the dart reversing involution takes (f, λ) 7→ (λf, λ), (f, ρ) 7→

25


Figure 3.5: Regular embedding of K5 in the torus appears in two ‘enantiomers’,in the category of ORMAPS they are not isomorphic.

(ρf, ρ), (f, τ) 7→ (τf, τ). Note that G is nothing but the dual of the barycentricsubdivision of M. The result follows.

Proposition 3.3 An nonoriented map M = (F ; λ, ρ, τ) is orientable if andonly if the associated 3-valent graph G is bipartite.

Homomorphisms of maps

Let M1 = (D1; R1, L1) and M2 = (D2; R2, L2) be two oriented maps. Ahomomorphism ϕ : M1 → M2 of oriented maps is a mapping ϕ : D1 → D2such that

ϕR1 = R2ϕ and ϕL1 = L2ϕ.

Analogously, a homomorphism ϕ : M1 →M2 of nonoriented mapsM1 = (F1; λ1, ρ1, τ1) and M2 = (F2; λ2, ρ2, τ2) is a mapping ϕ : F1 → F2 suchthat

ϕλ1 = λ2ϕ, ϕρ1 = ρ2ϕ and ϕτ1 = τ2ϕ.

2’

3’

1’

21

4 3

1 2

4’

4 3

Figure 3.6: Cube smoothly covering its halved quotient in the projective plane.

The properties of homomorphisms of both varieties of maps are similarexcept that homomorphisms of nonoriented maps ignore orientation. Everymap homomorphism induces an epimorphism of the corresponding monodromygroups. Indeed, it is not difficult to see that if ψ : (F1; λ1, ρ1, τ1) → (F2; λ2, ρ2, τ2)

26


is a map homomorphism then the assignment λ1 7→ λ2, ρ1 7→ ρ2, τ1 7→ τ2 ex-tends to an epimorphism ψ∗ called the induced epimorphism of the correspond-ing monodromy groups. Furthermore, transitive actions of the monodromygroups ensure that every map homomorphism is surjective and that it also in-duces an epimorphism of the underlying graphs. Topologically speaking, a maphomomorphism is a graph preserving branched covering projection of the sup-porting surfaces with branch points possibly at vertices, face centers or freeends of semiedges. Therefore we can say that a map M̃ covers M if there is ahomomorphism M̃ → M. A map homomorphism is smooth if it preserves thevalency of vertices, the length of faces and does not send a link or a loop ontoa semiedge.

With map homomorphisms we use also isomorphisms and automorphisms.The automorphism group Aut (M) of an oriented map M = (D; R, L) consistsof all permutations in the full symmetry group S(D) of D which commute withboth R and L. Similarly, the automorphism group Aut (M) of an nonorientedmap M = (F ; λ, ρ, τ) is formed by all permutations in the symmetry groupSym(F ) which commute with each of λ, ρ and τ . Hence, in both cases theautomorphism group is nothing but the centralizer of the monodromy group inthe full symmetry group of the supporting set of the map (cf. [91, Proposition3.3(i)]).

Since the action of the monodromy group Mon(M) is transitive, |Mon(M)| ≥|D(M)| for every oriented map M, and |Mon(M)| ≥ |F (M)| for every nonori-ented map M. If the equality is attained, then the monodromy group acts reg-ularly on the supporting set, and therefore the map is called orientably-regularor regular , respectively. As it will become to be clear later the automorphismgroup of an orientably regular map M acts regularly on darts of M, and sim-ilarly Aut (M) of a regular map M acts regularly on flags of M. Our use ofthe term regular map thus agrees with that of Gardiner et al. [62] and Wilson[200], but is not yet standard. For instance, Jones and Thornton [92] uses theterm “reflexible”, and White [192] calls such maps “reflexible symmetrical”. Onthe other hand, our orientably regular maps are called “regular” in Coxeter andMoser [51], “symmetrical” in [21] and [192], and “rotary” in Wilson [200].

