Highly Symmetric Maps and Dessins

Highly symmetric maps and dessins

Gareth A. Jones School of Mathematics University of Southampton Southampton SO17 1BJ, U.K. and Institute of Mathematics and Computer Science Matej Bel University Bansk´aBystrica, Slovakia [email protected]

Abstract These are lecture notes for a short course given at a summer school in Nov´ySmokovec, Slovakia, in September 2015. The main subject-matter concerns maps on surfaces, and related structures such as hypermaps and dessins d’enfants, together with their automorphisms. The lectures concentrate on the most symmetric objects in these categories, and outline techniques for enumerating and classifying them in terms of their embedded graph, their underlying surface, and their automorphism group.

1 Oriented maps and permutations

A map is an embedding of a graph G in a surface S, without crossings, so that the faces (connected components of S\G) are simply connected. For simplicity, let us first assume that the surface S is orientable, with a chosen orientation. There is a very efficient way of describing such a map using a pair of permutations, an idea first discovered by Hamilton [28] in 1856, though it has been rediscovered several times by others since then (see [18], for example). For full details, see [40].

1.1 Monodromy groups and automorphism groups Given an oriented map M, let Φ be the set of its arcs (directed edges). Define x to be the permutation of Φ which follows the chosen orientation around each incident vertex, so cycles of x correspond to vertices of M. Define y to be the permutation which reverses the direction of each arc, so cycles of y correspond to edges of M. (Notice that if we define z by xyz = 1 then cycles of z = (xy)−1 = y−1x−1 = yx−1 correspond to faces of M.)

1 orientation

αx αz α αy

Figure 1: The permutations x, y and z

Clearly y2 = 1, so let Γ be the group with the following equivalent presen- tations (in terms of generators and deﬁning relations)

Γ = hX,Y | Y 2 = 1i = hX,Y,Z | Y 2 = XYZ = 1i,

and deﬁne an action θ :Γ → Sym Φ (the symmetric group on Φ) of Γ on Φ by means of the homomorphism

X 7→ x, Y 7→ y (and so Z 7→ z).

Note that Γ can be expressed as the free product C∞ ∗ C2 of the groups

∼ 2 ∼ hX | −i = C∞ and hY | Y = 1i = C2.

Conversely, given any permutation representation of Γ, one can reconstruct an oriented map M with vertices, edges and faces corresponding to the cycles of X, Y and Z, incidence given by non-empty intersection, and orientation given by the cyclic order within cycles. Connected maps correspond to transitive representations, compact maps to ﬁnite representations, etc.

1 3 4 2

Figure 2: A map on the sphere

Example Let M be the map on the sphere on the left of Figure 2 (drawn, by stereographic projection, in the plane). If we number the arcs as on the right, we ﬁnd that

x = (1, 2, 3)(4) = (1, 2, 3) and y = (1, 2)(3, 4),

2 so that z = (1)(2, 3, 4) = (2, 3, 4) corresponding to two faces of valencies 1 and 3. (The edge with arcs 3 and 4 contributes two sides to the face containing it.) Warning 1 The permutations x, y and z are, in general, not automorphisms of the map or the graph, since they do not preserve incidence. They are simply instructions on how to join arcs together to form a map. Warning 2 A ﬁxed point of y corresponds to a free edge, or semi-edge, that is, an edge incident with a vertex at only one end, as in Figure 3: think of this as the quotient of an ordinary edge by the half-turn about its mid-point. In most of the examples we will consider, there will be no free edges, and hence y will have no ﬁxed points, so I will not spend much time on this point.)

Figure 3: A free edge

Example If we remove the vertex of valency 1 from the preceding example, as in Figure 4, there are just three arcs. Now we have

x = (1, 2, 3) and y = (1, 2)(3) = (1, 2),

so that z = (1)(2, 3) = (2, 3) corresponding to two faces of valencies 1 and 2. (The free edge contributes only one side to the face containing it.)

1 3 2

Figure 4: Another map on the sphere

1.2 Monodromy groups and automorphism groups

The permutation group G = hx, yi ≤ Sym Φ generated by x and y is called the monodromy group of M. The (orientation-preserving) automorphisms of M are the permutations of Φ which commute with x and y, so the (orientation- preserving) automorphism group A = Aut M is identiﬁed with the centraliser C(G) of G in Sym Φ, the group of all permutations commuting with those in G. Without loss of generality, we may assume from now on that G and S are connected (otherwise, consider connected components separately). This is

3 equivalent to assuming that G is transitive on Φ, so that the action of G on Φ is equivalent to its action on the cosets of the stabiliser Gα := {g ∈ G | αg = α} of some arc α ∈ Φ. This means that one can study maps by considering the internal structure of their monodromy groups. To make progress we need some terminology. A permutation group is semiregular, or fixed point free, if only the identity element fixes a point. It is regular if it is transitive and semiregular (so exactly one group element takes any given point to another). Just as any transitive action of a group can be identified with its action on the cosets of a point stabiliser, any regular action can be identified with its action on itself, either by right multiplication ρg : a 7→ ag or −1 by left multiplication λg : a 7→ g a. (We need inverses here so that, composing from left to right, λg ◦ λh = λgh.) For any group G, the regular permutation groups R(G) = {ρg | g ∈ G} and L(G) = {λg | g ∈ G} are isomorphic (to G), but they are distinct subgroups of Symm G unless G is abelian. They commute with each other, by the associative axiom for groups.

Lemma 1.1 Let a group G act transitively on a set Φ, and let A be its centraliser in Symm Φ. Then A acts semiregularly on Φ, and ∼ A = NG(Gα)/Gα for each α ∈ Φ.

−1 (Here NG(Gα) is the normaliser {h ∈ G | h Gαh = Gα} of Gα in G.) Proof. Suppose that a ∈ A, and a ﬁxes some α ∈ Φ. If β ∈ Φ then β = αg for some g ∈ G since G acts transitively on Φ. Then βa = αga = αag = αg = β. Thus a ﬁxes every β ∈ Φ, so a = 1 and hence A acts semiregularly on Φ.

Exercise. Finish the proof. (Hint: consider the points ﬁxed by Gα.) Lemma 1.1 implies that the automorphism group A of a connected map M acts semiregularly on the set Φ of arcs, and that ∼ ∼ A = NG(Gα)/Gα = NΓ(M)/M for any arc α ∈ Φ, where M = Γα (called a map subgroup of Γ) is the inverse image of Gα in Γ (both unique up to conjugation).

Exercise Find G, Gα, NG(Gα) and A, where G is a path graph with n edges and n + 1 vertices, embedded in the sphere (or, by stereographic projection, drawn in the plane).

Figure 5: A path graph

4 1.3 Orientably regular maps The most symmetric oriented maps M are the orientably regular maps (sometimes simply called regular, as in the classic book by Coxeter and Moser [12]), those for which A acts transitively on Φ. Lemma 1.1 then implies:

Lemma 1.2 If M is an oriented map then the following are equivalent: •M is an orientably regular map, • A is a regular permutation group, • G is a regular permutation group, • M is a normal subgroup of Γ. When these conditions are satisﬁed we have

A =∼ G =∼ Γ/M.

Exercise. Prove this lemma. Indeed, in the case of an orientably regular map, A and G can be identiﬁed with the left and right regular representation of the same group. The arcs are identiﬁed with the elements of G, and the vertices, edges and faces with the cosets ahxi, ahyi and ahzi (a ∈ G) of the cyclic subgroups hxi, hyi and hzi of G generated by x, y and z. The automorphism group A then consists of the left −1 ∼ multiplications λg : a 7→ g a for g ∈ G, forming a group L(G) = G.

Figure 6: The tetrahedron

Example. The platonic solids can all be regarded as orientably regular maps on the sphere. For the tetrahedron (embedding the complete graph K4 with triangular faces) we have ∼ ∼ ∼ A = G = Γ/M = A4, where A4 is the alternating group of degree 4, the group of even permutations of the vertices. (The odd permutations are excluded, since they reverse orientation — we will return to this later.) The 3-cycles in A4 correspond to the

5 rotations of order 3 around the vertices, and the double transpositions correspond to the rotations of order 2 around the midpoints of opposite edges. For the other platonic solids, the corresponding groups are the symmetric group S4 for the cube and its dual, the octahedron, and the alternating group A5 for the dodecahedron and its dual, the icosahedron. In an orientably regular map, all the vertices have the same valency n, and all the faces have the same valency (number of sides) m, so the generators of G satisfy xn = y2 = zm = xyz = 1 (and usually other relations), and the map is said to have type {m, n} in the notation of Coxeter and Moser [12]. (For a more general map the orders n and m of x and z are the least common multiples of the valencies of the vertices and faces.) Thus the tetrahedron has type {3, 3}, the cube and the octahedron have types {4, 3} and {3, 4}, and the dodecahedron and the icosahedron have types {5, 3} and {3, 5}. If M is an orientable regular map of type {m, n} then the numbers of vertices, edges and faces are |G| |G| |G| V = ,E = ,F = , n 2 m so the surface has Euler characteristic 1 1 1 χ = V − E + F = |G| − + (1) n 2 m and genus χ |G| 1 1 1 g = 1 − = 1 − − − . (2) 2 2 2 m n

Exercise. What are the possible types of orientably regular maps on the sphere and the torus? (Take m, n > 1 to avoid trivial cases.) An isomorphism between maps is an isomorphism of their embedded graphs, which extends to a homeomorphism between their underlying surfaces (preserving orientation if they are oriented). If the maps have monodromy groups G = hx, yi and G0 = hx0, y0i acting on sets Φ and Φ0 of arcs, this corresponds to a bijection Φ → Φ0, a 7→ a0 such that (ax)0 = a0x0 and (ay)0 = a0y0 for all a ∈ Φ, in which case there is an isomorphism G → G0, x 7→ x0, y 7→ y0. Morphisms are deﬁned similarly, as functions (not necessarily bijections) Φ → Φ0 which commute with the actions of Γ. In this way the permutation representations of Γ (indeed, of any group) form a category CΓ. (More about that later.) The following result shows that, in principle, one can study all oriented maps by restricting attention to the orientably regular maps and their automorphism groups: Theorem 1.3 Every oriented map M is isomorphic to the quotient N /S of an orientably regular map N , of the same type, by a subgroup S ≤ Aut N . If M is compact then N can also be chosen to be compact.

6 Proof. Let G be the monodromy group of M, and let M be a corresponding map subgroup of Γ. Let N be the core of M in Γ, that is, the intersection of the conjugates of M in Γ, or equivalently the kernel of the action of Γ on the cosets of M. Since N is a normal subgroup of Γ, it corresponds to an orientably regular map N , for which the monodromy group is the regular representation of G (=∼ Γ/N), with the same generators x, y and z so that N has the same type as M. Then M is isomorphic to the quotient of N by the subgroup S ≤ Aut N corresponding to the subgroup M/N ≤ Γ/N =∼ Aut N . If M is compact then M has finite index in Γ, and hence so does N, so N is compact. In fact, the core of M is the largest normal subgroup of Γ contained in M, so the map N constructed here is the ‘smallest’ orientably regular covering of M, in the sense that any other orientably regular covering of M also covers N . Example The map M of type {3, 3} in Figure 7 is clearly not orientably regular. As in the example in §1.1, we can number its arcs 1,..., 4 so that its defining permutations are x = (1, 2, 3) (fixing 4) and y = (1, 2)(3, 4), and hence the monodromy group G is hx, yi = A4. The orientably regular map N of type {3, 3} corresponding to G is the tetrahedron (see Figure 6), and we have M =∼ N /S ∼ for the subgroup S ≤ Aut N = A4 generated by a rotation of N of order 3.

Figure 7: A quotient of the tetrahedron

In the above example, M and N have the same genus, namely 0, but unfortunately the genus of N is usually much larger than that of M:

Exercise The permutations x = (1, 2, . . . , n) and y = (1, 2) in Sn deﬁne an oriented map M of genus 0. Draw M, and calculate the genus g of the corresponding orientably regular covering map N . What happens to g as n → ∞?

1.4 Chirality If M is any oriented map, its mirror image M is the map with the same set Φ of arcs, but with the permutations x and y replaced with their inverses (in fact, y = y−1 so it is sufficient to invert just x). A map is reflexible if it is isomorphic to its mirror image (equivalently, the monodromy group G has an automorphism inverting x and y), otherwise they form a chiral pair. An orientably regular map is fully regular (or simply regular) if it is also reflexible. The platonic solids (indeed all the orientably regular maps on the sphere) have this property, but there are examples on the torus, and on various surfaces of higher genus, which do not. Example. In each of the diagrams in Figure 9, identifying opposite sides of the outer square produces an orientably regular map of type {4, 4} on the torus (in

7 Figure 8: A chiral pair of maps

fact, an embedding of the complete graph K5, since the four vertices at the cor- ners are identiﬁed with each other). These two oriented maps are mirror images of each other, but they are not mutually isomorphic (see the next exercise), so they form form a chiral pair.

