Hecke Groups, Dessins d Enfants and the Archimedean Solids

Transcription

1 arxiv: v1 [math.ag] 9 Sep 2013 Hecke Groups Dessins d Enfants and the Archimedean Solids Yang-Hui He 1 and James Read 2 1 Department of Mathematics City University London Northampton Square London EC1V 0HB UK; School of Physics NanKai University Tianjin P.R. China; Merton College University of Oxford OX1 4JD UK hey@maths.ox.ac.uk 2 Oriel College University of Oxford OX1 4EW UK; Trinity College University of Cambridge CB2 1TQ UK jamr2@cam.ac.uk Abstract We show that every clean dessin d enfant can be associated with a conjugacy class of subgroups of a certain Hecke group or free product of Hecke groups. This dessin is naturally viewed as the Schreier coset graph for that class of subgroups while the Belyi map associated to the dessin is the map between the associated algebraic curve and P 1. With these points in mind we consider the well-studied Archimedean solids finding a representative the associated class of Hecke subgroups in each case by specifying a generating set for that representative before finally discussing the congruence properties of these subgroups. 1

2 Contents 1 Introduction 3 2 Dessins d Enfants and Hecke Groups The Modular Group Congruence Modular Subgroups Hecke Groups Congruence Subgroups of Hecke Groups Dessins d Enfants and Belyi Maps Schreier Coset Graphs Polygon Orientation in Schreier Coset Graphs Algebraic Curves Archimedean Solids Platonic and Archimedean Solids Hecke Subgroups for the Archimedean Solids Platonic Solids Prisms and Antiprisms Exceptional Archimedean Solids Summary

3 3.3 Congruence Archimedean Solids Example 1: The Truncated Tetrahedron Example 2: The 3-Prism Conclusions 45 A Algorithm for Computing the Generators for the Heck Subgroups 47 1 Introduction The Platonic solids the convex polyhedra with equivalent faces composed of congruent convex regular polygons have been known and studied by mathematicians and philosophers alike for millennia. More broadly another class of convex polyhedra which are also well-known to mathematicians are the Archimedean solids: the semiregular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices with no requirement that faces be equivalent. There are three categories of such Archimedean solids: (I) the Platonic solids; (II) two infinite series solutions - the prisms and anti-prisms; and (III) fourteen further exceptional cases. In [1] a novel approach to the Archimedean solids was taken by interpreting the graphs of these solids as clean dessins d enfants in the sense of Grothendieck. We recall that a dessin is a bipartite graph drawn on a Riemann surface and that clean is the criterion that all the nodes of one of the two possible colours have valency two [23]. The underlying Riemann surface we will choose throughout will be simply the sphere CP 1 onto which we can embed the solids; furthermore we will draw them in a planar projection. Now the planar graph of any polytope can be interpreted as a clean dessin 3

4 by inserting a black node into every edge of the graph and colouring every vertex white. Given any such clean dessin one can extract a remarkable array of group theoretic and algebraic properties. Indeed the motivation of [1] was to investigate some of these properties for the interesting class of highly symmetric dessins which correspond to the Archimedean solids. In a parallel vein trivalent clean dessins can be naturally associated to conjugacy classes of subgroups of the modular group in a manner we will soon describe; this further extends to a correspondence with K3 surfaces [4 5] and to gauge theories [6]. This paper constitutes an codification of the above directions into a unified outlook. While [1] focused on finding the Belyi maps corresponding to each of the dessins for the Archimedean solids here we shall approach these dessins from the group-theoretic angle discussed in [4] by considering their connections to subgroups of the famous modular group and a more general class of groups known as Hecke groups. Recall that the modular group Γ Γ (1) = PSL(2; Z) = SL (2; Z) / {±I} is the group of linear fractional transformations z az+b with a b c d Z and ad bc = 1. The cz+d presentation of Γ is x y x 2 = y 3 = I. A natural generalisation is to consider groups with presentation x y x 2 = y n = I : such groups which we shall denote H n are the Hecke groups. Clearly H 3 is the modular group. With this in mind every clean dessin d enfant can be associated to the conjugacy class of a subgroup of a certain Hecke group or free product of Hecke groups in the following way: black nodes are associated with elements of the cyclic group C 2 = x x 2 = I while n-valent white nodes are associated with elements of the cyclic group C n = y y n = I. For a dessin with white nodes with multiple valencies the different valency white nodes are associated with the different C n so that a clean dessin with white nodes of a single valency is associated with a conjugacy class of subgroups of a certain Hecke group while a clean dessin with white nodes of multiple valencies is associated with a conjugacy class of subgroup of a group that is isomorphic to the free product of the relevant Hecke groups. This association made each dessin can be interpreted as the Schreier coset graph of the corresponding Hecke group [18] which is the Cayley graph of the Hecke group quotiented by some finite index subgroup. From this coset graph the permutation 4