For each homomorphism ϕ : M1 → M2 of oriented maps there is the cor-responding homomorphism ϕ\ : M\1 → M\2 defined by ϕ\(x, i) = (ϕ(x), i). IfM1 = M2 = M, that is, ϕ is an automorphism, then this definition and theassignment ϕ 7→ ϕ\ yield the isomorphic embedding of Aut (M) → Aut (M\).This allows us to treat Aut (M) as a subgroup of Aut\(M\) and, consequently,speak that every orientable regular map is orientably-regular (but not necessar-ily vice versa). It is easy to see that the index |Aut\(M\) : Aut (M)| is at mosttwo. If it is two, then the map M is said to be reflexible, otherwise it is chiral .In the former case, there is an isomorphism ψ of the map M = (D; R, L) withits mirror image (D; R

−1, L) called a reflection of M. Clearly, ψ\ is an auto-

morphism that extends Aut (M) to Aut\(M\). Topologically speaking, orientedmap automorphisms preserve the chosen orientation of the supporting surfacewhereas reflections reverse it.

Transitivity of the action of the automorphism group of a regular (orientablyregular) map forces all the vertices to have the same valency and all the facesto have same size (covalency). We say that a map M has a type (p, q) if thecovalency of every face is p and the valency of every vertex of M is q for some

27


integers p, q. Generally, we can define the type of a map to be the couple (p, q)of integers, where p (q) is the least common multiple of covalencies (valencies)of the covalencies (valencies) of faces (vertices) of M.

Homomorphisms between regular maps

Given homomorphism M→N between oriented maps more can be said if oneof M, N , is a regular map. Let G ≤ Aut (M) = (D; R,L) be a subgroup ofthe automorphism group of an oriented map. For any x ∈ D denote [x] theorbit in the action of G containing x. Set R̄[x] = [Rx], L̄[x] = [Lx] for anyx ∈ D and denote D̄ = {[x];x ∈ D}. Then M̄ = (D̄; R̄, L̄) is a well definedoriented map and the natural assignment x 7→ [x] defines a map homomorphismM→ M̄. Homomorphisms arising in the above way are called regular, the groupG ≤ Aut (M) is called the group of covering transformations.

The following statement comes from [140].

Proposition 3.4 Let ϕ : M → N be a map homomorphism. If M is regularthen ϕ is regular. In particular, any homomorphism between regular maps isregular, and moreover, in this case the group of covering transformations is anormal subgroup of Aut(M).

3.2 Generalization to hypermaps, Walsh map ofa hypermap

A topological hypermap H is a cellular embedding of a connected trivalent graphX into a closed surface S such that the cells are 3-colored (say by black, greyand white colours) with adjacent cells having different colours. Numbering thecolours 0, 1 and 2, and labeling the edges of X with the missing adjacent cellnumber, we can define 3 fixed points free involutory permutations ri, i = 0, 1,2, on the set F of vertices of X; each ri switches the pairs of vertices connectedby i-edges (edges labeled i). The elements of F are called flags and the groupG generated by r0, r1 and r2 will be called the monodromy group1 Mon(H) ofthe hypermap H. The cells of H colored 0, 1 and 2 are called the hypervertices,hyperedges and hyperfaces, respectively. Since the graph X is connected, themonodromy group acts transitively on F and orbits of 〈r0, r1〉, 〈r1, r2〉 or 〈r0, r2〉on F determine hyperfaces, hypervertices and hyperedges, respectively. Theorder of the element k = ord(r0r1), m = ord(r1r2) and n = ord(r2r0) is calledthe valency of a hyperface, hypervertex and hyperedge, respectively. The triple(k, m, n) is called type of the hypermap.

Maps correspond to hypermaps satisfying condition (r0r2)2 = 1, or in otherwords, maps are hypermaps of type (p, q, 2) or of type (p, p, 1). Thus we canview the category of Maps as a subcategory of the category of Hypermaps whichis formed by 4-tuples (F ; r0, r1, r2), where ri (i = 0, 1, 2) are (fixed points free)involutory permutations generating the monodromy group Mon(H) acting tran-sitively on F . Similarly, the category of Oriented Hypermaps arises by relaxingthe condition L2 = 1 in the definition of an oriented map. More precisely,an oriented hypermap is a 3-tuple (D; R,L), where R and L are permutations

1This group has been called the monodromy group of H [96, 150], the connection group ofH [200] and the Ω-group of H [26].