Figure 9: A chiral pair of orientably regular torus embeddings of K5

Exercise. Show that the two maps in Figure 9 are orientably regular, and are not isomorphic to each other. What are their automorphism groups?

2 Classiﬁcation by graph: simple cases

One can try to classify orientably regular maps in several ways, in terms of their automorphism groups, their underlying surfaces, or their embedded graphs. In all three cases partial results are available, some obtained by applying deep mathematical techniques or extensive computer searches. In the case of classi- ﬁcation by graph, one can restrict attention to arc-transitive graphs, since the automorphism group of an orientably regular map acts transitively on its arcs. In [22], Gardiner, Nedela, Sir´aˇnandˇ Skovieraˇ have described a systematic way of doing this by characterising those groups of automorphisms of a graph which can arise. I shall start by considering the cases where the graph has at most two vertices, returning later to consider more complicated arc-transitive graphs.

2.1 Graphs with one vertex If a graph G has only one vertex, then any edge must be either a free edge or a loop, and if G is arc-transitive then they are all free edges or all loops. If G has n free edges, then in any orientably regular embedding, the generator x

8 must permute the n arcs cyclically, while y ﬁxes them, so the monodromy group G = hx, yi = hxi is a cyclic group of order n. The element z = (xy)−1 = x−1 has a single cycle on the arcs, so there is a single face, and the embedding is ∼ a ‘star map’ on the sphere, with automorphism group A = Cn (see Figure 10 for the case n = 8). We will therefore assume from now on that G has no free edges.

Figure 10: A star map with eight free edges

It follows that G is a bouquet of circles Bn, that is, a single vertex v with n edges, all loops, attached to v. If any oriented map M is an embedding of Bn, then the generator x of the monodromy group G permutes the 2n arcs in ∼ a single cycle, so G contains a subgroup hxi = C2n. If M is orientably regular then G acts regularly on Φ, so |G| = |Φ| = 2n and hence ∼ G = hxi = C2n. Now y has order 2, so it must be the unique element xn of this order, and hence −1 n−1 z = (xy) = x . Given n, this deﬁnes the map M up to isomorphism, so Bn ∼ has a unique orientably regular embedding, with automorphism group A = C2n. The face valency is the order of z, namely 2n m = , gcd(n − 1, 2n) so the number of faces is |G| F = = gcd(n − 1, 2n). m If n is even then n − 1 and 2n are coprime, so M has one 2n-gonal face and hence has genus n/2. If n is odd then gcd(n − 1, 2n) = 2, so M has two n-gonal faces and its genus is (n − 1)/2. In either case, M can be formed by placing a vertex at the centre of a 2n-gon, joining it hy edges to the mid-points of the sides, and then identifying pairs of opposite sides. (See Figure 11 for the case n = 4, and check that there is only one face, whereas there are two if n = 3.) By its uniqueness, M must be isomorphic to its mirror-image, so M is fully regular, with full automorphism group D2n (the symmetry group of the 2n-gon).

Theorem 2.1 For each n ≥ 1 the bouquet of circles Bn has a unique orientably regular embedding, namely as a map M of type {2n, 2n} and genus n/2, or of type {n, 2n} and genus (n−1)/2, as n is even or odd. This map is fully regular, with full automorphism group D2n.

9 Figure 11: The unique orientably regular embedding of B4

2.2 Graphs with two vertices Similar methods can be used to study regular embeddings of graphs with two vertices. A dipole is a graph with two vertices, and any number n of edges, all joining one vertex to the other. I will denote it by Dipn, rather than the more usual Dn, to distinguish it from the dihedral group of order 2n. The orientably regular embeddings of dipoles were classiﬁed by Nedela and Skovieraˇ in [57], as an application of their more general results on embeddings of double covers of graphs. Here I will take a more direct approach, extending the method used for bouquets of circles. If M is any orientably regular embedding of Dipn then the monodromy ∼ group G has order |G| = |Φ| = 2n and contains a subgroup hxi = Cn. Having index 2 in G, this subgroup hxi is normal. It is complemented by the subgroup ∼ ∼ ∼ hyi = C2, so G is a semidirect product of hxi = Cn by hyi = C2. The action x 7→ xy := y−1xy of y by conjugation on x uniquely determines this group, and hence also the embedding M, up to isomorphism. Since y has order 2, this action must take the form x 7→ xe 2 for some e ∈ Zn such that e = 1, so that n 2 y e ∼ G = Gn,e := hx, y | x = y = 1, x = x i = Cn o C2.

2 Conversely, any solution of e = 1 in Zn corresponds to such a semidirect product Gn,e, and hence to an orientably regular embedding M = Dn,e of Dipn. For a given n, diﬀerent solutions e give non-isomorphic groups Gn,e and maps Dn,e, so the number of such embeddings is equal to the number of such elements e ∈ Zn. (There is a simple explanation for the fact that e2 = 1: following the orientation, number the edges around one vertex with the elements 0, 1, 2, . . . , n − 1 of Zn. Then around the other vertex, following the same orientation, the edge labelled 0 is followed by the edge labelled e for some e ∈ Zn. By the n-fold rotational symmetry the next edge must be labelled 2e, and so on, giving the cyclic order 0, e, 2e, . . . , −e of edges around the second vertex. Now the 2-fold rotational symmetry, transposing the two vertices, shows that if we repeat this

10 step, starting at the second vertex, we ﬁnd that the cyclic order around the ﬁrst vertex must be 0, e2, 2e2,..., −e2. But this must coincide with the original cyclic order, so e2 = 1.) The following number-theoretic result, giving the number of solutions of 2 e = 1 in Zn, is well-known :

2 j+k Lemma 2.2 The number of elements e ∈ Zn satisfying e = 1 is 2 , where

f f1 fk n = 2 p1 . . . pk

for distinct odd primes p1, . . . , pk with each fi > 0, and j = 0, 1 or 2 as f < 2, f = 2 or f > 2.

Proof. When n is an odd prime power there are just two solutions e = ±1 of this equation, and when n = 2f there are 2j = 1, 2 or 4 solutions as f < 2, f = 2 or f > 2 (in this last case the solutions are ±1 and 2f−1 ± 1). It therefore follows from the Chinese Remainder Theorem that if n is as in the lemma then j+k the number of solutions in Zn is 2 . (See books on elementary number theory, such as [41], for full details.) This formula 2j+k gives the number of orientably regular embeddings M = Dn,e of Dipn. In each case the group G = Gn,e has an automorphism inverting x and y, so each map Dn,e is fully regular. Its face-valency m is the order of −1 2 y e+1 z = (xy) in Gn,e; since (xy) = xx = x we have 2n m = . gcd(e + 1, n)

The number of faces is therefore |G|/m = 2n/m = gcd(e + 1, n), and the Euler characteristic is χ = 2 − n + gcd(n, e + 1).

Example. If we take e = −1, so that m = 2, Dn,−1 is the hosohedron (or beach ball) of type {2, n}, a map on the sphere with n digonal faces (see Figure 12).

Figure 12: The beach ball Dn,−1

11 Example If we put e = 1 with n odd, so that m = 2n, we get a map Dn,1 of type {2n, n} and genus (n − 1)/2 formed by identifying opposite sides of a 2n-gon. However if n is even we get a map of type {n, n} of genus (n − 2)/2 with two n-gonal faces. (See Figure 13 for the cases n = 3 and n = 4, where in each case opposite sides of the outer polygon are identiﬁed to form a torus, and the two vertices are coloured black and white for clarity.)

Figure 13: The dipole embeddings D3,1 and D4,1

We have thus proved:

j+k Theorem 2.3 For each n ≥ 1 the dipole Dipn has 2 orientably regular embeddings Dn,e, where n is divisible by k distinct odd primes, j = 0, 1 or 2 as f 2 2 k n with f < 2, f = 2 or f > 2, and e is a solution of e = 1 in Zn. These maps have type {m, n} where m = 2n/ gcd(n, e + 1), and have automorphism group ∼ ∼ A = Gn,e = Cn o C2. They are fully regular with full automorphism group ∼ Gn,e o C2 = Dn o C2.

2.3 Some other families of graphs In the families of arc-transitive graphs considered so far, every member has at least one orientably regular embedding. This is not always the case. In the following theorem Biggs [3, 4], building on earlier work of Heﬀter [29], proved (a), and James and Jones [34] proved (b):

Theorem 2.4 (a) The complete graph Kn has an orientably regular embeddding if and only if n is a prime power. (b) If n = pe where p is prime then there are φ(n − 1)/e orientably regular embeddings of Kn.

(See Figure 9 for two examples for n = 5.)

12 Outline proof. Biggs constructed his maps as Cayley maps for the additive groups of finite fields Fn of order n (such a field exists if and only if n is a prime power, in which case it is unique up to isomorphism). The vertices are the elements of the field, and each vertex v is joined to the others in the cyclic order v +1, v +c, v +c2, . . . , v +cn−2 for some generator c of the (cyclic) multiplicative ∗ ∼ group Fn = Fn \{0} = Cn−1. This cyclic ordering of neighbours defines an embedding of Kn in an oriented surface. One can check that the affine group

A := AGL1(Fn) = {t 7→ at + b | a, b ∈ Fn, a 6= 0} acts as a group of automorphism of the resulting map M, regularly on the arcs, so M is orientably regular with automorphism group A. Conversely, if M is any orientably regular embedding of Kn, its automorphism group A acts regularly on the arcs, and hence sharply 2-transitively on the vertices. The sharply 2-transitive finite groups were classified by Zassen- haus [71], and those with cyclic point-stabilisers (as must happen here) are the affine groups over finite fields, so with a little more work it follows that the only examples are those produced by Biggs. The number of choices for a generator ∗ c for Fn is φ(n − 1), and one can show that two choices give isomorphic maps if and only if they are equivalent under the Galois group of the field. If n = pe this is a cyclic group of order e (generated by the Frobenius automorphism t 7→ tp), acting semiregularly on the generators c, so it has φ(n − 1)/e orbits, and hence this is the number of orientably regular embeddings M of Kn.

Thus inﬁnitely many of the graphs Kn have orientably regular embeddings, and inﬁnitely many do not, but asymptotically the latter predominate. Even worse, we have [38]:

Theorem 2.5 The line graph L(Kn) of Kn has an orientably regular embeddding if and only if n = 2, 3 or 4.

Exercise Find such an embedding for L(K4).

3 Classiﬁcation by genus

We will now consider classiﬁcation of orientably regular maps by their surface, or equivalently, in the compact orientable case, by their genus. If M is an orientably regular map of type {m, n} and genus g, with monodromy group G (and isomorphic orientation-preserving automorphism group A =∼ G), then equation (2) can be written as

|G| |G| 2g − 2 = (mn − 2m − 2n) = (m − 2)(n − 2) − 4. 2mn 2mn For a given (small) value of g, this equation can be used to classify the possible groups G and hence the maps M.

13 3.1 The sphere If g = 0 then (m − 2)(n − 2) < 4, so (assuming that m, n > 1 to avoid trivial cases) M must have type

{m, n} = {m, 2}, {2, n}, {3, 3}, {3, 4}, {4, 3}, {3, 5} or {5, 3},

with |G| = 2m, 2n, 12, 24, 24, 60 or 60 respectively. Since xn = zm = 1, we may add the relations Xn = Y m = 1 to the presentation for Γ, giving a quotient group

∆ = hX,Y,Z | Xn = Y 2 = Zm = XYZ = 1i, and then the required normal subgroups M of Γ, corresponding to orientably regular maps M of type {m, n}, will be the inverse images in Γ of normal subgroups N of ∆, with G =∼ ∆/N =∼ Γ/M. Now ∆ is the triangle group ∆(n, 2, m), generated by rotations through 2π/n, 2π/2 and 2π/m around the vertices of a spherical triangle with internal angles π/n, π/2 and π/m. These groups are known to have the orders given above for G, so N = 1, G =∼ ∆ and we can identify M with the platonic map of type {m, n} with automorphism ∼ group ∆ = Dm, Dn, A4, S4 (twice) or A5 (twice). By inspection these maps M are all fully regular (alternatively, each group ∆ has an automorphism inverting X and Y ), and the full automorphism group of M is the extended triangle group ∆[n, 2, m], generated by reﬂections in the sides of the corresponding spherical triangle.

3.2 The torus If g = 1 then (m−2)(n−2) = 4, so M has type {m, n} = {4, 4}, {3, 6} or {6, 3}, with this time no restriction on |G|. Example Suppose that m = n = 4, so all vertices and faces have valency 4. The elements x and z of the monodromy group G then have order 4, so we may add the relations X4 = Z4 = 1 to the presentation of Γ, giving the group

∆ = ∆(4, 2, 4) = hX,Y,Z | X4 = Y 2 = Z4 = XYZ = 1i.