5 representation for the associated conjugacy class of subgroups can be extracted. In this paper we shall use these data to compute the generators for a representative of the conjugacy class of Hecke subgroups corresponding to each of the exceptional Archimedean solids as explicit 2 2 matrices. Once we have obtained these results we shall moreover investigate whether any of these subgroups are so-called congruence subgroups. The structure of this paper is as follows. First in section 2 we present some technical details regarding Hecke groups and dessins d enfants. We show that it is possible to interpret each clean dessin as the Schreier coset graph for a conjugacy class of subgroups of a certain Hecke group or free product of Hecke groups. We then show that the Belyi map associated to the particular clean dessin in question is the map from the generalisation of the modular curve associated to each conjugacy class of subgroups of a Hecke group in question to P 1. In section 3 we find the permutations for the conjugacy classes of subgroups of Hecke groups corresponding to every Archimedean solid and provide explicit generating sets of matrices for representatives of those conjugacy classes of subgroups in the exceptional cases. Finally we remark on the congruence properties of these subgroups. 2 Dessins d Enfants and Hecke Groups In this section we first recall some essential details regarding both Hecke groups and clean dessins d enfants. We begin by considering the modular group Γ = H 3 as although this is isomorphic to only one particular Hecke group it is by far the most wellstudied and we shall draw upon the presented results at many points in the ensuing discussion. Subsequently we discuss Hecke groups more generally before moving on to consider clean dessins and their associated Belyi maps. With these well-known results in hand we then describe how every such clean dessin is isomorphic to the Schreier coset graph for a certain conjugacy class of subgroups of a Hecke group or free product of Hecke groups (henceforth we may refer to such conjugacy classes as classes of Hecke subgroups; where there is no ambiguity we may 5

6 use C to denote such classes). We then take a short digression in order to resolve some tangential questions regarding the construction of these Schreier coset graphs which have yet to be presented explicitly in the literature. Next we describe the connection between the Belyi maps associated to the clean dessins in question and the algebraic curves corresponding to precisely the same classes of Hecke subgroups as arose in the discussion of coset graphs. 2.1 The Modular Group To begin our discussion we recall some essential details regarding the modular group Γ. This is the group of linear fractional transformations Z z az+b with a b c d Z cz+d and ad bc = 1. It is generated by the transformations T and S defined by: T (z) = z + 1 S(z) = 1/z. (2.1) The presentation of Γ is S T S 2 = (ST ) 3 = I and we will later discuss the presentations of certain modular subgroups. The 2 2 matrices for S and T are as follows: ( ) ( ) T = S =. (2.2) Letting x = S and y = ST denote the elements of order 2 and 3 respectively we see that Γ is the free product of the cyclic groups C 2 = x x 2 = I and C 3 = y y 3 = I. It follows that 2 2 matrices for x and y are: ( ) ( ) x = y =. (2.3) With these basic details in hand we should consider some notable subgroups of Γ. 6

7 2.1.1 Congruence Modular Subgroups The most important subgroups of Γ are the so-called congruence subgroups defined by having the the entries in the generating matrices S and T obeying some modular arithmetic. Some conjugacy classes of congruence subgroups of particular note are the following: Principal congruence subgroups: Γ (m) := {A SL(2; Z) ; A ±I mod m} / {±I} ; Congruence subgroups of level m: subgroups of Γ containing Γ (m) but not any Γ (n) for n < m; Unipotent matrices: { Γ 1 (m) := A SL(2; Z) ; A ± ( ) 1 b 0 1 mod m } / {±I} ; Upper triangular matrices: {( ) } a b Γ 0 (m) := Γ ; c 0 mod m / {±I}. c d The congruence subgroups { ( Γ (m; m ) 1 + m d ɛ χ := ± for certain choices of m d ɛ χ. α ɛχ m γ χ dβ 1 + m ɛχ δ ) ; γ α mod χ } We note here that: Γ (m) Γ 1 (m) Γ 0 (m) Γ. (2.4) 7

8 In section 3 of this paper we shall remark on the connections between some specific conjugacy classes of congruence modular subgroups and the Archimedean solids. 2.2 Hecke Groups We can now extend our discussion of the modular group Γ = H 3 to the more general Hecke groups H n. The Hecke group H n has presentation x y x 2 = y n = I and is thus the free product of cyclic groups C 2 = x x 2 = I and C n = y y n = I. H n is generated by transformations T and S now defined by: T (z) = z + λ n S(z) = 1/z (2.5) where λ n is some real number to be determined. The 2 2 matrices for these S and T are: ( ) ( ) 1 λn 0 1 T = S =. (2.6) Letting x = S and y = ST as in our discussion of Γ we see that 2 2 matrices for x and y are: ( ) ( ) x = y =. (2.7) λ n For H n we clearly have (ST ) n = I thereby constraining λ n for a given n [12]. Diagonalizing y to compute y n explicitly places a constraint which allows for a solution of λ n we find that the following general expression as well as some important values for small n: λ n = cos ( ) 2π n In particular λ n are algebraic numbers. n λ n (2.8)