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(a) (b)

Figure 3.7: Regular embedding of the Fano plane in the torus gives rise to aregular hypermap of type (3, 3, 3).

acting on D such that the oriented monodromy group is transitive on D. Thenotions defined in the previous section extend from maps to hypermaps in anobvious way. For more information on hypermaps the reader is referred to [48].

The modified Euler formula for hypermaps reads as follows.

Theorem 3.5 (Euler formula for hypermaps) Let H = (F ; r0, r1, r2) be a hy-permap on a closed surface S of genus g having v hypervertices, e hyperedgesand f hyperfaces. Then

(i) v + e + f − |F |/2 = 2− 2g, if S is orientable,(ii) v + e + f − |F |/2 = 2− g, if S is nonorientable.

Walsh representation

An important and convenient way to visualize hypermaps was introduced byWalsh in [189]. Topologically, a map can be seen as a cellular embedding ofa graph in a closed surface and a hypermap as a cellular embedding of a hy-pergraph in a closed surface. Since hypergraphs are in a sense bipartite graphs(with one monochromatic set of vertices representing the hypervertices and theother monochromatic set of vertices representing the hyperedges) a hypermapcan be viewed as a bipartite map, as well. In fact, given any topological hyper-map H we can construct a topological bipartite map W (H), called the Walshbipartite map associated to H by taking first the dual of the underlying 3-valent map and then deleting the vertices (together with the edges attached tothem) lying inside the hyperfaces of H. The resulting map is bipartite with onemonochromatic set of vertices lying on the faces colored black, representing thehypervertices of H, and the other monochromatic set lying on the faces coloredgrey, representing the hyperedges.

This construction can be reversed: given any topological bipartite map B,where the vertices are bipartitioned in black and grey, we construct an associatedtopological hypermap W−1(B) = Tr(B∗) by truncating the dual map B∗; thefaces of the resulting 3-valent map Tr(B∗) contains the vertices and the face-centres of the original map and are henceforth 3-colorable black, grey and white,with all these colours meeting at each vertex of Tr(M∗). If B = W (H) is theWalsh bipartite map of an oriented hypermap H = (D; R, L) then R and L are

29


Figure 3.8: The Walsh map of the Fano plane embedding.

the respective rotations on the two bipartition sets of the dart set of B, so therotation of B is RL = LR.

3.3 Maps, hypermaps and groups

Schreier representations

In the previous section we have seen that maps and hypermaps can be repre-sented by means of two or three permutations satisfying some conditions. Theaim of this section is to show that one can study hypermaps as purely grouptheoretical objects. The idea emerges from the fact that every transitive permu-tation group is equivalent to a group acting on cosets by translation. Following[183, 184], we call these representations Schreier representations.

Schreier representations of oriented maps appear implicitly in Jones andSingerman [91]. Vince [183] developed a theory of Schreier representations of(hyper)maps on closed surfaces described by three involutions. Here we intro-duce Schreier representations of oriented hypermaps.

Let G be a finite group generated by two elements r and `. In other words,G is a finite quotient of some triangle group T+(k, m, n) = 〈r, `; `n = rm =(r`)k = 1〉. k,m and n being positive integers. Further let S be a subgroup ofG. Using the action of G on the set C = G/S of left cosets of S in G by the lefttranslation, we construct a hypermap A(G/S; r, `) whose monodromy group isa homomorphic image of G and the local monodromy group is a homomorphicimage of S. We take the cosets as darts of the hypermap and define the rotationR and the dart reversing involution L by setting

R(hS) = rhS, (3.1)L(hS) = `hS, (3.2)

respectively, hS being an arbitrary element of C. For the resulting hypermap(C; R, L) = A(G/S; r, `) we easily check that the assignment r 7→ R, ` 7→ Lextends to a homomorphism T+(k, m, n) → Mon(A(G/S; r, `)).