We then look for quotients G = ∆/N of ∆ by normal subgroups N such that the images x, y and z of X,Y and Z in G have orders 4, 2 and 4 (and not proper divisors of these). Such subgroups N then lift beck to the required normal map subgroups M of Γ, with Γ/M =∼ G. Now ∆ is the triangle group of type (4, 2, 4), generated by rotations through 2π/4, 2π/2 and 2π/4 in the vertices of a triangle in the euclidean plane with internal angles π/4, π/2 and π/4. Equivalently, ∆ is the orientation-preserving automorphism group of the square tessellation of C, with vertices at the ring of Gaussian integers Z[i] = {a + bi | a, b ∈ Z}, and edges parallel to the real

14 and imaginary axes: we can take X,Y and Z to be rotations of order 4, 2 and 4 around an incident vertex, edge-midpoint and face-centre, such as the vertices of the triangle shown in red in Figure 14.

Figure 14: The square tessellation of the plane

This group is the semidirect product of the normal translation group T =∼ Z2 ∼ by the subgroup ∆0 = C4 ﬁxing the vertex 0. The elements of ∆ \ T are all rotations of C, conjugate to a power of X,Y or Z, so N cannot contain any of these (otherwise x, y or z would have order less then 4, 2 or 4). Thus N ≤ T . We can identify T with the additive group of the ring Z[i], so that N is a subgroup of Z[i]. Since T is abelian, any subgroup N of T is normal in T , so it is normal in ∆ if and only it is normalised by ∆0. Since ∆0 is generated by multiplication of C by i, this is equivalent to iN = N, or equivalently to N being an ideal in Z[i]. Now Z[i] is a euclidean domain, hence a principal ideal domain, so N is a principal ideal, generated (as an ideal) by some Gaussian integer a + bi, and thus generated as an additive group by a + bi and i(a + bi) = −b + ai. If a + bi = 0 then N = {0} and M is the square tessellation of C. If a + bi 6= 0 then N has ﬁnite index

a b d = det = a2 + b2 −b a in T , and so |G| = |A| = 4d = 4(a2 + b2).

Then M, denoted by {4, 4}a,b in [12, §8.3], is the quotient of the square tessellation by N, with d vertices, 2d edges, and d faces. Two Gaussian integers yield isomorphic maps if and only if they are associates in the ring Z[i], that is, they diﬀer by multiplication by a power of i. (See Figure 9 for the non-isomorphic maps {4, 4}1,2 and {4, 4}2,1 .)

15 The map M = {4, 4}a,b is fully regular if and only if N is invariant under complex conjugation, that is, N = N. This happens if and only if a = 0, b = 0, ∼ or a = ±b, giving two families of fully regular maps, {4, 4}a,0 = {4, 4}0,a and ∼ {4, 4}a,a = {4, 4}a,−a. (See Figure 15 for {4, 4}3,0 and {4, 4}2,2 where, as usual, opposite sides of the outer square are identiﬁed to form a torus.) For all other a and b, the orientably regular maps {4, 4}a,b and {4, 4}b,a form a chiral psir, as in Figure 9.

Figure 15: The torus maps {4, 4}3,0 and {4, 4}2,2

Example The situation is similar for the torus maps of types {3, 6} and {6, 3}, except that now one uses the triangular and hexagonal tessellations of C (duals of each other), and the ring Z[ω] of Eisenstein integers, where ω = exp(2πi/3). See [12, §8.4] for details, and Figure 16 for an example of type {3, 6}, with opposite sides of the outer hexagon identiﬁed to form a torus.

Figure 16: A torus map of type {3, 6}

16 Exercise Count the edges of the map in Figure 16, and verify that it has Euler characteristic χ = 0. What is the embedded graph? Is the map chiral or fully regular? What is its automorphism group?

3.3 Surfaces of genus g > 1 Let A be the orientation-preserving automorphism group of an orientably regular map M of type {m, n} and genus g. We can rewrite equation (2) in the form 2g − 2 1 1 1 = − − , |A| 2 m n so for each ﬁxed g > 1, the order |A| is maximised when the right-hand side is minimised. For positive integers m and n, this number attains its least positive value, namely 1/42, when m = 3 and n = 7 or vice versa, giving the Hurwitz bound:

Theorem 3.1 If A is the automorphism group of an oriewntably regular map M of genus g > 1, then |A| ≤ 84(g − 1), attained if and only if M has type {3, 7} or {7, 3}.

(Actually, in 1893 Hurwitz [31] proved a rather stronger result, that the automorphism group of a compact Riemann surface of genus g > 1 has order at most 84(g − 1), but the method of proof implies the result for maps.)

Corollary 3.2 Given any g > 1, there are only ﬁnitely many orientably regular maps of genus g.

Proof. The monodromy group G of such a map is isomorphic to its automorphism group A, so |G| ≤ 84(g −1). There are only finitely many groups of order at most 84(g − 1), and each has only finitely many generating pairs x, y. This is in contrast with the situation for g = 0 and g = 1, where we have infinite families of orientably regular maps. It means that it is feasible, either by hand or by computer search, to attempt a complete classification of the orientably regular maps of a given genus g > 1. See the websites of Marston Conder [7] and PrimoˇzPotoˇcnik[60] for extensive computer-generated lists of maps, organised by their genus. The groups A attaining the Hurwitz bound are called Hurwitz groups. They are the non-identity finite quotient groups of the triangle group

∆ = ∆(3, 2, 7) = hX,Y,Z | X3 = Y 2 = Z7 = XYZ = 1i.

This group is generated by rotations through 2π/3, 2π/2 and 2π/7 in the vertices of a triangle in the hyperbolic plane H with internal angles π/3, π/2 and π/7. Equivalently, ∆ is the orientation-preserving automorphism group of a tessellation of H of type {7, 3} (or its dual, of type {3, 7}). (This is analogous to our use

17 of the triangle group of type (4, 2, 4) for the torus maps of type {4, 4}, except that now we have to use hyperbolic geometry rather than euclidean geometry, and there is no useful translation subgroup available.) Note that this group ∆ is perfect, that is, it has no non-trivial abelian quotients, or equivalently, it is equal to its commutator subgroup ∆0. To see this, note that if we add relations making the generators of ∆ commute, the group collapses to the identity (check this for yourself). It follows that every Hurwitz group A is also perfect, so the smallest Hurwitz groups are non-abelian finite simple groups, and all others are covering groups of them. Since the classification of finite simple groups (announced around 1980, but not completely proved until about 25 years later), a lot of effort has gone into determining which finite simple groups are Hurwitz groups. We first look for Hurwitz groups of the least possible genus, namely g = 2. Such a group would have order 84. Theorem 3.3 There is no perfect group A of order 84. Proof. Suppose that |A| = 84. Since 84 = 22.3.7, Sylow’s Theorems imply that the Sylow 7-subgroups of A have order 7, and the number n7 of them divides |A| and satisfies n7 ≡ 1 mod (7). The only possibility is that n7 = 1, so there is a unique (and hence normal) Sylow 7-subgroup S in A. The quotient group B := A/S has order 12, and if A is perfect then so is B. However, similar arguments using Sylow’s Theorems (give the details!) show that every group of order 12 has a normal Sylow 2- or 3-subgroup with a non-trivial abelian quotient group. Thus B cannot be perfect, and hence neither can A. Corollary 3.4 There is no Hurwitz group of genus g = 2. There is, however, a Hurwitz group of genus g = 3, an example due to Klein [46]. For any field F , the general linear group GLn(F ) is the group of all n × n invertible matrices with entries in F , and the special linear group SLn(F ) is the normal subgroup consisting of those matrices of determinant 1. Taking n = 2, the centre of SL2(F ) consists of the matrices ±I, and the quotient group SL2(F )/{±I} is the projective special linear group PSL2(F ), sometimes abbreviated to L2(F ). It is a simple group for all fields F with |F | > 3. Its elements are the pairs ±M of matrices M ∈ SL2(F ). If F is the finite field Fq of order q (necessarily a prime power), then this group, denoted by PSL2(q) 2 or L2(q), has order q(q − 1)/d where d = 1 or 2 as q is even or odd. Thus A := PSL2(7) is a simple group of order 168, and this has the form 84(g − 1) for g = 3. To show that A is a Hurwitz group we need to show that it is generated by elements x, y and z satisfying x3 = y2 = z7 = xyz = 1, so that there is an epimorphism ∆ → A given by X 7→ x, Y 7→ y, Z 7→ z. It is straightforward to check that the elements 1 1 0 1 1 1 x = ± , y = ± and z = ± −1 0 −1 0 0 1

18 of A satisfy these relations, so it remains to show that they generate A. The subgroup H := hx, y, zi is a Hurwitz group and if it is a proper subgroup of A it must have genus less that 3; but we have proved that there is no Hurwitz group of genus 2, so H = A. (Alternatively H has order divisible by 3.2.7 = 42, so it has index n = |A : H| ≤ 4 in A; if H is a proper subgroup then its core is a proper normal subgroup of index at most 4! = 24, contradicting the simplicity of A.) Thus PSL2(7) is a Hurwitz group. It is, in fact, the only Hurwitz group of genus 3, so it is the smallest Hurwitz group. It is the orientation-preserving automorphism group of an orientable regular map of type {7, 3} and genus 3, and also of the dual map of type {3, 7}. The ﬁrst of these is shown in Figure 17 as a map in the unit disc model of the hyperbolic plane, with the sides of the outer 14-gon identiﬁed according to their numbers to form a surface of genus 3.

Figure 17: Klein’s map of type {7, 3} and genus 3

The underlying surface arises as an algebraic curve of genus g = 3 with the maximum number 84(g−1) of automorphisms, namely Klein’s quartic curve [46], given in homogenous coordinates by x3y + y3z + z3x = 0. There is a whole book [48] devoted to geometric, combinatorial, algebraic, number-theoretic and even aesthetic aspects of this example. Later Macbeath [51] proved the following more general result:

Theorem 3.5 The group PSL2(q) is a Hurwitz group if and only if • q = 7, or • q = p for some prime p ≡ ±1 mod (7), or • q = p3 for some prime p ≡ ±2 or ±3 mod (7).

Dirichlet’s theorem on primes in arithmetic progressions shows that there are inﬁnitely many primes satisfying each of these congruences. Conder [6] proved that the alternating group An, which is simple for all n ≥ 5, is a Hurwitz

19 group for all n > 168, and it is also known to be a Hurwitz group for many smaller values of n. See [8] for a recent survey of Hurwitz groups. The most symmetric orientably regular maps of genus g = 2 are a dual pair ∼ of types {3, 8} and {8, 3}, with automorphism group A = GL2(3) of order 48. More generally, the ﬁnite quotients of the triangle group

∆ = ∆(3, 2, 8) = hX,Y,Z | X3 = Y 2 = Z8 = XYZ = 1i

are the automorphism groups A of orientably regular maps of type {8, 3} and genus g, with |A| = 48(g − 1). It is easy to check that the elements

−1 −1 1 0 1 1 x = , y = and z = −1 1 0 −1 −1 0

3 2 8 of GL2(3) satisfy x = y = z = xyz = 1, and that they generate this group, so there is an orientably regular map M of type {8, 3} and genus 2 with this automorphism group. This map, called the M¨obius-Kantormap, can be constructed as a double covering of the cube (a map of type {4, 3} with automorphism group ∼ S4 = P GL2(3) = GL2(3)/{±I}) branched over its six face-centres. It is shown in Figure 18, with opposite sides of the outer octagon identiﬁed to form an orientable surface of genus 2.

Figure 18: The M¨obius-Kantor map

3.4 The Accola-Maclachlan maps Extending what we have seen for genus 2 and 3, Accola [1] and Maclachlan [52] independently and simultaneously showed that for each genus g there exists an orientably regular map with a ‘reasonably large’ automorphism group:

Theorem 3.6 For each integer g ≥ 0 there is an orientably regular map of genus g with an orientation-preserving automorphism group of order 8(g + 1).