9 2.2.1 Congruence Subgroups of Hecke Groups By way of extension of the above discussion of subgroups of the modular group Γ it is useful to consider congruence subgroups of Hecke groups. As stated in [11] the Hecke groups are discrete subgroups of PSL (2 R); in fact the matrix entries are in Z [λ n ] the extension of the ring of integers by the algebraic number λ n. Note that: H n PSL (2 Z [λ n ]). (2.9) However unlike the special case of the modular group this inclusion is strict. With this point in mind we can define the congruence subgroups of Hecke groups in the following way [12]. Let I be an ideal of Z [λ n ]. We then define: {( ) } a b PSL (2 Z [λ n ] I) = PSL (2 Z [λ n ]) ; a 1 b c d 1 I. (2.10) c d By analogy we also define: {( ) } a b PSL 1 (2 Z [λ n ] I) = PSL (2 Z [λ n ]) ; a 1 c d 1 I c d {( ) } a b PSL 0 (2 Z [λ n ] I) = PSL (2 Z [λ n ]) ; c I. (2.11) c d Then we can define the congruence subgroups H n (m) H 1 n (m) and H 0 n (m) of H n as follows: H n (I) = PSL (2 Z [λ n ] I) H n ; H 1 n (I) = PSL 1 (2 Z [λ n ] I) H n ; H 0 n (I) = PSL 0 (2 Z [λ n ] I) H n. (2.12) As noted in [12] we clearly have: H n (I) H 1 n (I) H 0 n (I) H n. (2.13) 9

10 By analogy with our discussion of the modular group we define congruence subgroups of level m of the Hecke group H n as subgroups of H n containing H n (m) but not any H n (p) for p < m [11]. With these details regarding Hecke groups and their subgroups in hand we can now consider their connections to clean dessins d enfants. 2.3 Dessins d Enfants and Belyi Maps A dessin d enfant in the sense of Grothendieck is an ordered pair X D where X is an oriented compact topological surface and D X is a finite graph satisfying the following conditions [3]: 1. D is connected. 2. D is bipartite i.e. consists of only black and white nodes such that vertices connected by an edge have different colours. 3. X \ D is the union of finitely many topological discs which we call the faces. We can interpret any polytope as a dessin by inserting a black node into every edge and colouring all vertices white. This process of inserting into each edge a bivalent node of a certain colour is standard in the study of dessins d enfants and gives rise to socalled clean dessins i.e. those for which all the nodes of one of the two possible colours have valency two. An example of this procedure for the cube is shown in Figure 1. Now recall that there is a one-to-one correspondence between dessins d enfants and Belyi maps [3]. A Belyi map is a holomorphic map to P 1 ramified at only {0 1 } i.e. for which the only points x where d dx β (x) x = 0 are such that β ( x) {0 1 }. We can associate a Belyi map β (x) to a dessin via its ramification indices: the order of vanishing of the Taylor series for β (x) at x is the ramification index r β( x) {01 } (i) at that ith ramification point [4 6]. To draw the dessin from the map we mark one white node for the ith pre-image of 0 with r 0 (i) edges emanating therefrom; similarly we mark one black node for the jth pre-image of 1 with r 1 (j) edges. We connect the nodes with the edges joining only black with white such that each face is a polygon 10

11 (a): The planar graph for the cube. (b): The corresponding clean dessin. Figure 1: Interpreting the planar graph of the cube as a clean dessin. with 2r (k) sides [4]. The converse direction (from dessins to Belyi maps) is detailed in e.g. [5]. 2.4 Schreier Coset Graphs As already discussed the Hecke group Hn has presentation x y x 2 = y n = I and is thus the free product of cyclic groups C2 = x x 2 = I and Cn = y y n = I. Given the free product structure of Hn we see that its Cayley graph is an infinite free n-valent tree but with each node replaced by an oriented n-gon. Now for any finite index subgroup G of Hn we can quotient the Cayley graph to arrive at a finite graph by associating nodes to right cosets and edges between cosets which are related by action of a group element. In other words this graph encodes the permutation representation of Hn acting on the right cosets of G. This is called a Schreier coset graph sometimes also referred to in the literature as a Schreier-Cayley coset graph or simply a coset graph. There is a direct connection between the Schreier coset graphs and the dessins d enfants for each class of Hecke subgroups [18]: PROPOSITION 1 The dessins d enfants for a certain conjugacy class of subgroups 11

12 of a Hecke group H n can be constructed from the Schreier coset graphs by replacing each positively oriented n-gon with a white node and inserting a black node into every edge. Conversely the Schreier coset graphs can be constructed from the dessins by replacing each white node with a positively oriented n-gon and removing the black node from every edge. Let σ 0 and σ 1 denote the permutations induced by the respective actions of x and y on the cosets of each subgroup. We can find a third permutation σ by imposing the following condition thereby constructing a permutation triple: σ 0 σ 1 σ = 1. (2.14) The permutations σ 0 σ 1 and σ give the permutation representations of H n on the right cosets of each subgroup in question. As elements of the symmetric group σ 0 and σ 1 can be easily computed from the Schreier coset graphs by following the procedure elaborated in [17] i.e. by noting that the doubly directed edges represent an element x of order 2 while the positively oriented triangles represent an element y of order n. Since the graphs are connected the group generated by x and y is transitive on the vertices. Clearly σ 0 and σ 1 tell us which vertex of the coset graph is sent to which i.e. which coset of the modular subgroup in question is sent to which by the action of H n on the right cosets of this subgroup. Consider now a clean dessin d enfant with white nodes with m different valencies (in the above we assumed that all the white nodes of the dessin had the same valency). It is still possible to interpret this dessin as a Schreier coset graph though in this case the coset graph will correspond to a conjugacy class of subgroups of a group isomorphic to the free product of cyclic groups C 2 C k1... C km where the k i denote the different valencies of the white nodes of the dessin. Since to recall H n = C2 C n we see that dessins with multiple white node valencies correspond to conjugacy classes of subgroups of free products of Hecke groups. 12