Let H be a hypermap. A Schreier representation of H is an isomorphismH → A(G/S; r, `) for an appropriate group G = 〈r, `〉 and a subgroup S ≤ G, orsimply the hypermap A(G/S; r, `) itself. Given any hypermap H = (D; R, L),it is not difficult to find a Schreier representation for H. Indeed, we first fixany dart a of H and set G = Mon(H) = 〈R,L〉 and S = Mon(H, a), to be

30


the stabilizer of a. Then, for an arbitrary dart x we take any element h ∈ Hwith h(a) = x and label x by the coset hS ∈ C, thereby obtaining a labelingα(x) = hS. Observe that α is well-defined since for any two elements h and h′

of Mon(M) with h(a) = x = h′(a) we have hS = h′S. In fact, α is a bijection ofD(H) onto C. Clearly, α(Rx) = Rα(x) and α(Lx) = Lα(x) which means thatα : H → A(G/S; R, L) is the required isomorphism.

If we start from a given hypermapH, the Schreier representation we have justdescribed is in some sense best possible because the monodromy group Mon(H)is not merely a homomorphic image of G but is actually isomorphic to it. In thiscase we say that the Schreier representation is effective. In general, a Schreierrepresentation A(G/S; r, `) is effective if and only if G acts faithfully on C, i.e.,the translation by every non-identity element of G is a non-identity permutationof C. Elementary theory of group actions or straightforward computations yieldthat the latter occurs precisely when the subgroup

⋂h∈G

hSh−1, the core of S in

G, is trivial (cf. Rotman [160]).For an arbitrary Schreier representation A(G/S; r, `) of a hypermap H we

have Aut (H) ∼= NG(S)/S, where NG(S) is the normalizer of S in G (seeProposition 3.7 in [50]). In particular, if H is orientably regular we can takeG = Mon(H) and S = 1. Then Aut (A(G/1; r, `)) ∼= G ∼= Mon(A(G/1; r, `)),implying that Aut (H) ∼= Mon(H). Let us remark that the above isomorphismassigns the left translation by an element h ∈ G (representing a monodromy ofH) to the right translation ξh (representing an automorphism of H). Summingup we get the following theorem.

Theorem 3.6 Let H = (D; R, L) be an oriented hypermap. Then |Aut (H)| ≤|D| ≤ |Mon(H)| and the following conditions are equivalent:

(i) H is orientably regular,(ii) Mon(H) ∼= Aut (H),(iii) the action of Aut (H) on D is regular.In order to get a similar characterization of regular (nonoriented) hypermaps itsuffices to replace, in the above statement, an oriented hypermap by a hypermapand darts by flags.

Schreier representations provide a convenient tool to deal not only withautomorphisms but also with homomorphisms between hypermaps. If

G = 〈r, `; `n = rm = (r`)k = 1, . . . 〉is a finite quotient of the triangle group T+(k, m, n) and S ≤ S′ ≤ G are twosubgroups then the natural projection π : G/S → G/S′, hS 7→ hS′ (h ∈ G),is a homomorphism A(G/S; r, `) → A(G/S′; r, `). In fact, every hypermaphomomorphism ϕ : H1 → H2, where Hi = (Di; Ri, Li), is in the usual senseequivalent to an appropriate natural projection.

Generic hypermap

One consequence of these considerations is that every oriented hypermap isa quotient of a (finite) oriented regular hypermap. In fact, for every ori-ented hypermap H = (D; R, L) there exists a regular hypermap H# and a

31


homomorphism π : H# → H with the following universal property: for everyregular hypermap H̃ and a homomorphism ϕ : H̃ → H there is a homomor-phism ϕ′ : H̃ → H# such that ϕ = πϕ′. In terms of Schreier representations,the homomorphism π is equivalent to the natural projection A(G/1; R, L) →A(G/S; R, L) ∼= H where G = Mon(H) and S = Mon(H, a), is the stabilizerof some dart a ∈ D(H). We shall call the hypermap H# the generic regularhypermap over H and π : H# → H the generic homomorphism. It is obviousthat the induced homomorphism π∗ : Mon(H#) → Mon(H) is an isomorphismand that H# and H have the same type.