20 (Actually, they proved that there is a compact Riemann surface of genus g with such an automorphism group, but their proofs also apply to maps.) Proof. As in the construction of the M¨obius-Kantor map, we will construct the required map M as a double covering of a map on the sphere. Let N be the unique orientably regular map of type {m, 2} for some even m = 2k: this is the embedding of a circuit of m vertices and m edges in the sphere, with two m-gonal faces. We deﬁne M to be a double covering of N , branched over the m vertices. Thus each edge or face of N lifts homeomorphically to two edges or faces of M, whereas each vertex of N , of valency 2, lifts to a single vertex of M, of valency 4. The resulting map M has m vertices, 2m edges and four faces, so it has Euler characterstic χ = 4 − m and hence genus χ 1 − = k − 1. 2 By taking k = g + 1 we therefore obtain a map of the required genus g. We can label the arcs of N with the integers 1, 2,..., 2m so that its monodromy group (a dihedral group Dm) is generated by

x = (1, 2)(3, 4) ... (2m − 1, 2m) and y = (2, 3)(4, 5) ... (2m, 1),

with deﬁning relations x2 = y2 = (xy)m = 1. If we label the corresponding arcs on M, on the upper and lower sheets of the covering, as 1, 2,..., 2m and 1, 2,..., 2m, so that i and i cover the arc i of N , then the generators for the monodromy group of M are

x = (1, 2, 1, 2)(3, 4, 3, 4) ... (2m − 1, 2m, 2m − 1, 2m)

and y = (2, 3)(2, 3)(4, 5)(4, 5) ... (2m, 1)(2m, 1). It is easy to see that the group G = hx, yi they generate permutes the 4m arcs 1, 1,..., 2m, 2m transitively, so |G| ≥ 4m. On the other hand, x and y satisfy

x4 = y2 = (xy)m = 1 and x2y = yx2,

(the last equation following from the fact that x2 transposes pairs i and i); thus x2 commutes with both generators x and y of G, so it generates a normal (in 2 ∼ 2 fact, central) subgroup hx i = C2 of G, with G/hx i a quotient of Dm, giving |G| ≤ 4m. Thus |G| = 4m, so G permutes the arcs of M regularly and hence M is orientably regular, with automorphism group

A =∼ G = hx, y | x4 = y2 = (xy)m = 1, x2y = yx2i

of order 4m = 8k = 8(g + 1). Perhaps you are familiar with the method used in complex analysis to construct the Riemann surface of a many-valued function, by taking several copies

21 of the complex plane or Riemann sphere, one for each branch of the function, cutting them along lines between pairs of branch-points, and then joining different copies across the cuts. If so, you can visualise the above construction by taking two copies of N , cutting both along alternate edges, and then joining the two copies across these cuts. For most values of g there are also orientably regular maps of genus g with larger automorphism groups than these (for instance, we have already seen examples for g ≤ 3), but Accola and Maclachlan proved that for inﬁnitely many values of g these maps have the largest possible automorphism groups.

4 Permutational categories

Before considering the problem of classifying maps by their automorphism groups, I want to widen the scope to include other combinatorial objects, such as maps on all surfaces (possibly non-orientable, possibly with boundary), maps of a given valency, hypermaps, polytopes, etc. The reason is that essentially the same methods can be applied in all these cases. We have seen that maps on oriented surfaces (without boundary) can be identified with the permutation representations of the group 2 ∼ Γ = hX,Y | Y = 1i = C∞ ∗ C2 (acting on sets of arcs), and that interesting properties of such maps (finiteness, conectedness, type, automorphisms, etc) are equivalent to certain properties of the corresponding permutation representations. In various other categories C, mainly though not exclusively consisting of various maps on surfaces, the objects O can be identified with the permutation representations of a particular 0 group Γ = ΓC on sets Φ = ΦO associated with O, and the morphisms O → O correspond to the functions ΦO → ΦO0 commuting with the actions of Γ on these sets. Let us then call C a permutational category with parent group Γ. (More precisely, we require that C is equivalent to the category CΓ of permutation representations of Γ, meaning that there are functors from each category to the other, so that their composition, in either order, is naturally equivalent to the identity.) We have already seen several examples of this for oriented maps of type {m, n}, taking the parent group to be the triangle group ∆(n, 2, m) = hX,Y,Z | Xn = Y 2 = Zm = XYZ = 1i acting on sets of arcs: we did this for for spherical maps in §3.1, for torus maps of types {4, 4} in §3.2, and for Hurwitz maps of type {3, 7} or {7, 3} in §3.3. Here are some further examples (the details will follow later): • For maps on all surfaces, Γ is the extended triangle group 2 2 ∼ ∆[∞, 2, ∞] = hR0,R1,R2 | Ri = (R0R2) = 1i = V4 ∗ C2, acting on flags.

22 • For maps of type {m, n} on all surfaces, Γ is the extended triangle group

2 n 2 m ∆[n, 2, m] = hR0,R1,R2 | Ri = (R1R2) = (R0R2) = (R0R1) = 1i,

again acting on ﬂags.

2 • The parent group for hypermaps, without the relation (R0R2) = 1, is the extended triangle group

2 ∼ ∆[∞, ∞, ∞] = hR0,R1,R2 | Ri = 1i = C2 ∗ C2 ∗ C2

and for oriented hypermaps without boundary it is the subgroup ∼ ∼ ∆(∞, ∞, ∞) = C∞ ∗ C∞ = F2,

a free group of rank 2 generated by X = R1R2 and Y = R2R0. • For abstract polytopes of a given type one can use the corresponding string Coxeter group, again acting on ﬂags, though here one has to restrict attention to quotient groups satisfying the intersection property. • For coverings of a path-connected topological space X one uses the fundamental group π1X, acting on sheets of the covering, or more precisely on the ﬁbre over a base-point.

Many of the basic ideas introduced earlier for oriented maps extend in the obvious way to permutational categories C, as follows (see §1 for precise statements and proofs). Each object O ∈ C is the disjoint union of connected subobjects, one for each orbit of Γ on Φ. In the connected case, Γ is transitive and elements of Φ can be identiﬁed with cosets in Γ of a point-stabiliser M = Γφ for some φ ∈ Φ. The permutation group G induced on Φ by Γ is the monodromy group

G = Mon O = MonCO of O in C. The centraliser C(G) of G in Sym Φ is the automorphism group

A = Aut O = AutCO of O in C. (Note that an object O may have diﬀerent monodromy groups and automorphism groups in diﬀerent categories, for instance as an oriented or unoriented map.) Snce G is transitive on Φ, A acts semiregularly on Φ, and ∼ ∼ A = NΓ(M)/M = NG(Gφ)/Gφ.

The regular objects in C are the most symmetric ones, those for which A is transitive (and hence regular) on Φ; equivalently, M is a normal subgroup of Γ, with A =∼ Γ/M =∼ G.

23 Each object O ∈ C is the quotient of the universal object in C, a regular object corresponding to the regular representation of Γ (with the trivial map subgroup). Usually this object is infinite, but if M is finite then it is the quotient of some finite regular object N , corresponding to the core of M in Γ, by a group of automorphisms of N . This allows one to concentrate on the regular objects in C, together with their automorphism groups, or equivalently on the normal subgroups M of Γ, together with their quotient groups. The aim is to understand the set R(G) = RC(G) of regular objects O ∈ C with Aut O isomorphic to a given group G. For all the categories I will consider here, the parent group Γ is finitely generated. (Warning: a compact topological space can have an infinitely generated fundamental group!) If Γ is finitely generated and G is finite then the cardinality

r(G) := |R(G)| of this set is finite. This is because such regular objects O correspond to the kernels M of epimorphisms θ :Γ → G; each epimorphism is determined by the images in G of a finite set of generators of Γ, giving only finitely many possibilities for θ. I will show some methods for calculating r(G) in this case, as a first step towards classifying the objects in R(G).

5 Examples of permutational categories

First I will give more details about the examples of permutational categories mentioned above. This is just a brief summary; for further details, see, for example, [35, 45] for maps, and [33, 43] for hypermaps.

5.1 Maps on all surfaces The category M of all maps on surfaces (possibly non-orientable, possibly with boundary), with branched coverings of maps as its morphisms, is a permutational category, with parent group

2 2 Γ = ΓM = hR0,R1,R2 | Ri = (R0R2) = 1i (3)

acting as follows (see Tutte’s paper [67] for an early instance of this idea). For each map M ∈ M let Φ be the set of flags φ = (v, e, f) of M, where v, e and f are a mutually incident vertex, edge and face. For each φ ∈ Φ and each i = 0, 1, 2, there is at most one other flag φ0 with the same j-dimensional components as φ for each j 6= i (there may be none if M has non-empty boundary). Define 0 ri to be the permutation of Φ which transposes φ with φ if the latter exists, and fixes φ otherwise. Figures 19 and 20 illustrate these two cases, with the broken line in Figure 20 representing part of the boundary of the map. We define the monodromy group (in M) of M to be the subgroup

G := hr0, r1, r2i ≤ Sym Φ

24 f φr1 φ φr v 0 φr2 e φr0r2

Figure 19: Generators ri of G acting on a ﬂag φ = (v, e, f).

φr0 = φ φr = φ φr1 = φ 2

Figure 20: Flags ﬁxed by r0, r1 and r2.

generated by the permutations r0, r1 and r2. By their construction, they satisfy 2 2 ri = (r0r2) = 1, so there is a permutation representation of Γ on Φ given by

θ :Γ → G, Ri 7→ ri (i = 0, 1, 2). Conversely, given any permutation representation of Γ on a set Φ, one can construct a map M by taking the vertices, edges and faces to correspond to the ∼ ∼ ∼ orbits on Φ of the subgroups hR1,R2i = D∞, hR0,R2i = V4 and hR0,R1i = D∞, mutually incident when these orbits have non-empty intersection. More specifically, one can construct the barycentric subdivision B(M) of M by taking a set of triangles in bijective correspondence with Φ, each with edges labelled 0, 1 and 2, and joining two triangles along their edges labelled i whenever ri transposes the corresponding elements of Φ; the embedded graph is then the union of all the edges labelled 2 in B(M). As in the oriented case, we will assume that M is connected, or equivalently Γ acts transitively on Φ, in which case the stabilisers of flags form a conjugacy class of map subgroups M ≤ Γ. Finite maps correspond to subgroups M of finite index in Γ, and maps with non-empty boundary correspond to subgroups M containing a conjugate of a generator Ri (so that ri has fixed points in Φ). The automorphism group A = Aut M of M (in M) is the centraliser of G in Sym Φ, isomorphic to NΓ(M)/M. The map M is regular if A is transitive on Φ, or equivalently M is normal in Γ, in which case A =∼ G =∼ Γ/M. (Such maps are called ‘reflexible’ in [12]; the term ‘fully regular’ is also sometimes used, to distinguish these maps from the orientably regular maps considered earlier.) As in the oriented case, the group Γ decomposes as a free product, this time of the form ∼ hR0,R2i ∗ hR1i = V4 ∗ C2

25 where the factors are a Klein four-group and a cyclic group of order 2. This means that the various structure theorems for free products (see [50, 54], for example) can be applied to this group, as in [45]. One can also regard Γ as the extended triangle group ∆[∞, 2, ∞] of type (∞, 2, ∞), generated by reﬂections in the sides of a hyperbolic triangle with angles 0, π/2, 0 (see [43] for this and other similar realisations of parent groups). This is shown in red in Figure 21 using the disc model for the hyperbolic plane.

Figure 21: A hyperbolic triangle

For the subcategory M+ of oriented maps without boundary, we take the parent group ΓM+ to be the even subgroup of index 2 in ΓM, consisting of the elements of even word-length in the generators Ri. This is generated by the elements X = R1R2, Y = R2R0 = R0R2 and Z = R0R1, with deﬁning relations Y 2 = XYZ = 1, so it is simply the group 2 ∼ hX,Y | Y = 1i = ∆(∞, 2, ∞) = C∞ ∗ C2 introduced in §1. A map M ∈ M is orientable and without boundary if and only if its map subgroups M are contained in this subgroup ΓM+ . The (vertex-face) dual D(M) of M is a map on the same surface as M, with the roles of vertices and faces transposed. This corresponds to replacing the map subgroup M with its image under the automorphism δ of Γ given by

δ : R0 7→ R2,R1 7→ R1,R2 7→ R0. We define a Petrie walk in M to be a zig-zag walk which turns alternately first left and first right at successive vertices. More precisely, just as a face is an orbit of hR0,R1i on flags, this is an orbit of hR0R2,R1i. In a finite map these walks close up to form Petrie polygons. The Petrie dual P (M) of M is a map embedding the same graph as M, but with new faces bounded by the Petrie polygons of M. This operation P is induced by applying the automorphism

π : R0 7→ R0R2,R1 7→ R1,R2 7→ R2 of Γ to map subgroups M, so P 2 is the identity operation. Both of these operations D and P preserve regularity and automorphism groups, but (unlike D)

26 P may change the underlying surface, for instance by changing its orientability and by introducing or eliminating boundary components. Example If T denotes the tetrahedron, regarded as an embedding of the complete graph K4 in the sphere, then P (T ) is a regular embedding of K4 in the real projective plane, with three 4-gonal faces and ∼ ∼ Aut P (T ) = Aut T = S4 (see Figure 22, where opposite points of the boundary of the disc are identiﬁed). This map, which has type {4, 3}, can also be formed as the antipodal quotient of the cube.