13 (a): Γ 0 (8) (b): Γ 1 (5) Figure 2: Schreier cosets graphs for Γ 0 (8) and Γ 1 (5) with all positively oriented triangles. Figure 3: Schreier coset graph for Γ 0 (8) with one negatively oriented triangle Polygon Orientation in Schreier Coset Graphs As detailed above Schreier coset graphs are built from simple edges (x) and n-gons (y) which are assumed to be positively oriented. Reversal of n-gon orientation corresponds to applying the automorphism of H n that inverts each of the two generators x and y. At this point though it is worth answering an intriguing tangential question: to what conjugacy classes of subgroups do the coset graphs correspond when arbitrary n-gons are reversed in orientation? To answer this question first consider the Schreier coset graphs for the conjugacy classes of congruence subgroups of the modular group Γ 0 (8) and Γ 1 (5) as shown in Figure 2 (these coset graphs were originally presented in [13]; the associated dessins d enfants are drawn in [4]). Blue lines correspond to permutations σ 0 generated by x while red triangles correspond to permutations σ 1 generated by y and we assume initially that all such triangles are positively oriented. As stated in [17] these two coset graphs are not isomorphic as they have different circuits. What happens if we swap the orientation of an arbitrary triangle in one of these coset graphs? For example what happens if we choose the third-from-left triangle in the coset graph for Γ 0 (8) to be negatively oriented rather than positively oriented? Such a modified coset graph is drawn in Figure 3. 13

14 To answer this question we read the relevant permutations off of the original and altered coset graph. In the original case the corresponding conjugacy class of modular subgroups Γ 0 (8) is generated by the permutations: σ 0 : (1 2) (3 4) (6 9) (5 7) (8 10) (11 12) σ 1 : (1 2 3) (4 5 6) (7 8 9) ( ) (2.15) In the second altered case the corresponding conjugacy class of modular subgroups is that generated by the permutations: σ 0 : (1 2) (3 4) (6 9) (5 7) (8 10) (11 12) σ 1 : (1 2 3) (4 5 6) (7 9 8) ( ) (2.16) Note that the triple (7 8 9) has changed to (7 9 8): this is the only change. Now at this point we can use the GAP [16] algorithm presented in section 3.2 of [4] to compute a generating set of matrices for a representative of the conjugacy class of subgroups corresponding to our original and modified coset graphs taking the above permutations as input. Of course we would need to modify the algorithm in order to study classes of Hecke subgroups more generally; we present this modified code explicitly in the Appendix. In the original case such a generating set is: {( ) ( ) ( )} (2.17) From the definition of Γ 0 (m) we see that these are generators for a representative of the conjugacy class of congruence subgroups Γ 0 (8) as expected. In the modified case a generating set is given by: {( ) ( ) ( )} (2.18) Brief reflection on these matrices suffices to make apparent that these are generators of a representative of the conjugacy class of congruence subgroups Γ 1 (5). By reversing 14

15 the orientation of the chosen triangle in the coset graph for Γ 0 (8) we have obtained the permutations for Γ 1 (5). Hence this modified coset graph no longer corresponds to Γ 0 (8) but rather to Γ 1 (5). The reasons for this are easily stated in general terms: swapping n-gon orientations in Schreier coset graphs changes the permutations defining the associated conjugacy class of subgroups of the relevant Hecke group. These permutations need not define the same conjugacy class of subgroups as the original permutations. In fact in general they will not and will instead define the subgroup whose Schreier coset graph with all positively oriented n-gons has the same cycles as the modified coset graph with arbitrary n-gon orientations. This final point is nicely viewed in geometrical terms for our specific example: by choosing the thirdfrom-left triangle in the coset graph for Γ 0 (8) to be negatively oriented we obtain a coset graph which has the same cycles as that for Γ 1 (5) when all the triangles are chosen to be positively oriented. 2.5 Algebraic Curves Let p be prime and let Γ (p) Γ (1) be the subgroup of matrices congruent to (plus or minus) the identity modulo p. Then Γ (p) acts on the completed upper half-plane H and the quotient X (p) = Γ (p) \ H can be given the structure of a Riemann surface known as a modular curve. Defining the subgroup G = Γ (1) /Γ (p) Aut (X (p)) the j-map j : X (p) X (p) /G = P 1 is a Galois cover ramified (with suitable normalisation) at {0 1 }. j is a Belyi map from X (p) to P 1 associated to a dessin drawn on the Riemann surface X (p). Generalising now from Γ = H 3 to all Hecke groups H n we first note that as detailed in [3 11] the quotient space H/H n is a sphere with one puncture and two elliptic fixed points of order 2 and n. Therefore all Hecke groups H n can be considered as triangle groups and the Hecke surface H/H n is a Riemann surface [3 11]. Clearly we are interested in the class of algebraic curves defined in analogy to the modular curve above with the property that there exists a Hecke subgroup G Aut (X) such that the map X X/G = P 1 is a Galois cover ramified at {0 1 }. 15