Construction of generic hypermap

To construct the generic hypermap H# = (D#; R#, L#) for an oriented hyper-map H = (D; R, L) it is sufficient to set D# = Mon(H), R#(x) = Rx, andL#(x) = Lx for any x ∈ D#. Observe that the automorphisms of H# arejust the right translations of D# = Mon(H) by the elements of Mon(H), and soH# is indeed an orientably-regular hypermap. Similarly, ifH = (F ; λ, ρ, τ) is annonoriented hypermap then the generic regular hypermapH+ = (F+; λ+, ρ+, τ+)over H can be constructed by setting F+ = Mon(H), λ+(x) = λx, ρ+(x) = ρxand τ+(x) = τx, for any x ∈ F+. Again, the hypermap automorphisms aregiven by the right translations of F+ by the elements of Mon(H) = F+.

It is obvious that if H is orientable, then so is H+. Moreover, the hypermapH+, as a topological hypermap, smoothly covers (H#)\.

In the following sections we shall see that maps or Walsh bipartite hypermaprepresentations combined with the generic hypermap construction provide aconvenient tool for construction regular hypermaps satisfying certain constrains(see for instance [90, 151]). Let us note that there is an interesting relationshipbetween hypermaps and algebraic curves ( see Section 5). Via this relationship,actions of Galois groups of algebraic number fields on maps on surfaces areinvestigated (see [71, 96, 95, 165]). In this context the base maps are (followingGrothendieck) called dessigns d’enfants.

Example 3.7 Figure 3.9 shows spherical maps M1, . . .M5 which generic mapsare the five Platonic solids. The respective (oriented) monodromy groups are A4,S4, S4, A5 and A5. Dessign d’enfants M6 and M7 represent regular maps onsurfaces with higher genera. The associated monodromy groups are the projectivelinear group PSL(2, 7) and Mathieu group M12, respectively. The generic mapM#6 is known as the dual of the Klein’s triangulation of the surface of genus 3and this map is the smallest Hurwitz map (see the following Section).

Maps and hypermaps from triangle groups

The above theory of Schreier representations apply without any problem toinfinite hypermaps as well. It follows that oriented maps and hypermaps ofgiven type (k, m, n) can be described as quotients of the universal orientedhypermap of type (k, m, n) which (oriented) monodromy group is T+(k,m, n).This is the even-word subgroup of the triangle group

T (k,m, n) = 〈r0, r1, r2; r20 = r21 = r22 = (r1r2)k = (r0r1)m = (r2r0)n = 1〉,

32


M4

M3M2M1

M5

M6 M7

Figure 3.9: Dessign d’enfants

which is the monodromy group of the universal hypermap A(T (k, m, n); r0, r1, r2)for the category of (nonoriented) hypermaps of type (k,m, n). It follows that

T+(k, m, n) = 〈R, L; (RL)k = Ln = Rm = 1〉.

Note that the universal maps of type (k,m) with the monodromy group T (k,m, 2)are the well known tessellations of the sphere, plane or hyperbolic plane byk-gons (m of them meeting at each vertex) provided the expression 1k +

1m

is greater, equal or less than 12 , respectively. It is well-known that a free 2-generator group can be represented as a matrix-group [159, pages 48-49]. Denoteby ν = 2cos(π/k), η = 2cos(π/m) and ξ = 2cos(π/n). Set

r =

1 ξ νξ + η0 −1 −ν0 ν ν2 − 1

(3.3)

` =

−1 −ξ 0ξ ξ2 − 1 0η ηξ + ν 1

. (3.4)

Then the assignment R 7→ r and L 7→ ` extends to a group monomorphism.Hence triangle groups are matrix groups, see [171, 172, 1] for more details.It follows that (hyper)maps can viewed as quotients of certain matrix groups,while regular (hyper)maps correspond to factor groups of the matrix groupsrepresenting the respective triangle groups. This approach to maps and hyper-maps is used to study regular maps and hypermaps of large planar width, seeSection 3.4.