Figure 22: The tetrahedron and its Petrie dual

(James [32] proved that the only non-orientable regular embeddings of Kn are the maps denoted in [12] by {6, 2}3, {4, 3}3, {3, 5}5 and {5, 5}3 for n = 3, 4, 6 and 6. The ﬁrst three are the antipodal quotients of a circuit of length 6, the cube and the icosahedron, all in the real projective plane, while the last map, of type {5, 5} and characteristic −3, is the Petrie dual of the latter, both having automorphism group A5.) In [69], Wilson showed that D and P generate a group of operations on maps isomorphic to S3 (see also the paper [49] by Lins): the third involution in this group is the ‘opposite’ operation DPD = PDP , which cuts each map apart along its edges, and then rejoins faces along their common edges but with the reverse identiﬁcations. In [45], Jones and Thornton showed that this group can be interpreted as the outer automorphism group Out Γ = Aut Γ/Inn Γ =∼ ∼ ∼ Aut V4 = S3 of the group Γ = V4 ∗ C2 acting on conjugacy classes of subgroups of Γ. See [33, 39, 58] for other categories with similar interpretations of the corresponding groups Out Γ as groups of operations,

5.2 Other categories of maps Other triangle groups act as parent groups for related categories of maps. In the case of M, the permutation r1r2 rotates ﬂags around their incident vertices, with k valencies corresponding to its cycle-lengths, so by adding the relation (R1R2) = 1 to the presentation (3) we get the parent group 2 2 k ΓMk = hR0,R1,R2 | Ri = (R0R2) = (R1R2) = 1i = ∆[k, 2, ∞].

27 for the category Mk of k-valent maps, or more precisely those with all vertex- valencies dividing k. This group is the extended triangle group ∆[k, 2, ∞], or equivalently the free product ∼ hR1,R2i ∗hR2i hR0,R2i = Dk ∗C2 V4 ∼ ∼ of hR1,R2i = Dk and hR0,R2i = V4, amalgamating their common subgroup ∼ hR2i = C2. (See [50, 54] for free products with amalgamation.) Similarly, since r0r1 rotates ﬂags around their incident faces, the isomor- k phic group ∆[∞, 2, k], given by adding the relation (R0R1) = 1 to the presentation of ΓM, is the parent group for the dual category of maps with all face- + valencies dividing k. For the subcategory Mk of oriented maps with all vertex- ∼ valencies dividing k we use the ordinary triangle group ∆(k, 2, ∞) = Ck ∗ C2, the orientation-preserving subgroup of index 2 in ∆[k, 2, ∞], and similarly for its dual category. The case k = 3 is particularly interesting here, partly for applications to cubic and triangular maps, and also because the parent groups ∆(3, 2, ∞) and ∆[3, 2, ∞] are isomorphic to the modular group PSL2(Z) and the extended modular group P GL2(Z), important in number theory and other areas of mathematics.

5.3 Categories of hypermaps A hypermap is an embedding of a hypergraph in a surface. A graph can be regarded as a set of points, called vertices, with a set of subsets of these, called edges, each containing at most two elements; if we relax this last condition, and allow any non-zero number of elements, we obtain a hypergraph: a set of elements, called hypervertices or points, together with a set of non-empty subsets of these, called hyperedges, or blocks. There are many examples: graphs them- selves, of course, but also block designs, for instance. It is useful to represent a hypergraph by means of its Levi graph: this is a bipartite graph, with black and white vertices representing points and blocks, and edges between them in- dicating incidence. Conversely, any bipartite graph, with a speciﬁc black and white colouring of its vertices, represents a hypergraph.

Figure 23: The Fano plane and the Heawood graph

28 Example The Fano plane is the projective plane over the ﬁeld F2. It has seven 3 points, corresponding to the non-zero elements of the vector space F2, and seven 3 blocks, or lines, corresponding to the 2-dimensional subspaces of F2. Each point is in three lines and each line contains three points, so the corresponding Levi graph, known as the Heawood graph, is a bipartite cubic graph with 14 vertices (see Figure 23). Hypermaps can be deﬁned and represented in many ways, but the simplest is via their Walsh representation [68], as maps consisting of bipartite (or more precisely, 2-coloured) graphs embedded in surfaces. Any map can be regarded as a hypermap, simply by colouring its vertices black (if they aren’t already!), and placing a white vertex at the midpoint of each edge (or at the free end of each semi-edge). Example Figure 24 shows the tetrahedron, as a map and as a hypermap.

Figure 24: The tetrahedron as a map and as a hypermap

Figure 25: The Heawood graph as a torus hypermap

29 Example The orientably regular torus embedding M of K7 in Figure 16 is 2-face colourable (alternate triangular faces point up or down), so the dual map D(M) is bipartite: see Figure 25, with M now shown in red, with dotted edges. The graph embedded by D(M) is the Heawood graph, so if its vertices are 2- coloured black and white we obtain a hypermap which embeds the Fano plane in the torus, with black and white vertices representing the points and lines. Let H be the category of all hypermaps. Since hyperedges may have any 2 valency, not necessarily dividing 2, we delete the relation (R0R2) = 1 from (3), so the parent group is

2 ∼ ΓH = hR0,R1,R2 | Ri = 1i = ∆[∞, ∞, ∞] = C2 ∗ C2 ∗ C2, (4) again permuting ﬂags. We say that a hypermap has type (l, m, n) if l, m and n are the orders of the induced permutations r1r2, r2r0 and r0r1. Then the extended triangle group

2 l m n ∆[l, m, n] = hR0,R1,R2 | Ri = (R1R2) = (R0R2) = (R0R1) = 1i is the parent group for the category of hypermaps of type dividing (l, m, n), that is, of type (l0, m0, n0) where the valencies l0, m0 and n0 divide l, m and n. The category H+ of oriented hypermaps (without boundary) has as its parent group ΓH+ the even subgroup of index 2 in ΓH, generated by X = R1R2, Y = R2R0 and Z = R0R1 with deﬁning relation XYZ = 1. This is the group ∼ ∼ ∆(∞, ∞, ∞) = hX,Y,Z | XYZ = 1i = C∞ ∗ C∞ = F2, a free group of rank 2. We can take Φ to be the set of edges in the Walsh bipartite map, with X and Y rotating edges around their incident black and white vertices by following the chosen orientation. The triangle group

∆(l, m, n) = hX,Y,Z | Xl = Y m = Zn = XYZ = 1i, the even subgroup of index 2 in ∆[l, m, n], is the parent group for the category of oriented hypermaps of type dividing (l, m, n).

C objects parent group triangle group M maps V4 ∗ C2 ∆[∞, 2, ∞] + M oriented maps C∞ ∗ C2 ∆(∞, 2, ∞)

Mk k-valent maps Dk ∗C2 V4 ∆[k, 2, ∞] H hypermaps C2 ∗ C2 ∗ C2 ∆[∞, ∞, ∞] + H oriented hypermaps C∞ ∗ C∞ = F2 ∆(∞, ∞, ∞)

Table 1: Some categories and their parent groups.

Table 1 shows some of the more important permutational categories of maps and hypermaps, with their parent groups expressed as free products and as triangle groups.

30 Finite oriented hypermaps are sometimes called dessins d’enfants (that is, children’s drawing), or simply dessins; later I will explain their appearance in Grothendeck’s programme [24, 26], linking Galois theory, algebraic numbers and Riemann surfaces.

5.4 Polytopes

One can generalise the category M of all maps to the category Pd of d-dimensional polytopes for each d ≥ 2 by using as its parent group the string Coxeter group of rank d + 1 given by 2 2 Γ = hR0,...,Rd | Ri = 1, (RiRj) = 1 whenever |i − j| ≥ 2i. (Note that putting d = 2 gives the group Γ used in §5.1.) As with all Coxeter groups, Γ can be described by means of its Coxeter graph (see Figure 26), which is simply a representation of its presentation.

∞ ∞ ∞

R0 R1 R2 Rd−1 Rd

Figure 26: The Coxeter graph for Γ

For a general Coxeter group, the vertices correspond to the generators Ri, 2 all satisfying Ri = 1. Non-adjacent vertices represent generators Ri and Rj 2 satisfying (RiRj) = 1, or equivalently commuting, RiRj = RjRi. An edge k labelled k between vertices Ri and Rj represents a relation (RiRj) = 1, or the absence of such a relation when k = ∞. Here Γ acts on the set Φ of flags of a polytope, with Ri changing (in the only possible way) the i-dimensional components of flags, while fixing their j- dimensional components for all j 6= i. The resulting permutations ri of the flags generate the monodromy group G ≤ Sym Φ, and the automorphism group A of the polytope is the centraliser of G in Sym Φ. The regular polytopes are those for which A acts transitively (and hence regularly) on Φ. If one wants to restrict attention to polytopes with Schläflisymbol {p1, . . . , pd} (an analogue pi of the type of a map), one adds the relations (Ri−1Ri) = 1 for i = 1, . . . , d, giving the Coxeter group corresponding to the Coxeter graph in Figure 27.

p1 p2 pd

R0 R1 R2 Rd−1 Rd

Figure 27: The Coxeter graph for the Schl¨aﬂisymbol {p1, . . . , pd}

For geometric reasons, one has to restrict attention to those quotient groups G of Γ which satisfy the intersection condition: this requires that if S and

31 T are any subsets of the set {r0, . . . , rd} of standard generators of G, then hS ∩ T i = hSi ∩ hT i. (The inclusion hS ∩ T i ≤ hSi ∩ hT i is trivially satisﬁed, but the reverse inclusion may fail.) See the book [53] by McMullen and Schulte for an excellent introduction to abstract regular polytopes. There are many analogies between the theories of maps and polytopes, and applications of ideas from each theory to the other.

5.5 Coverings of topological spaces If X is a suitably ‘nice’ topological space (specifically, if it is path connected, locally path connected, and semilocally simply connected, see [56, Ch. 13]), then classical covering space theory tells us that the connected unbranched coverings β : Y → X of X correspond to the conjugacy classes of subgroups of the fundamental group Γ = π1X. It follows that they form a permutational category C with the fundamental group Γ as parent group. This group permutes −1 the fibre Φ = β (x0) ⊂ Y of β over a chosen base-point x0 ∈ X by lifting closed paths γ from x0 to itself in X to pathsγ ˜ in Y ; starting at any point φ ∈ Φ, such a path finishes at a point φ0 ∈ Φ, so γ induces a permutation φ 7→ φ0 of Φ which depends only on the homotopy class [γ] ∈ Γ of γ. The regular coverings β of X (those induced by a group A acting on Y , so that X =∼ Y/A) correspond to the normal subgroups N of Γ, with covering group A = Aut β =∼ Γ/N. If X is also a compact Hausdorff space (for instance, a compact manifold or orbifold), then Γ is finitely generated [56, p. 500], so enumeration of coverings is feasible. Example Grothendieck’s dessins d’enfants (see §9) correspond to the finite coverings of a sphere minus three points, so their parent group is the fundamental group Γ = F2 of this space, with generators X,Y and Z such that XYZ = 1 inducing the monodromy permutations at the three punctures. Warning (not for those of a nervous disposition). The Hawaiian earring, the union X ⊂ R2 of an infinite sequence of circles, of radii 2−n and centred −n at (2 , 0) for n ∈ N, is a compact Hausdorff space, but π1X is not finitely generated. Even worse, π1X is uncountable, so it is not countably generated, and hence not generated by what looks in Figure 28 like the obvious infinite set of elements. The problem here is that X is not semilocally simply connected.