16 With this in mind let X be a compact Riemann surface which admits a regular tessellation by hyperbolic n-gons. Such a surface admits reconstruction from a torsionfree subgroup G of finite index in a Hecke group H n in the following way. The compact surface X = X G obtained from the group G has finitely many cusps added to the surface H/G one for each G-orbit of boundary points at which the stabiliser is non-trivial. The holomorphic projection π : H/G H/H n = C induced from the identity map on the universal covering extends to a Belyi map β : X P 1 from the compactification with each cusp of X mapped by β to the single cusp { } of H n which compactifies [9] the plane H/H n to C = P 1. The connection to dessins is then as follows: choose any particular clean dessin d enfant. We have already described how this dessin can be interpreted as a Schreier coset graph for a particular Hecke group H n (or free product of Hecke groups). In addition though the above results demonstrate that the Belyi rational map associated to the dessin is precisely the map X P 1 with each cusp of X mapped by β to the single cusp { } of H n. 3 Archimedean Solids With the results of the previous section in hand we can proceed to apply the theory to the so-called Archimedean solids. In this section we first give a technical definition of these geometrical entities. We then identify the Hecke group for which each of the Archimedean solids has a corresponding conjugacy class of subgroups before giving the permutation representations σ 0 and σ 1 for each corresponding class. We then (in every case where possible) find a representative of these classes of subgroups associated to each Archimedean solid by specifying a generating set for that representative following the procedure in [4]. Finally we follow up some residual questions from [4] investigating whether the trivalent Archimedean solids when interpreted as dessins d enfants correspond in general to conjugacy classes of congruence subgroups of the modular group Γ. 16

17 3.1 Platonic and Archimedean Solids The Platonic solids are well-known to us and are the regular convex polyhedra. They are the tetrahedron cube octahedron dodecahedron and icosahedron. In order to introduce the wider class of so-called Archimedean solids consider planar graphs without loops and with vertices of degree k > 2. Following [1] let us call the list of numbers (f 1 f 2... f k ) where the f i are the number of edges of the adjacent faces taken in the counter-clockwise direction around the vertex the type of that vertex. Two such lists are equivalent if one can be obtained from the other by (a) making a cyclic shift and (b) inverting the order of the f i. A solid is called Archimedean if the types of all its vertices are equivalent [1]. We emphasize a subtlety here. As stated in the Introduction one informal way of defining the Archimedean solids is as the semi-regular convex polyhedra composed of two or more types of regular polygons meeting in identical vertices with no requirement that faces be equivalent. Identical vertices is usually taken to mean that for any two vertices there must be an isometry of the entire solid that takes one vertex to the other. Sometimes however it is only required that the faces that meet at one vertex are related isometrically to the faces that meet at the other. On the former definition the so-called pseudorhombicuboctahedron otherwise known as the elongated square gyrobicupola is not considered an Archimedean solid; on the latter it is. The formal definition of the Archimedean solids above corresponds to the second broader definition of identical vertices; following [1] it is this definition which we shall use. For a more extended discussion of this point the reader is referred to [19]. It turns out the the solids which satisfy the above Archimedean condition are: I The five Platonic solids; II Two infinite series (the prisms and anti-prisms); and III Fourteen exceptional solutions 1. 1 An interesting aside: It is well-known that the five Platonic solids correspond to three symmetry groups which in turn are related to the exceptional Lie algebras E 678 and furthermore to Arnold s simple surface singularities which have no deformations [20]. It is curious that the next order gen- 17