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Hypermap subgroups

We can go even one step further. Let us denote by

G = T (∞,∞,∞) = 〈r0, r1, r2; r20 = r21 = r22 = 1〉,

the free product of three two-element groups. Since the monodromy group ofany hypermap H is a finite quotient of G we can identify every hypermap withthe algebraic hypermap A(G/S; r0, r1, r2) for some S ≤ G of finite index. Thesubgroup S is called the hypermap subgroup. Consequently, one can study hy-permaps via the subgroups of G of finite index. The facts listed in the followingstatement are well-known between map- and hypermap experts (see [48, 50]).

Theorem 3.8 Let H, H1 and H2 be hypermaps, and let G = 〈r0, r1, r2; r20 =r21 = r

22 = 1〉.

(a) H1 covers H2 if and only if there are S1 ≤ S2 ≤ G such that H1 ∼=A(G/S1; r0, r1, r2) and H2 ∼= A(G/S2; r0, r1, r2),

(b) H1 ∼= H2 if and only if the corresponding hypermap subgroups are conju-gate in G,

(c) H is orientable if and only if its hypermap subgroup is contained in theeven-word subgroup G+ ≤ G,

(d) the hypermap subgroup of the nonoriented generic hypermap for a hyper-map given by hypermap subgroup S ≤ G is the largest normal subgroupcontained in S. In particular, regular hypermaps correspond to normalsubgroups of G.

Using the algebraic representation via hypermap subgroups one can handlemany problems. For instance, it is straightforward that given two hypermapsH1, H2 with the respective subgroups S1, S2, the intersection S1∩S2 defines thesmallest common cover for bothH1 andH2. Because of many advantages the in-vestigation of maps and hypermaps via the corresponding hypermap subgroupsposses, sometimes a hypermap itself is identified with its hypermap subgroup(see [15, 61]).

Product of hypermaps

The smallest common cover of given hypermaps H1 = (F1; r0, r1, r2) and H2 =(F2; q0, q1, q2) can be viewed as a product of two hypermaps. The questionarises whether we can construct the product explicitly, or in other words, canwe derive the monodromy group of the product in terms of the monodromygroups of factors? The most natural approach is to set F = F1 × F2 andpi(x, y) = (rix, qiy) for any (x, y) ∈ F and i = 0, 1, 2. Unfortunately, thehypermapH = (F ; p0, p1, p2) is, in general, not correctly defined since the actionof Mon(H1) ×Mon(H2) may not be transitive on F . Necessary and sufficientcondition ([33, 25]) follows.

Theorem 3.9 [33] Let H1, H2 be hypermaps with the hypermap subgroups S1and S2. Then the monodromy group of the smallest common cover for H1, H2is the direct product Mon(H1)×Mon(H2) if and only if T (∞,∞,∞) = 〈S1, S2〉.

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The above theorem deals with a special instance of a general situation. LetH = U/H and K = U/K be algebraic hypermaps. Set H ∨ K = U/(H ∩ K)and H ∧ K = U/〈H, K〉. The hypermaps H ∨ K, H ∧ K will be called join andintersection of H and K, respectively. If both hypermaps are regular then thehypermap subgroup is unique and join and intersection are well-defined binaryoperators in the category of regular hypermaps. Joins and intersections arestudied in [29]. Join and intersection of regular hypermaps can be constructedusing the following operation introduced by Wilson in [203].

Let A = 〈r0, . . . , rk〉 and B = 〈s0, . . . , sk〉 be two k-generated groups. Let usdefine their monodromy product A×mB to be the subgroup of the direct productgenerated by (ri, si), where i = 0, 1, . . . , k. Note that S. Wilson calls it theparallel product in [203]. Further, denote by π1 : A×mB → A, π2 : A×mB → Bthe natural projections erasing the second and first coordinate, respectively. Thefollowing theorem relates the above operations.