Figure 28: The Hawaiian earring

32 6 Counting regular objects

From now on, C will denote a permutational category with parent group Γ, always assumed to be finitely generated so that enumeration is feasible. For each group G, there is a natural bijection between the set R(G) = ∼ RC(G) of (isomorphism classes of) regular objects O ∈ C with Aut O = G ∼ and the set N (G) = NΓ(G) of normal subgroups N of Γ with Γ/N = G. These normal subgroups are the kernels of the epimorphisms Γ → G. Two such epimorphisms θ1, θ2 have the same kernel if and only if θ2 = θ1 ◦ α for some automorphism α of G, so there is a bijection between N (G) and the set of orbits of Aut G, acting by composition on the set Epi(Γ,G) of epimorphisms θ :Γ → G. This action of Aut G is semiregular, since only the identity automorphism of G fixes an epimorphism. Now if G is finite then so is Epi(Γ,G), since each epimorphism Γ → G is uniquely determined by the images in G of a finite set of generators of Γ. In this case the sets R(G) and N (G) have the same finite cardinality

|Epi(Γ,G)| r(G) = r (G) = |R(G)| = |N (G)| = . (5) C |Aut G|

Example In some cases one can use direct counting. For example, let C = M+, 2 ∼ so that Γ = hX,Y | Y = 1i = C∞ ∗ C2, and take G = A4, so we are counting orientably regular maps with automorphism group isomorphic to A4. We need first to count epimorphisms Γ → G. Recall that, apart from the identity element, A4 has three elements (ij)(kl) of order 2 and eight elements (ijk) of order 3; the elements of order 2, together with the identity, form a normal subgroup V4. Since A4 is not cyclic, the images x and y of X and Y in G cannot be the identity. Thus y must have order 2, giving three choices. If x also has order 2 then x and y are both elements of the proper subgroup V4, so they do not generate G. Hence x must have order 3, giving eight choices. Since A4 has no proper subgroups of order 6, each of these 3 × 8 choices gives an epimorphism, so |Epi(Γ,G)| = 24. ∼ Now Aut A4 = S4, so |Aut G| = 24 and hence r(G) = 24/24 = 1. Thus there ∼ is a unique orientably regular map with automorphism group G = A4. It is, of course, the tetrahedron, corresponding to the unique normal subgroup M of Γ ∼ with Γ/M = A4. For most finite groups G, this na¨ıve approach in inadequate. In [27], Hall developed a method for counting epimorphisms onto G by first counting homomorphisms (generally an easier task) to subgroups of H ≤ G, and then using Möbiusinversion in the lattice Λ(G) of subgroups of G. Each homomorphism Γ → G is an epimorphism onto a unique subgroup H ≤ G, so X |Hom(Γ,G)| = |Epi(Γ,H)|. (6) H≤G We need to evaluate |Epi(Γ,G)|, so we would like to transpose the roles of the operators Epi and Hom in this equation, in order to express |Epi(Γ,G)| in terms

33 of |Hom(Γ,H)| for subgroups H ≤ G. Unfortunately, X |Epi(Γ,G)| 6= |Hom(Γ,H)| H≤G unless G is the trivial group, but Hall showed that if we include a small correction factor on the right-hand side we obtain the desired equation. In fact, his results, which generalise the Möbiusinversion formula in elementary number theory, have wider application than this. First let µG be the Möbiusfunction on the subgroup lattice Λ(G) of G, defined recursively by X µG(K) = δH,G, (7) K≥H where δH,G is the Kronecker delta, equal to 1 or 0 as H = G or not. Thus X µG(G) = 1 and µG(H) = − µG(K) if H < G. K>H

Example Figure 29 shows the subgroup lattice for the group G = A4, with the isomorphism type of each subgroup H on the left, and the corresponding value µG(H) on the right.

H = A4 µG(H) = 1

V4 −1

C3 −1

C2 0

1 4

Figure 29: The M¨obiusfunction for A4

Now let σ and φ be any functions from isomorphism classes of finite groups to such that C X σ(G) = φ(H) (8) H≤G for all finite groups G (see equation (6), for example). Then a simple calculation with double summation (do it yourself!) gives the Möbiusinversion formula for G, namely X φ(G) = µG(H)σ(H) (9) H≤G

34 (This is a group-theoretic analogue of the inclusion-exclusion principle, which |G\H| applies to the lattice of all subsets of a set G, so that µG(H) = (−1) for each H ⊆ G.) Applying this to equation (6) gives X |Epi(Γ,G)| = µG(H)|Hom(Γ,H)|. (10) H≤G This proves the ﬁrst part of the following theorem; the second follows easily.

Theorem 6.1 If C is a permutational category with a ﬁnitely generated parent group Γ, and G is a ﬁnite group, then the number r(G) of isomorphism classes of regular objects O ∈ C with Aut O =∼ G is given by 1 X r(G) = µ (H)|Hom(Γ,H)|. (11) |Aut G| G H≤G

The number m(G) of isomorphism classes of objects O ∈ C with Mon O =∼ G is given by m(G) = r(G)c(G), (12) where c(G) is the number of conjugacy classes of subgroups of G with trivial core.

+ ∼ Example Let C = H , the category of oriented hypermaps, so that Γ = F2. Then |Hom(Γ,H)| = |H|2 for any ﬁnite group H, since a homomorphism Γ → H is uniquely determined by an arbitrary choice of elements x, y ∈ H as images of free generators X and Y for Γ. If we take G = A4, then using the values of ∼ µG(H) computed earlier for H = A4, V4, C3, C2 and 1, we ﬁnd that there are 1 96 r(A ) = 122 − 42 − 4.32 + 0.22 + 4.12 = = 4 4 24 24

regular objects with automorphism group A4. There are three hypermaps, all on the sphere, obtained from the tetrahedron T : there is T itself, regarded as a hypermap of type (3, 2, 3) as in Figure 24, its black-white dual, of type (2, 3, 3), and another of type (3, 3, 2) with black and white vertices at the vertices and face-centres of T . The fourth, of type (3, 3, 3), is the Petrie dual of the cube, a torus embedding of Q3 as a bipartite map, shown in Figure 30 with opposite sides of the outer hexagon identiﬁed in the usual way. There are m(A4) = 3 conjugacy classes of subgroup of A4 with trivial core, namely those isomorphic to C3, C2 and 1. Hence, in addition to these four regular objects, there are eight non-regular objects with monodromy group A4, namely their quotients by subgroups of order 3 or 2 in their automorphism group. These are all easily drawn (exercise!). In order to apply equation (11) to a speciﬁc pair C and G, we need to know the following:

• |Aut G|,

35 Figure 30: A hypermap of type (3, 3, 3) on the torus

• µG(H) for each H ≤ G, and

• |Hom(Γ,H)| for each H ≤ G such that µG(H) 6= 0.

The ﬁrst is often well-known or easily calculated: for example, one can look at [11] or [69] if G is a ﬁnite simple group. Finding µG(H) and |Hom(Γ,H)| is usually much harder, with only partial results available.

7 Evaluating the M¨obiusfunction

The Möbiusfunction µG has been calculated by hand for several infinite families of groups G, and by computer for a number of individual groups: for the latter, see the database of subgroup lattices described by Connor and Leemans in [9], available at [10]. Here I will briefly describe some infinite families.

7.1 Simple examples ∼ ∼ Example If G = Cn then there is a unique subgroup H = Cm for each m dividing n. As shown by Hall [27] we have n µ (H) = µ , G m where µ is the classical M¨obiusfunction on N, deﬁned recursively by X µ(m) = δn,1 m|n

so that µ(n) = (−1)k if n is a product of k distinct primes, and µ(n) = 0 otherwise. Indeed, here µ can be regarded as the Möbiusfunction on the lattice of subgroups of finite index in the infinite cyclic group Z.

36 Exercise Prove the above result of Hall, and extend it to the ﬁnite dihedral groups, as in [37]. Example If G is an elementary abelian group of order pd for some prime p, one can be regard G as a d-dimensional vector space over Fp, so that its subgroups H are the linear subspaces. For each codimension k = 0, 1, . . . , d, the number of these is given by the Gaussian binomial coeﬃcient

d (pd − 1)(pd−1 − 1) ... (pd−k+1 − 1) = . k k k−1 p (p − 1)(p − 1) ... (p − 1) Hall [27] showed that they satisfy

k k(k−1)/2 µG(H) = (−1) p .

In [27], Hall also showed that in any ﬁnite group G, µG(H) = 0 unless H is an intersection of maximal subgroups of G. This means that we can restrict attention to those subgroups containing the Frattini subgroup Φ(G) of G, the intersection of all its maximal subgroups. Example If G is a d-generator ﬁnite p-group then Φ(G) is the subgroup G0Gp generated by the commutators and p-th powers in G (see standard textbooks on group theory), and G/Φ(G) is an elementary abelian p-group of order pd. The subgroups of H ≤ G with µG(H) 6= 0 all contain Φ(G), and correspond to the subgroups of G/Φ(G), with µG(H) given by the preceding example.

If G = G1 × G2 where G1 and G2 are ﬁnite groups of mutually coprime orders, each subgroup H of G has the form H = H1×H2 for a unique pair of subgroups Hi ≤ Gi. Hall showed that in this situation µG(H) = µG1 (H1)µG2 (H2). Example If G is a ﬁnite nilpotent group then G is the direct product of its Sylow subgroups, which are p-groups for the various primes p dividing |G|, so the preceding examples show how to compute µG.

7.2 Non-nilpotent groups However, calculating the M¨obiusfunction for non-nilpotent groups is generally much harder than this. Here are some examples.

Example Dickson described the subgroups of the groups PSL2(q), for all prime powers q, in [13, Ch. XII]. Using this, Hall [27] calculated the Möbius function µG for the simple groups G = PSL2(p) for all primes p ≥ 5. Even in this sim- pler case, the function is rather complicated to describe, depending on various congruences satisfied by p mod (5) and mod (8). In the smallest case, PSL2(5) is isomorphic to the icosahedral group A5. This has five subgroups A4 and V4, six subgroups D5 and C5, ten subgroups D3 and C3, fifteen subgroups C2, and the identity subgroup. Hall showed that for this group, equation (9) takes the form

φ(A5) = σ(A5) − 5σ(A4) − 6σ(D5) − 10σ(D3) + 20σ(C3) + 60σ(C2) − 60σ(1).

37 + Example Let C = H , so that Γ = F2. Putting

σ(H) = |Hom(Γ,H)| = |H|2 for each subgroup H on the right-hand side of the above equation gives

|Epi(Γ,A5)| = φ(A5) = 2280.

Dividing this by |Aut A5| = |S5| = 120 we get r(A5) = 19, so there are 19 orientably regular hypermaps with automorphism group A5. (These have been described, together with their analogues for the other platonic groups, by Breda and Jones in [5].) An equivalent statement is that there are 19 normal subgroups of the free group F2 with quotient group A5, so by considering their intersection we get the surprising corollary that the direct product of 19 copies of A5 is a 2-generator group.

In [14] Downs extended Hall’s calculation of µG to the groups PSL2(q) and e P GL2(q) for all prime powers q; he gave a proof for the case q = 2 in [15], together with a statement of the results for odd q. Some applications of his results to counting maps, etc, can be found in [16]. Example Typical results for odd q are that if C = M (the category of all maps) then 1 X e r(PSL (pe)) = µ pf (pf − a) 2 8e f f|e for all p > 2 and odd e > 1, where a = 2 or 4 as p ≡ 1 or −1 mod (4), and that if C = M3 (cubic maps) then

3 X e r(P GL (pe)) = µ (pf − 1) 2 4e f f

for p > 3 and e > 1, where the sum is over all f dividing e with e/f odd. The Suzuki groups Sz(2e), for odd e > 1, are a family of non-abelian e ﬁnite simple groups with properties similar to those of PSL2(2 ) (see [11, 66, 69]). Downs and Jones have given the M¨obius function for these groups, with applications to enumeration, in [17]. Example Typical results for G = Sz(2e) are that if C = H+ (oriented hypermaps) then 1 X e r(G) = µ 2f (24f − 23f − 9), e f f|e and that if C = M (all maps) then

1 X e r(G) = µ (2f − 1)(2f − 2). e f f|e

38 Hubard and Leemans [30] have also obtained this last formula by more direct means, counting involutions in G. A naive explanation of this formula us that P f|e µ(e/f) ... represents Möbiusinversion (algebraic inclusion-exclusion) over f subgroups Sz(2 ) defined over subfields of the field of definition F2e , that the summand (2f − 1)(2f − 2) counts choices of ordered pairs of distinct involutions in an elementary abelian subgroup of order 2f in Sz(2f ), and that dividing by e represents counting orbits of the outer automorphism group of G, isomorphic ∼ to Gal F2e = Ce. Example The ‘small’ Ree groups Re(3e), for odd e > 1, are another family of non-abelian finite simple groups with similar properties [61, 69]. Analogous results for them have recently been obtained by Pierro [59].

8 Counting homomorphisms

The third ingredient needed in order to apply equation (11) to a finite group G is the ability to count the homomorphisms Γ → H for all subgroups H of G such that µG(H) 6= 0. If Γ has a presentation with generators Xi (i ∈ I) and defining relations Rj(Xi) = 1 (j ∈ J), this is equivalent to counting the solutions (xi) in each such H of the simultaneous equations Rj(xi) = 1 (j ∈ J). Example If Γ is a free product of cyclic groups then this is easy: one simply needs to map generators of the cyclic free factors of Γ to elements of the appropriate orders in H, so all one needs to know is the distribution of orders of elements of H. For many groups H this is straightforward. In many other cases, such as extended triangle groups, Γ does not have a presentation which leads to such simple calculations. However, there are a few cases where the required information can be obtained from the character table of H. Character tables of many finite groups are now available, either in printed form (see [11], for example), or online. (Before continuing, let me briefly summarise what character tables are. If H is a finite group, a representation of H will mean a homomorphism ρ : H → GL(V ), where V is an n-dimensional vector space over C (other fields are allowed, but here C is the most convenient). Two representations of H are equivalent if they differ by an isomorphism of vector spaces commuting with the actions of H. We say that ρ is irreducible if there are no H-invariant subspaces of V other than V and 0. Then the number of irreducible representations of H (up to equivalence) is equal to the number c of conjugacy classes of H. The character χ = χρ of ρ is the function H → C, h 7→ tr ρ(h) sending each h ∈ H to the trace of ρ(h). Since conjugate elements are represented by similar matrices, characters are constant on conjugacy classes. The character table of H is a c×c array, with columns indexed by the conjugacy classes of H, rows indexed by the irreducible characters of H, and entries given by the values of these characters (always algebraic numbers) on elements of the corresponding classes.)