18 Name V Vertex type F E Sym Trivalent? Hecke Group I Tetrahedron 4 (3 3 3) 4 6 T d Yes Γ = H 3 I Cube 8 (4 4 4) 6 12 O h Yes Γ = H 3 I Octahedron 6 ( ) 8 12 O h No H 4 I Dodecahedron 20 (5 5 5) I h Yes Γ = H 3 I Icosahedron 12 ( ) I h No H 5 II n-prism 2n (4 4 n) n + 2 3n D nh Yes Γ = H 3 II n-antiprism 2n (3 3 3 n) 2n + 2 4n D nd No H 4 III Truncated tetrahedron 12 (3 6 6) 8 18 T d Yes Γ = H 3 III Truncated cube 24 (3 8 8) O h Yes Γ = H 3 III Truncated octahedron 24 (4 6 6) O h Yes Γ = H 3 III Truncated isocahedron 60 (5 6 6) I h Yes Γ = H 3 III Truncated dodecahedron 60 ( ) I h Yes Γ = H 3 III Truncated cuboctahedron 48 (4 6 8) O h Yes Γ = H 3 III Truncated icosidodecahedron 120 (4 6 10) I h Yes Γ = H 3 III Cuboctahedron 12 ( ) O h No H 4 III Icosidodecahedron 30 ( ) I h No H 4 III Rhombicuboctahedron 24 ( ) O h No H 4 III Rhombicosidodecahedron 60 ( ) I h No H 4 III Pseudorhombicuboctahedron 24 ( ) D 4d No H 4 III Snub cube 24 ( ) O No H 5 III Snub dodecahedron 60 ( ) I No H 5 Table 1: The Archimedean solids. Type I are the 5 famous Platonic solids. Type II are the two infinite families of the prisms and anti-prisms. Type III are the 14 exceptional solids. We record the number of vertices V faces F and edges E for each. The vertex type is the number of edges which the adjacent faces to the vertex counter-clockwisely have; the fact that that all vertices have the same vertex type is the definition of Archimedean. Sym is the symmetry group of the solid given in standard Schönflies notation. We also indicate whether the vertices are all trivalent in which case the solid yields a planar Schreier coset graph for a conjugacy class of subgroups of the modular group; if not it can be accommodated by a more general Hecke group H n. All these solids are listed in Table 1 and are drawn in Figures 4 to 6 (for the n-prisms and n-antiprisms we give examples for small n). All these images were constructed using Mathematica Version 8.0 [15]. For completeness we also give the symmetry groups of each of these solids in the column labelled Sym in this table. These are given in standard so-called Schönflies notation. We recognize the standard tetrahedral octahedral and icosahedral groups T A 4 O S 4 I A 5 (3.1) eralization of the simple singularities which has exactly 1 deformation modulus have precisely 14 exceptional cases. We conjecture therefore some connection to the exceptional Archimedean solids. 18

19 as well as the dihedral group D n. The extra subscript d and h denote extra symmetries about a horizontal (h) mirror plane or a diagonal (d) plane. Tetrahedron Cube Octahedron Dodecahedron Icosahedron Figure 4: Type I Archimedean solids the five famous Platonic solids. 3-Prism 4-Prism 5-Prism 6-Prism 7-Prism 3-Antiprism 4-Antiprism 5-Antiprism 6-Antiprism 7-Antiprism Figure 5: The two infinite families of the type II Archimedean solids the n-prisms and the n-antiprisms for n = The 4-prism is simply the square and the 3-anti-prism is the octahedron. Clearly we can interpret all the Archimedean solids (and indeed as mentioned every other polytope) as clean dessins d enfants by inserting a black node into every edge and colouring every vertex white. We are not the first to interpret the Archimedean solids in this way: in [1] the authors do precisely this computing the associated Belyi map in each case; thus the work in this section can be viewed as complimentary mate- 19

20 Truncated tetrahedron Truncated cube Truncated octahedron Truncated icosahedron Truncated dodecahedron Truncated cuboctahedron Truncated icosidodecahedron Cuboctahedron Icosidodecahedron Rhombicuboctahedron Rhombicosidodecahedron Pseudorhombicuboctahedron Snub cube Snub dodecahedron Figure 6: Type III Archimedean solids: the 14 remaining exceptional cases. Note that in some references the Pseudorhombicuboctahedron is not included because there is no isometry of the entire solid which takes each vertex to another. rial to that work. By the correspondences detailed in the previous section we should be able to associate a class of Hecke subgroups to every Archimedean solid. This task shall be undertaken in the following subsection. 20

21 3.2 Hecke Subgroups for the Archimedean Solids Can the Archimedean solids all be associated with conjugacy classes of subgroups of the modular group and are those subgroups congruence? The answer to the first question is straightforwardly no: only the trivalent solids are such that interpreted as dessins they are subgroups of the modular group which is isomorphic to the free product of cyclic groups C 2 and C 3 as detailed above. Hence only those Archimedean solids labelled Yes in the column Trivalent? of Table 1 can be associated with subgroups of the modular group. The rest can be associated with conjugacy classes of subgroups of other Hecke groups however. For example the cuboctahedron is 4-valent [1] and can thus be associated with a conjugacy class of subgroups of the Hecke group H 4 = C2 C 4. The Hecke group related to each of the Archimedean solids in this way is presented in the final column of Table 1. In the remainder of this subsection we give the permutations σ 0 and σ 1 for the conjugacy class of subgroups of the relevant Hecke group for each Archimedean solid. We also find explicitly a representative of the conjugacy class of subgroups of the relevant Hecke group to which each of the Archimedean solids corresponds (bar the two infinite series cases). The GAP [16] algorithm to achieve this task is presented in the Appendix of this paper. Since the Platonic solids have already been partly studied in e.g. [4] we divide the task into three sections based upon the three categories of Archimedean solid presented in the previous subsection Platonic Solids Interpreted as clean dessins d enfants the tetrahedron cube and dodecahedron correspond to the conjugacy classes of principal congruence subgroups Γ (3) Γ (4) and Γ (5) respectively. This can be verified explicitly from the dessins d enfants presented in [4] where we can also see that the index of the associated class of modular subgroups is twice the number of edges of the Platonic solid in question. For convenience these results are tabulated in Table 2. Now reflection on Table 1 suffices to show that the remaining two Platonic solids - the octahedron and the icosahedron - will not correspond to subgroups of Γ but rather to H 4 and H 5 respectively. What are these subgroups? 21