Theorem 3.10 [29] Let H = (A; r0, r1, r2) and K = (B; s0, s1, s2) be regularhypermaps. Then Mon(H ∨ K) = Mon(H) ×m Mon(K) and Mon(H ∧ K) =Mon(H)×m Mon(K)/Ker π2Ker π1.

Two regular hypermaps H, K will be called orthogonal if the respectivehypermap subgroups generate T (∞,∞,∞) = 〈H, K〉. It follows that orthogo-nality of hypermaps is the necessary and sufficient condition under which themonodromy product is simple the direct product of the monodromy groups offactors. Joins and intersections of maps and hypermaps allow us to constructmaps and hypermaps satisfying some special properties, for instance, self-dualor totally self-dual hypermaps, reflexible hypermaps. These operations play animportant role in the study of the phenomenon of chirality which we are goingto deal with in the next paragraph.

Chirality index

Orientable hypermaps split into two families: family of reflexible and family ofchiral hypermaps. Reflexible hypermap is mirror symmetric which means thatthe two associated oriented hypermaps (D; R, L) and (D; R−1, L−1) are iso-morphic. Topologically speaking, a reflexible hypermap admits an orientationreversing self-homeomorphism of the supporting surface preserving the embed-ded graph and colours of faces. Note that S. Wilson [196] and some other authorsuse the term reflexible to denote (reflexible) regular hypermap. An orientablehypermap which is not reflexible is chiral (or mirror asymmetric).

From the first point of view the “chirality” seems to be a binary invariantfor the category of orientable hypermaps. Surprisingly, it turns out [32] thatone can measure by a group, called the chirality group of a hypermap, of howmuch given orientable hypermap deviates from being mirror symmetric. For thesimplicity let us restrict to orientably regular hypermaps now. Let us denoteby H∆ the largest (reflexible) regular hypermap covered by H and by H∆ thesmallest (reflexible) regular hypermap covering H. It follows that H∆ = H∨Hrand H∆ = H∧Hr, where Hr denotes the mirror image of H. With the help ofhypermap subgroup representation the following result is proved in [32].

Theorem 3.11 [32] Let H be an orientably regular hypermap. Then there existsa finite group G = X(H) of size κ such that H∆ is a smooth κ-fold cover of

35


Figure 3.10: The smallest chiral orientably regular map is toroidal {4, 4}2,1

H, and H is a κ-fold cover of H∆. Moreover, both coverings are regular, withgroups of covering transformations isomorphic to G.

The above defined group X(H) is called the chirality group while its size κ isthe chirality index. It follows that X(H) is trivial if and only if H is reflexible.Structure of chirality groups is studied in [32] in a more detail. It is proved therethat every abelian group is the chirality group of an oriented regular hypermap.On the other hand, many non-abelian groups including symmetric groups anddihedral groups cannot serve as chirality groups.

Two-generator groups

Finite groups generated by two involutions coincide with dihedral groups. Anyfinite 2-generator group can be interpreted as a monodromy group of a regularoriented hypermap, while groups generated by three involutions give rise to(nonoriented) regular hypermaps. A lot of finite groups belong to one or to theother above mentioned classes of groups. The following related problem can befound in the Kourovka notebook.

Problem 3.1 [114, Problem 7.30] Characterize the finite simple groups gener-ated by three involutions, two of which commute.