39 8.1 Triangle groups The following theorem is very useful in connection with triangle groups:

Theorem 8.1 Let X , Y and Z be conjugacy classes in a ﬁnite group H. Then the number of solutions of the equation xyz = 1 in H, with x ∈ X , y ∈ Y and z ∈ Z is |X |.|Y|.|Z| X χ(x)χ(y)χ(z) (13) |H| χ(1) χ

where xi ∈ X , y ∈ Y and z ∈ Z, and χ ranges over the irreducible complex characters of H.

(In this formula, |X |.|Y|.|Z| is the number of choices of triples x ∈ X , y ∈ Y and z ∈ Z. If the values of products were uniformly distributed in groups, one could then divide by |H| to ﬁnd the number of triples with xyz = 1. However, groups are rarely as conveniently structured as this, and one can regard the remaining summation as an error term, compensating for the non-uniform distribution of products in H.) Now if Γ is a triangle group

∆(l, m, n) = hX,Y,Z | Xl = Y m = Zn = XYZ = 1i,

then homomorphisms Γ → H correspond to choices of elements x, y, z ∈ H, from conjugacy classes of elements of orders dividing l, m and n, satisfying xyz = 1. (In the case of triangle groups with inﬁnite periods, we regard all integers as dividing ∞.) The character table for H gives the values of the irreducible characters χ on all the conjugacy classes of H, so with this information (and the size of the classes) one can compute the numbers given by this theorem for all appropriate choices of the classes X , Y and Z in H, and then take their sum to obtain |Hom(Γ,H)|. This method has often been used in connection with various categories of oriented maps and hypermaps, where the parent group is a triangle group ∆(l, m, n): see [36] and [42], for instance.

Example The√ character table of A5 is shown in Table 2, where we define λ, µ = (1 ± 5)/2. The columns are headed by representatives of the five conjugacy classes: note that there are two conjugacy classes of elements of order 5 (they merge to form a single class in S5). The first row corresponds to the 1-dimensional principal representation of A5. The second can be regarded 3 3 as the natural representation on R as the icosahedral√ group, extended to C . This representation is defined over the field Q( 5), and the third row corresponds to the algebraic√ √ conjugate of the second, obtained by applying the field automorphism 5 7→ − 5 (equivalently, conjugating by an odd permutation). The fourth and fifth rows represent irreducible summands, of dimensions 4 and 5, of the doubly transitive permutation representations of A5 of degree 5 (the natural representation) and 6 (as PSL2(5)).

40 1 (12)(34) (123) (12345) (12354) 1 1 1 1 1 3 1 0 λ µ 3 1 0 µ λ 4 0 1 -1 -1 5 1 -1 0 0

Table 2: The character table of A5.

Let us use this table to count the orientably regular hypermaps of type ∼ (3, 3, 5) with orientation-preserving automorphism group G = A5. This is equivalent to counting the normal subgroups of Γ := ∆(3, 3, 5) with quotient group A5. In applying the character formula, there is only one choice for the classes X and Y containing elements x and y of order 3, and there are two choices for the class Z containing the element z of order 5. In each case the character formula shows that these classes contain 60 triples x, y, z satisfying xyz = 1, so the total number of such triples is 120. Each triple generates A5, since no proper sub- ∼ group contains elements of order 3 and 5. Since Aut A5 = S5, the number of normal subgroups is therefore 120/120 = 1, so there is a unique orientably regular hypermap. It has genus 5, and it can be constructed as a double covering of the dodecahedron branched over its twelve face-centres; the resulting map is bipartite, so we can colour its vertices alternately black and white.

8.2 Surface groups If Γ is the fundamental group

g Y Πg = π1Sg = hAi,Bi (i = 1, . . . , g) | [Ai,Bi] = 1i i=1 of a compact orientable surface Sg of genus g ≥ 1, then the following theorem of Frobenius [19] and Mednykh [55], is useful in counting homomorphisms Γ → H, and hence in counting regular covering of Sg with a given covering group G: Theorem 8.2 If H is any ﬁnite group then

2g−1 X 2−2g |Hom(Πg,H)| = |H| χ(1) , (14) χ where χ ranges over the irreducible complex characters of H. Similarly, if Γ is the fundamental group

g − Y 2 Πg = hAi (i = 1, . . . , g) | Ai = 1i i=1 of a non-orientable surface of genus g ≥ 1, the following result of Frobenius and Schur [20] can be used:

41 Theorem 8.3 If H is a ﬁnite group then

− g−1 X g 2−g |Hom(Πg ,H)| = |H| cχχ(1) , (15) χ

where χ ranges over the irreducible complex characters of H.

Here 1 X c := χ(h2) χ |H| h∈H is the Frobenius-Schur indicator of χ, equal to 1, −1 or 0 as χ is the character of a real representation (one which can be realised over R), the real character of a non-real representation, or a non-real character. These two theorems have been applied to count various surface coverings in [37].

9 Dessins d’enfants

In this final section, I want to show how maps (more specifically. finite oriented bipartite maps) can provide an important link between other areas of mathematics, such as Riemann surface theory, algebraic geometry, algebraic number theory and Galois theory. Recall that hypermaps can be regarded as bipartite maps, with a specific black and white colouring of their vertices. In the oriented case, the parent group is a free group ∼ Γ = hX,Y | −i = F2 of rank 2. Given any oriented bipartite map B, let Φ be the set of its edges. Then X and Y act on Φ by rotating edges around their incident black and white vertices, following the orientation of the surface. The resulting permutations x and y of Φ generate the monodromy group G of B, and the automorphism group A of B (preserving the orientation and vertex-colouring) is the centraliser of G in Sym Φ. The regular objects B in this category are those for which A acts transitively on the edges, in which case A =∼ G =∼ Γ/M, where M is the corresponding map subgroup, normal in Γ. Example The cube, regarded as a bipartite map on the sphere, is regular, with ∼ automorphism group A = A4 acting regularly on the 12 edges. The universal object in this category, corresponding to the map subgroup M = 1 of Γ, is the universal bipartite map B∞. This can be drawn in the upper half-plane H, as shown partially in Figure 31. The vertices are the rational numbers a/b (with a and b mutually coprime integers) such that b is odd, coloured black or white as a is even or odd. These are ideal vertices, on the boundary R ∪ {∞} of H in the Riemann sphere. Two vertices a/b and c/d are adjacent if and only if ad − bc = ±1, in which case the edge between them is a hyperbolic geodesic, that is, a euclidean circle meeting the boundary perpendicularly. (The resulting graph is a subgraph of the Farey graph, where all rational numbers are

42 vertices, with the same rule deﬁning the edges; in B∞ the rational numbers with b even play the role of the face-centres, including ∞ = 1/0 as the centre of the ‘upper’ face in Figure 31.) (Singerman gives more details about this, together with universal tessellations associated with other categories of maps, in [64].)

0 1 1 2 3 2 4 1 6 4 7 8 5 9 2 Q 1 5 3 5 5 3 5 1 5 3 5 5 3 5 1

Figure 31: The universal bipartite map B∞

One can identify Γ with the automorphism group of B∞, so that the standard generators of Γ act as the Möbiustransformations z z − 2 X : z 7→ and Y : z 7→ , −2z + 1 2z − 3 fixing the black and white vertices at 0 and 1 and in each case cyclically rotating the edges around these vertices. Each oriented bipartite map B is isomorphic to B∞/M for some corresponding map subgroup M of Γ. Now H is a Riemann surface (that is, it has compatible local complex coordinates, as a subset of C), so B, as a quotient of B∞, inherits the structure of a Riemann surface. It is not compact, since there are punctures at the vertices of B, corresponding to the fact that the vertices of B∞ are not in H. However, if B is a finite map (that is, M has finite index in Γ) then one can compactly H/M by filling in the punctures and defining local coordinates appropriately, so that B becomes a compact Riemann surface. It is natural to ask which compact Riemann surfaces arise in this way from bipartite maps. They must form a rather small subset, since Teichmüllerthe- ory tells us that there are uncountable many mutually non-isomorphic Riemann surfaces for each genus g > 0 (corresponding, for example, to different shaped lattices in C in the case g = 1), whereas there are only countably many finite bipartite maps. Remarkably, a theorem of Bely˘ı[2], intended originally as a lemma for realising certain finite groups as Galois groups, provides a characterisation of these Riemann surfaces in terms of algebraic numbers. Riemann showed that, in modern terminology, the categories of compact Riemann surfaces and complex algebraic curves are naturally equivalent. Put simply, each compact Riemann surface can be regarded as a complex curve, that is, a 1-dimensional projective algebraic variety, defined by polynomials with complex coefficients, and conversely each such curve has the structure of a compact Riemann surface. Bely˘ı’sTheorem, as reinterpreted by Grothendieck [26],

43 states that a compact Riemann surface is obtained from a bipartite map if and only if it is defined over the field Q of algebraic numbers, that is, by polynomials with coefficients in Q. Example Let B be the standard embedding of the complete bipartite graph Kn,n (see Figure 32 for the cases n = 3 and 4). This is a Cayley map for the additive group Z2n with respect to the generators 1, 3, 5,..., 2n−1 in that cyclic order, so that the neighbours of each vertex v, following the orientation, are v + 1, v +3, v +5, . . . , v +2n−1. The vertices are coloured black or white as they are even or odd. There are n faces, all 2n-gons, so the genus is (n − 1)(n − 2)/2. As ∼ an oriented bipartite map B is regular, with automorphism group A = Cn × Cn; one can realise this group as Zn ⊕ Zn, with each element (i, j) acting by adding 2i and 2j to the even and odd integers. The map subgroup for B is the normal subgroup M of Γ generated by the commutators and nth powers.

B A

(a) (b)

B A

Figure 32: The standard embeddings of K3,3 and K4,4

One can show that the projective algebraic curve corresponding to B is the Fermat curve, given in homogeneous coordinates by the equation

xn + yn = zn,

or equivalently, if we multiply z by an nth root of −1, by the more symmetric equation xn + yn + zn = 0. The black and white vertices are the points where x = 0 or y = 0. The automorphism group A of B can be realised by multiplying the variables by nth roots of 1. The automorphism group of the curve is larger: it is a semidirect product of A by a group S3 permuting the variables (and thus permuting black and white vertices and face-centres). This curve is visibly deﬁned over Q — in fact, over Q, by either of the above equations.

Example The map M on the left of Figure 33 is not bipartite, but it can be converted into a bipartite map B, or equivalently a hypermap, by placing a

44 M B

Figure 33: Dessins with monodromy group M12 white vertex at the midpoint of each edge, as explained in §5.3 and as shown on the right of Figure 33. A program such as GAP [21] will show that the monodromy group G of B, permuting its twelve edges, is isomorphic to the Mathieu group M12, one of the five sporadic simple groups discovered by Mathieu. This is a 5-transitive simple group of degree 12 and order 95040, and it is also the automorphism group of the canonical regular covering B˜ of B, a bipartite map of genus√ 3601. One can show that these dessins are defined over the quadratic field Q( −11); however, the defining polynomials are not at all easy to write down, involving some very complicated elements of this field. It was the fact that such childish drawings as this can encode such rich and sophisticated mathematical structures which led Grathendieck to call these maps dessins d’enfants (children’s drawings). From now on I shall call them simply dessins.

