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 con- centrate on the most symmetric objects in these categories, and outline techniques for enumerating and classifying them in terms of their embed- ded 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 n 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 j Y 2 = 1i = hX; Y; Z j 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 C1 ∗ C2 of the groups ∼ 2 ∼ hX j −i = C1 and hY j 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 := fg 2 G j αg = αg of some arc α 2 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 iden- tified 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) = fρg j g 2 Gg and L(G) = fλg j g 2 Gg 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 cen- traliser in Symm Φ. Then A acts semiregularly on Φ, and ∼ A = NG(Gα)=Gα for each α 2 Φ. −1 (Here NG(Gα) is the normaliser fh 2 G j h Gαh = Gαg of Gα in G.) Proof. Suppose that a 2 A, and a fixes some α 2 Φ. If β 2 Φ then β = αg for some g 2 G since G acts transitively on Φ. Then βa = αga = αag = αg = β. Thus a fixes every β 2 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 2 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 (some- times 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 2 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 2 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 orien- tation | 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 corre- spond 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.