Note that a regular map (F ; λ, ρ, τ) with a simple monodromy group is necessar-ily non-orientable. A similar problem was considered by Malle, Saxl and Weigel[134]. They proved that every non-abelian simple group can be generated bytwo elements, one of them being of order two. In other words, every non-abeliansimple group is a monodromy group of some oriented regular map. Interestingproblem of classification of all possible couples R, L (L2 = 1) of generators (upto the action of Aut (G)) for a given group G arises. Having in mind the aboveresult, the solution of the problem for the class of (finite) simple groups meansto prove a refinement of the classification of simple groups. Although, in generalthis problem seems to be intractable, at least for some groups it can be solved,see for instance Sah [161]. Moreover, using the Hall’s counting principle [75]one can calculate the number of nonisomorphic pairs of generators in terms of

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group characters (see [59, 93]). Regular hypermaps which monodromy coincidewith the monodromy groups of the five Platonic solids are classified in [27]

3.4 Regular maps of large planar width and resid-ual finiteness of triangle groups

While regularity (in either arithmetical or group-theoretical sense) is an obviousproperty of any generalization of the Platonic solids, it is somewhat less obviouswhether or how their planarity should be generalized. The idea is to replace theglobal planarity of the Platonic solids by a certain local variant of this notion.This should guarantee that a sufficiently “large” neighbourhood of each face issimply connected. Recent works in topological graph theory (cf. [145, 157, 158,179]) suggest the following concept as a convenient measure of local planarity.A map M on a closed surface S other than the 2-sphere is said to have planarwidth at least k, w(M) ≥ k, if every non-contractible simple closed curve onS intersects the underlying graph of M in at least k points. Planar width(most often called “face-width” or “representativity”) has recently received aconsiderable attention as an important tool for the study of graph embeddingson surfaces [146, Chapter 5]. The following theorem presents the main resultof [151]. Its proof consists in construction of a certain planar map Mw(p, q) oftype (p, q) for which the generic map construction applies.

Theorem 3.12 [151] For every pair of integers p ≥ 3 and q ≥ 3 such that1/p + 1/q ≤ 1/2 and for every integer k ≥ 2 there exists an orientable regularmap M of type (p, q) with planar width w(M) ≥ k. Moreover, we can requirethe map M to be reflexible.

This theorem has several predecessors in the literature.

Grünbaum’s problem

In 1976, Grünbaum [72] asked if for every pair of positive integers p and q with1/p + 1/q < 1/2 (i.e., in the hyperbolic case) there are infinitely many finiteregular maps of type (p, q). He also remarked, however, that it was not evenknown whether for such p and q there was at least one map of that type. Thequestion was answered in the affirmative by Vince [184] (1983) within a moregeneral framework of higher-dimensional analogues of regular maps. His proof,based on a theorem of Mal’cev saying that every finitely generated matrix groupis residually finite (see, e.g., Kaplansky [103]), was non-elementary and non-constructive. Constructive proofs of Vince’s theorem were subsequently givenby Gray and Wilson [63] and Wilson [199, 202] along with some refinements.Further constructions of regular maps of each type (p, q) have recently beengiven by Jendrol’ et al. [90] and Archdeacon et al. [6]. Perhaps the mostelementary affirmative solution of the Grunbaum’s problem is obtained in [90]giving the dessing’s d’enfants for each hyperbolic type (p, q) and deriving therespective generic covering maps, see Figure 3.11, where p = qt + r, for some0 ≤ r < q and t ≥ 1.

Parallel to this development there is another line of research which is closelyrelated to our main theorem. The bridge between the two streams is the obser-vation that an orientable regular map of type (p, q) exists if and only if there is

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p−2

p−2

r

q−r

r

q−r−1q−2q−2q−2q−1

a) b)

c)

Figure 3.11: Dessigns d’enfants of hyperbolic types

a finite group with presentation G = 〈x, y; xq = y2 = (xy)p = 1, . . . 〉 formingthe oriented monodromy group of the oriented regular map A(G; x, y). Withthis relationship in mind, the solution of the above Grünbaum’s (p, q)-problemcan be derived from an old result (1902) of Miller [144] (rediscovered by Fox[60] in 1952) which states:

Theorem 3.13 (Miller [144]) For any three integers p, q, and n, all greaterthan 1, there exist infinitely many pairs of permutations α

Group Actions on Graphs, Maps and SurfacesDoc. RNDr. Roman Nedela, CSc. Group Actions on Graphs,...

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