22 Table 2: The tetrahedron cube and dodecahedron with associated conjugacy classes of modular subgroups C and dessins. We can answer this question by finding a generating set for a representative of the corresponding class in each case. Octahedron: Permutations σ0 and σ1 for the conjugacy class of subgroups corresponding to the octahedron are: σ0 : (1 5)(6 11)(4 12)(2 13)(3 23)(8 14)(7 18)(10 19)(9 22)(16 24)(15 17)(20 21) σ1 : ( )( )( )( )( )( ). (3.2) Using the procedure detailed in the Appendix generators for a representative of the conjugacy class of subgroups of H4 for the octahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( )}. (3.3) 22 Name Tetrahedron Cube Dodecahedron Corresponding C Γ (3) Γ (4) Γ (5) Dessin

23 Comparison with our results concerning congruence subgroups of Hecke groups in section above suffices to show that the octahedron corresponds to the principal congruence subgroup H 4 (3). We remark here that as stated in [12] the matrices of H 4 are of the following two types: ( ) a b 2 ( a 2 b ) c 2 d c d 2 (3.4) The elements of the first type form a subgroup of index 2 in H 4 ; all the subgroups of H 4 corresponding to the Archimedean solids will turn out to be subgroups of this subgroup. Icosahedron: The elements of H 5 need not fall into such simple categories as those for H 4 as given in equation (3.4) above. The consequence is that the elements of the matrices of the respesentiative generating set for the icosahedron snub cube and snub dodecahedron (to be discussed below) which correspond to conjugacy classes of subgroups of H 5 are too complicated to present on paper though they are given in the document which accompanies this paper. For this reason in these three cases we resort in this paper proper to providing just the permutations σ 0 and σ 1. Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the icosahedron are: σ 0 : (1 6)(10 11)(5 15)(2 21)(3 16)(4 43)(7 22)(8 27)(9 33)(12 34)(13 37)(14 42) (17 25)(23 26)(28 32)(35 36)(38 41)(20 44)(18 46)(24 47)(30 48)(29 52)(31 53) (40 54)(39 57)(45 58)(19 59)(50 60)(55 56)(49 51) σ 1 : ( )( )( )( )( ) ( )( )( )( ) ( )( )( ). (3.5) 23

24 3.2.2 Prisms and Antiprisms The prisms and antiprisms form infinite series of Archimedean solids. Since all prisms are trivalent all correspond to a conjugacy class of subgroups of Γ; since all antiprisms are 4-valent all correspond to a conjugacy class of subgroups of H 4. Given that the prisms and antiprisms form infinite series we cannot write down a generating set of a representative of the conjugacy class of subgroups for each. Nevertheless we can construct general expressions for the permutations σ 0 and σ 1 for the prisms and antiprisms. These expressions can be used to find the permutations σ 0 and σ 1 for any particular (anti)prism of interest. These permutations can in turn be input into the GAP code presented in the Appendix in order to compute a generating set for a representative of the relevant class of Hecke subgroups for any particular (anti)prism of interest. Prisms: Call the n-gon faced prism the n-prism. The n-prism has the following permutations σ 0 and σ 1 for x and y respectively: σ 0 : (3n 2 3) (3n 1 6n 2) (6n 1 3n + 3) n 2 (3i + 1 3i + 6) (3i + 2 3n + 3i + 1) (3n + 3i + 2 3n + 3i + 6) σ 1 : i=0 n+2 (3i + 1 3i + 2 3i + 3). (3.6) i=0 This renders it a trivial matter to find a generating set for a representative of the relevant conjugacy class of modular subgroups for any prism and moreover to check whether any n-prism corresponds to a conjugacy class of congruence subgroups of Γ (a topic to which we shall return in the following subsection). Antiprisms: Call the n-gon faced anti-prism the n-antiprism. The n-antiprism has the following permutations σ 0 and σ 1 for x and y respectively: 24

25 σ 0 : (4n 3 4) (4n 2 8n 2) (3 8n 1) (4n + 1 8n) n 2 (4i + 1 4i + 8) (4i + 2 4n + 4i + 2) (4i + 7 4n + 4i + 3) (4n + 4i + 4 4n + 4i + 5) σ 1 : i=0 n+2 (4i + 1 4i + 2 4i + 3 4i + 4). (3.7) i=0 Again this renders it straightforward to find a generating set for a representative of the relevant conjugacy class of subgroups for any antiprism Exceptional Archimedean Solids In addition to the Platonic solids and the prisms and antiprisms there remain fourteen exceptional Archimedean solids as given in Table 1. Here we give the permutations and compute a generating set for a representative of the corresponding class of Hecke subgroups. Truncated tetrahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated tetrahedron are: σ 0 : (1 32)(3 4)(2 7)(5 9)(6 24)(8 10)(11 13)(12 18)(15 16)(17 19)(14 28)(29 31) (33 34)(30 36)(26 35)(20 25)(23 27)(21 22) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( ). (3.8) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated tetrahedron are: 25