Let G denote the absolute Galois group, that is, the Galois group Gal Q/Q of the field extension Q ⊃ Q, or more simply the group of field automorphisms of Q. This group has a natural action on polynomials with coefficients in Q, and hence on algebraic curves defined over Q, so there is an induced action of G on dessins. Grothendieck observed that this action is faithful: in other words, each non-identity element of G sends at least one dessin to a non-isomorphic dessin. Now, in a sense I will describe shortly, G can be thought of as encoding the whole of the Galois theory of algebraic number fields, so, at least in principle, one can ‘see’ the whole of this theory through the action of G on dessins. Examples To see a non-trivial action of G on dessins, there is no point in looking√ at the Fermat dessins: being defined over Q, are fixed by G. The field K =√Q( −11)√ has a Galois group of order 2, generated by the automorphism α : −11 7→ − −11, and each element of G induces either the identity or α on K. Now α is simply complex conjugation, so it is hardly surprising to find that the orbits of B and B˜ under G consist of just these dessins and their mirror images. For a less trivial illustration we use an example taken from [47]. The three

45 Figure 34: Three dessins on the sphere

dessins in Figure 34 form an orbit of G. They are defined over an algebraic number field with Galois group S3. The first two are clearly mirror images of each other, whereas the third is combinatorially quite different. Notice that, although the three dessins are mutually non-isomorphic, they share the same numerical data, such as genus, numbers and valencies of black and white vertices, faces, etc. These, together with the monodromy and automorphism groups G and A, are all invariants of G, as proved by Jones and Streit in [44]. These last two examples show how elements of G can be described in terms of their restrictions to various Galois extensions K of Q. (These are the finite normal extensions of Q, or equivalently the splitting field of polynomials in Q[t].) Indeed, since Q is the union of all such extension fields K, each element g ∈ G is uniquely determined by its restrictions gK to these fields, so we can think of g as a ‘vector’ (gK ), with a component gK ∈ GK := Gal K/Q for each Galois extension K of Q. These components can be arbitrary elements of GK , except that they must be compatible with inclusions between Galois extensions: if K and L are Galois extensions of Q, and K ⊇ L, then L is invariant under GK , and the restriction of gK to L must be gL. Thus we have an system of groups GK and restriction homomorphisms ρK,L : GK → GL, and G is its inverse limit: this is the subgroup of the cartesian product of all the groups GK consisting of those (gK ) such that gK ρK,L = gL whenever K ⊇ L. Every automorphism of L extends, in at least one way, to an automorphism of K, so the homomorphisms ρK,L are all epimorphisms, or projections; since the groups GK are all finite, we say that G is a profinite group, meaning a projective limit of finite groups. There has been much interest in profinite groups in recent years. They arise naturally in the Galois theory of infinite field extensions (as above), and they also provide an effectve link between finite and infinite group theory. I will close with two conjectures about this particular profinite group, one still open and the other recently proved. If we put the discrete topology on the groups GK , then since they are all compact so is their cartesian product, by Tychonoff’s Theorem. As a closed subgroup, G inherits a compact topology, called the Krull topology. This extra structure is important, since the Galois correspondence gives a pairing between subfields of Q and closed subgroups of G. Then Hilbert’s conjecture that every finite group arises as the Galois group of some finite extension of Q is equivalent to the conjecture that every finite group is the quotient of G by some closed

46 subgroup. This has been proved for many classes of finite groups, but the general conjecture is still open. Although it has long been known that G acts faithfully on various classes of dessins (such as plane trees, as proved by Schneps [62], or those of a given genus, as proved by Girondo and González-Diez[23]), the dessins used to construct these faithful actions have almost exclusively been highly non-regular (see those in Figure 34, for example). Recently, González-Diezand Jaikin- Zapirain [25] have used deep results from the theory of profinite groups to prove a long-standing conjecture of Catanese that G acts faithfully on regular dessins. Indeed, their method shows that it is faithful on regular dessins of any given hyperbolic type, such as those of type {3, 2, 7} corresponding to Hurwitz groups. Once again, this illustrates the importance of regular objects in various categories.

Acknowledgement. The author is grateful for support from the project: Mo- bility — enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education coﬁnanced by the European Social Fund.

References

[1] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131, 398–408 (1968). [2] G. V. Bely˘ı,On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser Mat. 43 (1979), 267–276, 479. [3] N. L. Biggs, Automorphisms of imbedded graphs, J. Combin. Theory Ser. B 11 (1971), 132–138. [4] N. L. Biggs, Classification of complete maps on orientable surfaces, Rend. Mat. (6) 4 (1971), 645–655. [5] A. J. Breda d’Azevedo and G. A. Jones, Platonic hypermaps, Beiträge Algebra Geom. 42 (2001), 1–37. [6] M. D. E. Conder, Generators for alternating and symmetric groups, J. Lond. Math. Soc. 22 (1980), 75–86. [7] M. D. E. Conder, Regular maps and hypermaps of Euler characteristic −1 to −200, J. Combin. Theory Ser. B 99 (2009), 455–459, with associated lists of computational data available at http://www.math.auckland.ac.nz/∼conder/hypermaps.html. [8] M. D. E. Conder, An update on Hurwitz groups, Groups Complexity Cryp- tology 2 (2010), 35–49.

47 [9] T. Connor and D. Leemans, An atlas of subgroup lattices of ﬁnite almost simple groups, Ars Math. Contemp. 8 (2015), 259–266. [10] T. Connor and D. Leemans, An atlas of subgroup lattices of ﬁnite almost simple groups, http://homepages.ulb.ac.be/∼tconnor/atlaslat/. [11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Clarendon Press, Oxford, 1985. [12] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, Berlin – Heidelberg – New York, 1980. [13] L. E. Dickson, Linear Groups, Dover, New York, 1958. [14] M. L. N. Downs, PhD thesis, University of Southampton, Southampton, UK, 1988.

[15] M. L. N. Downs, The Möbiusfunction of PSL2(q), with application to the maximal normal subgroups of the modular group, J. London Math. Soc. 43 (1991), 61–75. [16] M. L. N. Downs and G. A. Jones, Enumerating regular objects with a given automorphism group, Discrete Math. 64 (1987), 299–302. [17] M. L. N. Downs and G. A. Jones, MöbiusInversion in Suzuki Groups and Enumeration of Regular Objects, Proc. SIGMAP 2014, to appear. [18] J. R. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7 (1960), 646. [19] F. G. Frobenius, Uber¨ Gruppencharaktere, Sitzber. Königlich Preuss. Akad. Wiss. Berlin (1896), 985–1021. [20] G. Frobenius and I. Schur, Uber¨ die reellen Darstellungen der endlichen Gruppen, Sitzber. Königlich Preuss. Akad. Wiss. Berlin (1906), 186–208. [21] The GAP Group: GAP – Groups, Algorithms, and Programming. Version 4.7.6 (2014), http://www.gap-system.org. [22] A. Gardiner, R. Nedela, J. Siráˇnˇ and M. Skoviera,ˇ Characterisation of graphs which underlie regular maps on closed surfaces, J. London Math. Soc. (2) 59 (1999), 100–108. [23] E. Girondo and G. González-Diez,A note on the action of the absolute Galois group on dessins, Bull. Lond. Math. Soc. 39, 721–723 (2007). [24] E. Girondo and G. González-Diez, Introduction to Compact Riemann Sur- faces and Dessins d’Enfants, London Math. Soc. Student Texts 79, Camb- bridge University Press, Cambridge, 2011.

48 [25] G. González-Diezand A. Jaikin-Zapirain, The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, Proc. London Math. Soc., to appear. [26] A. Grothendieck, Esquisse d’un Programme, in Geometric Galois Ac- tions I, Around Grothendieck’s Esquisse d’un Programme (ed. P. Lochak and L. Schneps) London Math. Soc. Lecture Note Ser. 242 (Cambridge University Press, Cambridge, 1997), 5–48. [27] P. Hall, The Eulerian functions of a group, Q. J. Math. 7 (1936), 134–151. [28] W. R. Hamilton, Letter to John T. Graves on the Icosian, 17 October 1856, in The Mathematical Papers of Sir William Rowan Hamilton, Vol. III, Algebra, pp. 612–625, ed. H. Halberstam and R. E. Ingram, Cambridge Univ. Press, Cambridge, 1967. [29] L. Heffter, Uber¨ metazyklische Gruppen und Nachbarconfigurationen, Math. Ann. 50 (1898), 261–268. [30] I. Hubard and D. Leemans, Chiral polytopes and Suzuki simple groups, in Rigidity and Symmetry, 155–175, Fields Inst. Commun., 70, Springer, New York, 2014. [31] A. Hurwitz, Uber¨ algebraische Gebilde mit eindeutige Transformationen in sich, Math. Ann. 41 (1893), 403–442. [32] L. D. James, Imbeddings of the complete graph, Ars Combon. 16-B (1983), 57–72. [33] L. D. James, Operations on hypermaps, and outer automorphisms, Euro- pean J. Combin. 9 (1988), 551–560. [34] L. D. James and G. A. Jones, Regular orientable imbeddings of complete graphs, J. Combin. Theory Ser. B 39 (1985), 353–367. [35] G. A. Jones, Graph imbeddings, groups and Riemann surfaces, in Algebraic Methods in Graph Theory I, II, Szeged 1978, Colloq. Math. Soc. János Bolyai 25, North-Holland, Amsterdam, 1981. [36] G. A. Jones, Ree groups and Riemann surfaces, J. Algebra 165 (1994), 41–62. [37] G. A. Jones, Enumeration of homomorphisms and surface-coverings, Q. J. Math. 46 (1995), 485–507. [38] G. A. Jones, Automorphisms and regular embeddings of merged Johnson graphs, European J. Combin. 26 (2000), 417–435. [39] G. A. Jones, Regular dessins with a given automorphism group, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, 245– 260, Contemp. Math., 629, Amer. Math. Soc., Providence, RI, 2014.

49 [40] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. Lond. Math. Soc. (3) 37 (1978), 273–307. [41] G. A. Jones and J. M. Jones, Elementary Number Theory, Springer, Lon- don. [42] G. A. Jones and S. A. Silver, Suzuki groups and surfaces, J. London Math. Soc. (2) 48 (1993), 117–125. [43] G. A. Jones and D. Singerman, Maps, hypermaps and triangle groups., in The Grothendieck theory of dessins d’enfants (Luminy, 1993), 115–145, London Math. Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cam- bridge, 1994. [44] G. A. Jones and M. Streit, Galois groups, monodromy groups and car- tographic groups, in L. Schneps and P. Lochak (eds.), Geometric Galois Actions 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, pp. 25–65, London Math. Soc. Lecture Note Ser. 243, Cambridge University Press, Cambridge, 1997. [45] G. A. Jones and J. S. Thornton, Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B 35 (1983), 93–103. [46] F. Klein, Uber¨ die Transformationen siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1878/79), 428–497. [47] S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, Springer, Berlin / Heidelberg / New York, 2004. [48] S. Levy (ed.), The Eightfold Way: The Beauty of Klein’s Quartic Curve, MSRI Publ., 1999. [49] S. Lins, Graph-encoded maps, J. Combin. Theory Ser. B 32 (1982), 171– 181. [50] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer- Verlag, Berlin – Heidelberg – New York, 1977. [51] A. M. Macbeath, Generators of the linear fractional groups, in: Number Theory (Houston 1967), ed. W. J. Leveque and E. G. Straus, Proc. Sym- pos. Pure Math. 12, Amer. Math. Soc., Providence, RI, 1969, 14–32. [52] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, J. Lond. Math. Soc. 44, 265–272 (1969). [53] P. McMullen and E. Schulte, Abstract Regular Polytopes, Cambridge Uni- versity Press, Cambridge, 2002. [54] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover, New York, 1976.

50 [55] A. D. Mednykh, Determination of the number of nonequivalent coverings over a compact Riemann surface, Dokl. Akad. Nauk SSSR 239 (1978), 269– 271 (Russian); Soviet Math. Dokl. 19 (1978), 318–320 (English transl.). [56] J. R. Munkres, Topology (2nd ed.), Prentice Hall, Upper Saddle River NJ, 2000. [57] R. Nedela and M. Skoviera,ˇ Regular maps of canonical double coverings of graphs J. Combin. Theory Ser. B 67 (1996), 249–277. [58] R. Nedela and M. Skoviera,ˇ Exponents of orientable maps, Proc. London Math. Soc. 75 (1997), 1–31. [59] E. Pierro, The M¨obiusfunction of the small Ree groups, arXiv:1410.8702[math.GR]. [60] P. Potoˇcnik,Census of rotary maps. http://www.fmf.uni-lj.si/∼potocnik/work.htm. [61] R. Ree, A family of simple groups associated with the simple Lie algebra of type (G2), Amer. J. Math. 83 (1961), 432–462. [62] L. Schneps, Dessins d’enfants on the Riemann sphere, in L. Schneps (ed.), The Grothendieck Theory of Dessins d’Enfants, pp. 47–77, London Math. Soc. Lecture Note Ser. 200, Cambridge University Press, Cambridge, 1994. [63] J-P. Serre, Topics in Galois Theory (2nd ed.), A. K. Peters, Wellesley MA, 2008. [64] D. Singerman, Universal tessellations, Revista Mat. Complut. 1, 111–123 (1988). [65] J. Sir´aˇn,Howˇ symmetric can maps on surfaces be?, in Surveys in Combina- torics 2013, 161–238, London Math. Soc. Lecture Note Ser., 409, Cambridge Univ. Press, Cambridge, 2013. [66] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. [67] W. T. Tutte, What is a map?, in New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), pp. 309–325, Academic Press, New York, 1973. [68] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory Ser. B 18 (1975), 155–163. [69] R. A. Wilson, The Finite Simple Groups, Graduate Texts in Math. 251, Springer, London, 2009.

51 [70] S. E. Wilson, Operators over regular maps, Paciﬁc J. Math. 81 (1979), 559–568. [71] H. Zassenhaus, Kennzeichnung endlicher linearer Gruppen als Permuta- tionsgruppen, Abh. Math. Sem. Hamburg 11 (1936), 17–40.