26 {( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.9) Truncated cube: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated cube are: σ 0 : (3 4)(2 7)(5 9)(60 61)(63 55)(57 58)(20 22)(21 27)(24 25)(11 13)(12 18) (15 16)(39 45)(43 42)(38 40)(30 31)(29 35)(32 34)(66 72)(65 68)(67 70) (48 49)(47 53)(50 52)(1 59)(62 64)(69 54)(51 6)(41 46)(8 10)(56 23) (36 71)(33 37)(17 44)(14 19)(26 28) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( ). (3.10) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated cube are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.11)

27 Truncated octahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated octahedron are: σ 0 : (1 11)(2 4)(5 8)(9 10)(12 13)(3 36)(6 56)(7 65)(35 26)(32 34)(29 31)(27 28) (25 17)(16 14)(15 24)(21 22)(18 19)(23 68)(70 69)(66 67)(62 64)(63 72)(59 61) (57 58)(51 60)(50 52)(54 55)(49 48)(45 71)(20 42)(30 37)(39 40)(38 47)(44 46) (41 43)(33 53) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( ). (3.12) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated octahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.13) Truncated icosahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated icosahedron are: σ 0 : (1 14)(11 13)(8 10)(5 7)(2 4)(3 16)(15 28)(12 26)(9 22)(6 20)(19 35)(32 34) (17 31)(18 58)(56 60)(29 55)(30 53)(50 52)(27 49)(25 47)(45 46)(44 23) (24 41)(38 40)(21 37)(36 73)(33 64)(59 61)(57 110)(54 107)(51 98)(48 95) 27

28 (43 86)(42 83)(39 77)(71 75)(70 68)(65 67)(66 62)(63 116)( ) ( )( )( )( )(100 99)(96 97)(92 94)(89 91) (87 88)(84 85)(80 82)(119 79)(78 118)(74 76)( )(72 164)(69 158) ( )( )( )( )(93 137)(131 90)(128 81)( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ). (3.14) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated icosahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 28

29 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.15) Truncated dodecahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated dodecahedron are: σ 0 : (1 29)(3 32)(30 31)(27 44)(23 25)(24 43)(21 41)(18 40)(17 19)(11 13)(15 39) (12 38)(5 7)(9 35)(6 34)(2 4)(26 28)(22 20)(14 16)(8 10)(36 62)(33 46)(45 107) (42 92)(37 77)(61 59)(60 66)(63 64)(47 49)(51 120)(48 119)( )( ) ( )(90 96)(93 94)(89 91)(74 76)(78 79)(75 81)(58 56)(50 52)(55 53) (54 122)(57 123)( )( )( )( )( )( ) ( )( )(95 97)(98 100)( )(99 130)( )(86 88)(80 82) (83 85)(87 128)(84 127)( )(71 73)(65 67)(68 70)(72 126)(69 125) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( ) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) 29

30 ( )( )( )( )( ) ( )( )( )( )( ). (3.16) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated dodecahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.17) Truncated cuboctahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated cuboctahedron are: σ 0 :(1 23)(24 20)(21 17)(18 14)(15 12)(10 9)(7 6)(4 3)(2 25)(22 29)(19 32) (16 35)(13 38)(11 41)(8 44)(5 46)(26 28)(30 55)(56 58)(31 59)(33 34)(36 62) (63 64)(65 37)(39 40)(42 67)(68 70)(43 71)(45 48)(47 49)(50 52)(27 53) (54 77)(51 73)(76 74)(78 79)(75 119)(57 86)(60 89)(88 87)(85 83)(90 91) (61 98)(66 101)(100 99)(95 97)( )(72 113)(69 110)( )( ) 30

31 ( )(80 82)(84 128)( )(81 125)( )( )(96 134)(92 94) (93 131)( )( )( )( )( )( )( ) ( )( )( )( ) σ 1 :(1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( ). (3.18) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated cuboctahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (3.19) Truncated icosidodecahedron: Permutations σ 0 and σ 1 for the conjugacy class of subgroups corresponding to the truncated icosidodecahedron are: 31

32 σ 0 : (1 29)(26 28)(23 25)(20 22)(17 19)(15 16)(11 14)(8 10)(5 7)(2 4)(3 31)(30 59) (27 56)(24 52)(21 50)(18 47)(13 44)(12 41)(9 38)(6 34)(33 60)(58 89)(88 86) (85 57)(53 55)(83 54)(80 82)(79 51)(48 49)(46 77)(75 76)(45 74)(42 43)(40 71) (68 70)(39 67)(35 37)(36 65)(64 62)(61 32)(90 119)(87 116)(84 113)(81 109) (78 107)(73 104)(72 101)(69 98)(66 95)(63 91)(93 121)( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )(99 100)(97 134)( )( ) ( )(96 124)(92 94)( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) σ 1 : (1 2 3)(4 5 6)(7 8 9)( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( ) 32

33 ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ). (3.20) Generators for a representative of the conjugacy class of subgroups of Γ for the truncated icosidodecahedron are: {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )