Coxeter-Petrie Complexes of Regular Maps

Kevin Anderson Department of Mathematics, Missouri Western State College St. Joseph, MO 64507, USA andersk@griﬀon.mwsc.edu and David B. Surowski Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA [email protected] ∗

∗European J. Combinatorics, 23 (2002), 861–880. Running Head: Coxeter-Petrie Complexes

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David Surowski Department of Mathematics Cardwell Hall Kansas State University Manhattan, KS 66506-2602 Abstract

Coxeter-Petrie complexes naturally arise as thin diagram geometries whose rank- three residues contain all of the dual forms of a regular algebraic map M. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. G. A. Jones and J. S. Thornton have shown that these involutory duality operations generate the symmetric group S3, giving in all six dual forms, and whose source is the outer automorphism group of the inﬁnite triangle group generated by involutions s1, s2, s3, subject to the additional relation s1s3 = s3s1. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s1, s3, s1s3. These involutions together with the involution s2 can be taken to deﬁne the nodes of a Coxeter diagram of shape D4 (with the involution s2 at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank-3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map. 1 Introduction: Algebraic Maps and (Quasi)thin Chamber Systems

In this introductory section, we gather together the deﬁnitions necessary to realize an algebraic map as a thin chamber complex. In Section 2 we recall the duality operations of an algebraic map in the sense of G. A. Jones and J. S. Thornton [5], and construct, for a given regular algebraic map, the corresponding Coxeter-Petrie complex as a rank 4 thin chamber complex having as rank 3 residues the various dual forms of the given map. The monodromy group of such a complex is a rank 4 Coxeter group with added “Petrie relations.” In Section 3 the structure of the monodromy groups of the Coxeter-Petrie complexes corresponding to the Platonic solids is given, and in Section 4 presentations of the monodromy groups of the Coxeter-Petrie complexes corresponding to regular aﬃne maps are given.

A chamber system C = (C, ∼i | i ∈ I) consists of a set C (of chambers), together with a family of equivalence relations ∼i, i ∈ I. The cardinality of I is called the rank of the chamber system. Arguably the most often quoted example of a rank n chamber system is the following. Let V be an (n + 1)-dimensional vector space over the field F. Recall that a flag in V is a collection F = {W1,W2,...,Wl} of subspaces with W1 ⊆ W2 ⊆ · · · ⊆ Wl.A maximal flag in V is a flag F = {V1,V2,...,Vn}, where dim Vj = j, j = 1, 2, . . . , n. We then can define a chamber system over the set 0 I = {1, 2, . . . , n} to consist of the maximal flags F in V and where F ∼j F when the constituent subspaces of F and F 0 are the same, except possibly in dimension j. Notice that in the above example, if F is the finite field of order q, and if F is 0 0 a fixed maximal flag, then F ∼j F for precisely q + 1 maximal flags F in V . In general, a chamber system C = (C, ∼i | i ∈ I) is called quasithin if all of the ∼i- equivalence classes have cardinality at most 2 and is called thin if all such classes have cardinality exactly 2. Thus, the “flag complex” given above is not quasithin—instead, it is called thick. When the chamber system is quasithin, then an alternative, but equivalent definition can be given as follows. Namely, A quasithin chamber system C =

(C, ai | i ∈ I) consists of a set C of chambers and a family of involutory permutations

1 2

si, i ∈ I on C. Thus, in this case, the ∼i-equivalence classes are the haii-orbits in C.

Note that the quasithin chamber system C = (C, ai | i ∈ I) is thin precisely when each of the monodromy involutions ai, i ∈ I acts without fixed points on C (in general, the union of the fixed points of the monodromy involutions si, i ∈ I is called the boundary of C and denoted ∂(C)). As above, the cardinality of the index set I is the rank of C. The prototypical thin chamber system of rank n is modeled on the above example, as follows. Instead of an (n + 1)-dimensional vector space V , start with a set S of cardinality n + 1. Again, define flags (and maximal flags) in S in terms of subsets of S, by analogy with taking subspaces of V . Thus, if F is a maximal flag in S, it is readily seen that other than itself, F is ∼i equivalent to exactly one other maximal flag F 0 in S, for each i ∈ I . Thus, the corresponding involution satisfies 0 ai(F ) = F .

Let C = (C, ∼i | i ∈ I), D = (D, ≈i | i ∈ I) be chamber systems over the set I, and let φ : C → D be a mapping. We say that φ is a morphism from C to D, and write φ : 0 0 C → D, if c ∼i c in C implies that cφ ≈i c φ. If C = (C, ai| i ∈ I), D = (D, bi| i ∈ I) are quasithin chamber systems over I, a morphism φ : C → D of chamber systems is then simply a mapping φ : C → D satisfying (aic)φ ∈ {cφ, bi(cφ)} for all i ∈ I, and all c ∈ C. An invertible morphism φ : C → D is an isomorphism, and an isomorphism φ : C → C is called an automorphism of C. The set of all such is clearly a group under composition and is denoted Aut(C). It is easy to check that when C = (C, ai| i ∈ I) is quasithin, then

Aut(C) = {bijections φ : C → C | (aic)φ = ai(cφ), for all i ∈ I and all c ∈ C}.

If C = (C, ai| i ∈ I) is a quasithin chamber system, we set G = Mon(C) = hai| i ∈

Ii, the group generated by the involutions ai, i ∈ I, and call it the monodromy group of C. More generally, if C is a chamber system (not necessarily quasithin) such that each ∼i-equivalence class is finite for each i ∈ I, one defines the Hecke algebra H(C; F) of C, over the field F, as follows. First, let V be the F-vector space with basis c ∈ C, 3

and for each i ∈ I, let Ti : V → V , be the linear transformation deﬁned by setting

X 0 Ti(c) = c . 0 0 c ∼ic, c 6=c

Then the Hecke algebra H(C; F) is the F-algebra generated by Ti, i ∈ I. Therefore, if C is a quasithin chamber system, we see that the Hecke algebra is just the group algebra of the monodromy group.

We say that the chamber system C = (C, ∼i | i ∈ I) is connected if and only if the transitive closure of the equivalence relations ∼i, i ∈ I represents C as a single equivalence class. Equivalently, if c, c0 are chambers, then there is a “path” from c to c0 of the form

0 c = c0 ∼i1 c1 ∼i2 c2 ∼ i3 · · · ∼ik ck = c .

In case C is a quasithin chamber system, this can be stated more succinctly simply by stating that the monodromy group acts transitively on the set C of chambers.

Let G be a group with set of involutory generators ai ∈ I, and let H be a subgroup of G. We may form the connected quasithin chamber system C(G/H) = (G/H, ai, i ∈

I), where the monodromy involutions ai, i ∈ I act on G/H via left multiplication.

The following is easy:

Lemma 1.1. Let C = (C, ai, i ∈ I) be a connected quasithin chamber system. If c ∈ C is a ﬁxed chamber, and if H is the stabilizer in G = Mon(C) of c, then the mapping φ : C(G/H) → C, gH 7→ g(c) is an isomorphism of quasithin chamber systems over I.

Henceforth, we shall stick to quasithin chamber systems, even though the deﬁ- nitions and many of the results are valid more generally. Let C = (C, ai, i ∈ I) be a connected, quasithin chamber system over I, with monodromy group G. Fix a 4 chamber c ∈ C, and let H be the stabilizer of c in G. By the above lemma, we may identify C with C(G/H) = (G/H, ai , i ∈ I). It is trivial to verify that if N = NG(H), and if n ∈ N, then the mapping C(G/H) → C(G/H) given by gH 7→ gnH is an automorphism of C(G/H). It is easy to verify that every autmorphism of C(G/H) arises in this way, from which it follows easily that

∼ Aut(C) = NG(H)/H, thereby elucidating the relationship between the monodromy and automorphism groups of the connected, quasithin chamber system C.

Therefore, the connected quasithin chamber system C has a transitive automorphism group if and only if the monodromy group acts regularly on the chambers of C, in which case the automorphism group does, as well. Therefore, we say that the quasithin chamber system C = (C, ai| i ∈ I) is regular if and only if the monodromy group acts regularly on the set C of chambers of C. In this case the automorphism group also acts regularly on the chambers C and is isomorphic with the monodromy group, as the two groups are then the right and left regular representations of the same group.

Let C = (C, ai| i ∈ I) be a quasithin chamber complex with monodromy group

G = hai | i ∈ Ii. For any subset J ⊆ I, we deﬁne the parabolic subgroup GJ = haj | j ∈

Ji ≤ G. Correspondingly, a residue of type J ⊆ I in C is simply one of the GJ -orbits in C. For any i ∈ I, the varieties of type i ∈ I are the residues of type I − {i}.

A quasithin rank 3 chamber system C = (C, a1, a2, a3) is called a hypermap or an algebraic hypermap. If, in addition a1a3 = a3a1, we call C a map, or sometimes an algebraic map. In the context of maps and hypermaps, we typically refer to the underlying chambers as blades. The category of algebraic hypermaps and morphisms is denoted HMap, and the category of algebraic maps and morphisms is denoted Map. If G = ha, b, ci is a group generated by involutions such that ac = ca, and if H ≤ G 5 is a subgroup, we deﬁne the connected map M(G/H, a, b, c) to have set of blades G/H and monodromy involutions a, b, c (acting on G/H). Note that M(G/H, a, b, c) is thin precisely when the subgroup H contains no conjugate of a, b, or c.

If M = (B, a, b, c) is a map, we have varieties in M as follows:

vertices: These are the hb, ci-orbits in B;

edges: These are the ha, ci-orbits in B;

faces: These are the ha, bi-orbits in B.

If M is a map, and if v ⊆ B is any variety, let Gv be the corresponding parabolic subgroup of the monodromy group G. Thus, Gv = hx, yi (a dihedral group), where x, y ∈ {a, b, c} and where v = Gvb (the Gv-orbit of B) for some blade b ∈ B. We set + + Gv = hxyi; then it’s easy to see that either Gv acts transitively on v (which happens + in case v contains an element of ∂(M); note that |v ∩ ∂(M)| ≤ 2), or Gv acts in two orbits of the same cardinality. In either case, we say that the valency of v is the size + of any Gv -orbit in v (which might be infinite). We define the vertex valency of the map M to be the least common multiple of the valencies of each of the vertices. Similarly, the face valencies and edge valencies are defined. Note that if the map is , thin then the edge valency is always 2.

1.1 Diagrams and Rank-2 Residues

In this subsection, we consider the notion of Coxeter diagrams associated with thin chamber systems. To this end, let C = (C, ai | i ∈ I) be a connected thin chamber complex of rank card(I) ≥ 2, and let G = hai | i ∈ Ii be its monodromy group. Let

J ⊆ I be a subset of cardinality 2 with corresponding parabolic subgroup GJ . Since

GJ is dihedral we may, exactly as above, deﬁne the valency of each type-J residue in C. Continuing to assume that J has cardinality 2, we deﬁne the J-valency of C to be the least common multiple of the valencies of varieties of type J. Thus, if

J = {i, j} ⊆ I of cardinality 2, we set mij to be the J-valency of C. Correspondingly, 6 we can associate a Coxeter diagram, whose nodes correspond with the elements i ∈ I, and where for each pair i 6= j in I we draw an edge from node i to node j marked mij.

We adopt the usual convention that when mij = 2, then no edge is drawn from node i to node j. We mention in passing that the notion of diagrams attached to geometries (and later adapted to chamber systems) was ﬁrst formalized by Francis Buekenhout [2], in an attempt to understand non-Lie type geometries, especially those arising from sporadic simple groups. When this process is applied to an algebraic map M = (B, a, b, c), then the nodes of the Coxeter diagram correspond to the involutions a, b, c; since ac = ca, one arrives at a diagram of the form

k l

ag bg c g where k, l are the face and vertex valencies, respectively, of M. Sometimes, to em- phasize the roles of vertices and faces, we exhibit the diagram thus:

k l

verticesg edgesg faces g

More speciﬁcally, the underlying map of the tetrahedron has Coxeter diagram

3 3

verticesg edgesg faces g that of the cube (hexahedron) and its dual, the octahedron, have the respective diagrams

4 3 3 4

verticesg edgesg faces g verticesg edgesg faces g 7

2 Duality and the Coxeter-Petrie Complex

Let M = (B, a, b, c) be a connected map, and set

2 2 2 2 ∆ = hs1, s2, s3 | s1 = s2 = s3 = (s1s3) = 1i.

Clearly, there is a surjective homomorphism ∆ → G, given by s1 7→ a, s2 7→ b, s3 7→ c. As a result, we have that ∆ acts on the left on the set B of blades of M. If we ﬁx a blade β ∈ B, and let H∆ be the stabilizer in ∆ of β, then we may identify M as ∞ ∞ ∞ an orbit map M /H∆, where M is the regular map M = (∆, s1, s2, s3). Fur- 0 0 −1 thermore, if H∆ ≤ ∆ and is conjugate to H∆, say H∆ = τ H∆τ, then the mapping 0 0 ∆/H∆ → ∆/H∆ given by xH∆ 7→ xτH∆ is a well-deﬁned isomorphism of maps. Con- 0 0 versely, if H∆ ≤ ∆ is a subgroup, and if there is a bijection ∆/H∆ → ∆/H∆ realizing an isomorphism of maps (i.e., is a bijection that commutes with left multiplication 0 by elements of ∆), then H∆ and H∆ are conjugate in ∆. Therefore, one seeks to σ produce new maps from old by the process (∆/H∆, s1, s2, s3) 7→ (∆/H∆, s1, s2, s3), where σ : ∆ → ∆ is an outer automorphism.

G. A. Jones and J. S. Thornton [5] showed that the group of outer automorphisms of ∆ is isomorphic with S3, the symmetric group on three symbols. In fact, in Theorem

1 of this paper, they showed that any permutation of s1, s3, s1s3 uniquely determines an outer automorphism of ∆ that ﬁxes s2.

Thus, let M be a connected map, represented as M = (∆/H∆, s1, s2, s3) as above, σ σ let σ ∈ Out(∆), and form the new map M = (∆/H∆, s1, s2, s3). Note ﬁrst that σ ∼ σ σ σ σ −1 σ σ M = (∆/H∆, s1, s2, s3), where γ = σγσ , γ ∈ ∆, via the mapping γH∆ 7→ γH∆. The map Mσ is called the σ-dual of M. If Mσ ∼= M we say that M is σ–self-dual. σ In particular, if σ corresponds to the transposition s1 ↔ s3, then M is called the classical dual of M. If σ is the transposition s1 ↔ s1s3, then M is called the Petrie dual of M and is denoted p(M). Therefore, we see that the various dual forms of the given map M = (B, a, b, c) can be exhibited as follows:

M = (B, a, b, c); 8

M∗ = (B, c, b, a)(Classical dual); p(M) = (B, ac, b, c)(Petrie dual); p(M∗) = (B, ac, b, a); (p(M))∗ = (B, c, b, ac); p((p(M))∗) = (B, a, b, ac) = (p(M∗))∗;

Note that the classical dual of M = (B, a, b, c), obtained by interchanging the involutory generators a and c, is tantamount to reversing the roles of vertices and faces in M. In this way, it can be seen that the tetrahedron can be seen as self-dual and that the octahedron and cube (hexahedron) are dual to each other. However, there is an additional species of objects in the map M, the so-called Petrie polygons, which occur as the orbits in B of the subgroup Gp = hb, aci. A more geometrical description of the Petrie polygons is as follows. Any Petrie polygon is uniquely determined by two adjacent edges e1, e2 bounding a common face; the subsequent edges e3, e4,... are uniquely determined by the requirement that any two consecutive edges must bound a common face, but that no three consecutive edges can bound a common face. Thus, while the tetrahedron is self-dual relative to classical duality, we can exhibit the diagram corresponding to the Petrie dual of the tetrahedron via

4 3

verticesg edgesg Petrie g polygons

To see geometrically that Petrie polygons in the tetrahedron have length 4, we indicate one such below:

¡A ¡ u ¡ A A ¡ 6 ¡ A A ¡ AU ¡ A @ A ¡ u @ ¡ A @ A ¡ @ ¡ A @A ¡ @A u u 9

It is clear that the monodromy group of each of the dual forms of M agrees with that of M. As a result, we see that M is regular if and only if each of its duals is regular. However the same does not apply to orientability. Indeed, the Petrie dual of the cube is orientable of genus 1, having eight vertices, 12 edges and four faces. On the other hand, the Petrie dual of the octahedron is nonorientable of genus 4. To this end, it’s an easy exercise to show that if M has odd face valency, then p(M) is necessarily non-orientable. More generally, in [7] it was observed—and is easy to prove—that p(M) is orientable if and only if the underlying graph (the vertices and edges of M) is bipartite.

As already noted above, the Petrie dual of the tetrahedron must be non-orientable. In fact it is doubly covered by the cube and hence tesselates the real projective plane with three 4-gonal faces. As for the duals of the cube M, exhibited through their diagrams, one has

4 3 3 4

g Mg g g Mg∗ g

6 3 3 6

g p(Mg) g g (p(Mg))∗ g

6 4 4 6

g p(Mg∗) g g (p(Mg∗))∗ g

We turn now to the deﬁnition of the Coxeter-Petrie complex of the regular algebraic map M = (B, a, b, c). Thus, let M have vertex valency l, face valency k and Petrie polygon valency m. For example, the values (k, l, m) for the Platonic 10 solids (tetrahedron, octahedron, cube, icosahedron, and dodecahedron) are respectively (3, 3, 4), (4, 3, 6), (3, 4, 6), (5, 3, 10) and (3, 5, 10). In general, when M is regular, we may identify l, k, and m as the orders of the elements bc, ab, and bac, respectively. As a result, the Coxeter diagrams attached to M and its Petrie dual p(M) are

k l m l

g Mg g g p(gM) g

We shall synthesize the above information into a thin, rank-4 chamber system CP(M) with associated diagram

@ k g @ @ m l g g

g Furthermore, we require the corresponding rank-4 chamber complex to have the prop- erty that each of the rank-3 residues (obtained by omitting an outer node of the Cox- eter diagram), viewed as algebraic maps, is isomorphic with one of the dual forms of M. Finally, if N = M† is one of the dual forms of M, then we will ﬁnd N (or N ∗) as one of the rank-3 residues of CP(M).

Let W be the Coxeter group associated with the diagram

s1 @ k g @ @ m l sg2 sg4

s3 g Therefore, W is generated by elements s1, s2, s3, s4, subject to the Coxeter relations, which we collectively denote by C:

2 2 2 2 2 2 2 k l m s1 = s2 = s3 = s4 = (s1s3) = (s1s4) = (s3s4) = (s1s2) = (s2s3) = (s2s4) = 1. 11

Inside W are the rank-3 parabolic subgroups Wi = WI−{i}, i = 1, 2, 3, 4, where

I = {1, 2, 3, 4}. Furthermore, by [8, Corollary 2.4], we know that each Wi is a Coxeter group with the indicated relations. Therefore, we have surjective homomorphisms

θi : Wi → G, i = 1, 3, 4, deﬁned as follows:

θ1 : s2 7→ b, s3 7→ c, s4 7→ ac,

θ3 : s1 7→ a, s2 7→ b, s4 7→ ac,

θ4 : s1 7→ a, s2 7→ b, s3 7→ c.

For i = 1, 3, 4, set Ki = ker θi and chose a subset Ri ⊂ Wi such that Ki is the normal closure in Wi of Ri. Set R = R1 ∪ R3 ∪ R4, and deﬁne the Coxeter-Petrie group of M to be given by

G = G(M) = hs1, s2, s3, s4 | C ∪ Ri, where C is above set of Coxeter relations. Note that by construction, we have a surjective homomorphism

G → G, s1 7→ a, s2 7→ b, s3 7→ c, s4 7→ ac; furthermore, this homomorphism restricts to isomorphisms Wi → G, i = 1, 3, 4. Note that by construction, for any regular map M and any dual form M† of M, we have that G(M) ∼= G(M†).

For convenience, we record the following result.

Lemma 2.1. Let M be a regular map with monodromy group G = ha, b, ci, where o(ab) = k, o(bc) = l, and let m = o(bac). If G(M) = hs1, s2, s3, s4 | C ∪ R i, where k R = R1 ∪ R3 ∪ R4, then the relation (s2s3s4) = 1 is implied by those contained in l R1, and the relation (s2s1s4) = 1 is implied by those contained in R3.

Proof. Indeed, θ1(s2s3s4) = bcac = ba, which has order k in G. Similarly,

θ3(s2s1s4) = bc, which has order l in G. 12

3 The Coxeter-Petrie Complexes of the Platonic Solids.

In this section, we shall consider in more detail the structure of the Coxeter-Petrie group of a Platonic solid. Thus, the map in question is M = (B, a, b, c), where a, b, c satisﬁes the deﬁning relations given by the Coxeter diagram:

k l

ag bg c g and where {k, l} = {3, 3}, {3, 4}, {3, 5}, i.e., M is the tetrahedron, octahedron (cube), or the icosahedron (dodecahedron), respectively. In turn, the Coxeter-Petrie group has generators s1, s2, s3, s4 satisfying the Coxeter relations depicted in the diagram

s1 @ k g @ @ m l sg2 sg4

s3 g together with the additional relations R1, R3, R4 described in Section 2.

Theorem 3.1. Let M be a Platonic solid with monodromy group G = ha, b, ci, vertex valency l, face valency k, and length of Petrie polygons m. Then the corresponding Coxeter-Petrie group has the presentation

G = hs1, s2, s3, s4 | C ∪ R i,

l k where R consists precisely of the two relations (s2s1s4) = (s2s3s4) = 1.

Proof. We must determine the kernels of the surjections of the parabolic subgroups

θi : Wi → ha, b, ci. Since hs1, s2, s3i → ha, b, ci is already an isomorphism, we see that k R4 = ∅. Furthermore, from Lemma 2.1, we conclude that the relations (s2s3s4) = 1 l and (s2s1s4) = 1 necessarily hold in G. We shall show that no other relations are k required. Let N1 denote the normal closure in W1 of (s2s3s4) ; similarly let N3 13

l denote the normal closure in W3 of (s2s1s4) . Thus we have surjective mappings ∼ Wi/Ni → G (= W4), i = 1, 3. However, the mappings W4 → Wi/Ni, i = 1, 3 given by

s1 7→ s3s4, s2 7→ s2, s3 7→ s3,

s1 7→ s1, s2 7→ s2, s3 7→ s1s4, are easily checked to determine well-deﬁned homomorphisms, and are inverse to the homomorphisms Wi/Ni → W4. The result follows.

Next, we calculate upper bounds on the orders of the Coxeter-Petrie groups corresponding to the Platonic solids. Note that it is suﬃcient to consider one solid from each dual pair.

Theorem 3.2. Let M be a Platonic solid, and let G = G(M) be its Coxeter-Petrie group. Then

1. |G| ≤ 96 if M is the tetrahedron;

2. |G| ≤ 384 if M is the octahedron;

3. |G| ≤ 3840 if M is the icosahedron.

Proof. We consider the above results in turn.

Tetrahedron. Rather than carry out a coset enumeration (which is fairly easy in this case), we take a more explicit approach. The results of this approach will be important in actually constructing models for the Coxeter-Petrie group. In

the present case, we have G = hs1, s2, s3, s4i with the Coxeter relations inferred from the diagram

s1 @ 3 g @ @ 4 3 sg2 sg4

s3 g 14

3 3 together with the additional relations (s2s1s4) = (s2s3s4) = 1. Deﬁne the elements x, y ∈ G by setting x = s1s3s4, y = s2xs2; note that since s1, s3, s4 commute, x2 = y2 = 1 in G. Deﬁne the subgroup N of G by setting N = hx, yi. We state and prove a number of claims; bear in mind that if i, j, k is any 3 permutation of 1, 2, 4 or of 2, 3, 4, then we have that (sisjsk) = 1.

(1) N is elementary abelian. Indeed, note that

xy = s1s3s4s2(s1s3s4)s2

= s1s3s4s2(s3s4s1)s2 ( since s1, s3, s4 commute)

= s1(s3s4s2s3s4)s1s2 3 = s1(s2s4s3s2)s1s2 ( since (s3s4s2) = 1)

= s1s2s4s3(s2s1s2) 3 = s1s2s4s3(s1s2s1) ( since (s2s1) = 1)

= s1s2(s4s3s1)s2s1

= s1s2(s1s3s4)s2s1 ( since s1, s3, s4 commute)

= (s1s2s1)s3s4s2s1 3 = (s2s1s2)s3s4s2s1 ( since (s1s2) = 1)

= s2s1(s2s3s4s2)s1 3 = s2s1(s4s3s2s4s3)s1 ( since (s2s3s4) = 1)

= s2s1(s4s3)s2s4s3s1

= s2s1(s3s4)s2(s4s3s1) ( since s3, s4 commute)

= s2s1s3s4s2(s1s3s4) ( since s1, s3, s4 commute) = yx, proving the result.

(2) N E G. This is carried out in steps.

(a) s1xs1 = x, s1ys1 = xy. The ﬁrst statement is, of course, obvious. As for the second, 15

s1ys1 = (s1s2s1)s3s4s2s1

= (s2s1s2)s3s4s2s1

= s2s1(s2s3s4s2)s1 3 = s2s1(s4s3s2s4s3)s1 ( since (s2s3s4) = 1)

= s2s1(s4s3)s2s4s3s1

= s2s1(s3s4)s2(s4s3s1)

= s2s1s3s4s2(s1s3s4) = yx = xy.

(b) s2xs2 = y, s2ys2 = x. This follows from the deﬁnitions of x, y.

s3ys3 = s3s2s1(s3s4)s2s3

= s3s2s1(s4s3)s2s3

= s3s2s1s4(s3s2s3) 3 = s3s2s1s4(s2s3s2) ( since (s3s2) = 1)

= s3(s2s1s4s2)s3s2 3 = s3(s4s1s2s4s1)s3s2 ( since (s2s1s4) = 1)

= (s3s4s1)s2(s4s1s3)s2

= (s1s3s4)s2(s1s3s4)s2 = xy.

(d) As a result of the above, we see that the elements s1, s2, s3 normalize

N. However, since s1s3s4 = x ∈ N, we conclude that s4 also normalizes N, and so N E G.

Furthermore, since s4 ≡ s1s3 ( modulo N), we see that G/N is a homomorphic image of the Coxeter group of order 24 corresponding to the diagram

3 3

g g g Therefore, we conclude that [G : N] ≤ 24; since N is elementary abelian and generated by x, y, we have that |N| ≤ 4. From this it follows that |G| ≤ 24 · 4 = 96. 16

Octahedron. Here, the Coxeter-Petrie group G = hs1, s2, s3, s4i has Coxeter relations taken from the diagram

s1 @ 3 g @ @ 6 4 sg2 sg4

s3 g 3 4 together with the Petrie relations (s2s3s4) = (s2s1s4) = 1.

3 Lemma 3.3. The element (s2s4) is in the center of G.

Proof. We set Gi, i = 1, 3, 4 to be the obviously deﬁned rank-3 parabolic subgroups in G; as already noted above, these are isomorphic with the corre- ∼ sponding parabolic subgroups of the Coxeter group W: Gi = Wi, i = 1, 3, 4. 3 We take as known the result that (s2s1s3) is the involution that generates the ∼ ∼ ∼ center of G4 = W4 = W (B3) = S4 × Z2, where W (B3) is the Coxeter group of

type B3. Next we have an isomorphism G4 → G3 given by

s1 7→ s1, s2 7→ s2, s3 7→ s1s4,

3 3 3 under which (s2s1s3) 7→ (s2s4) . Therefore, it follows that (s2s4) is central in

G3. Similarly, we have an isomorphism G4 → G1, given by

s1 7→ s3s4, s2 7→ s2, s3 7→ s3;

3 3 3 under this isomorphism (s2s1s3) 7→ (s2s4) . Therefore, (s2s4) is also central in 3 G1, which implies that (s2s4) is central in G.

By analogy with the previous case, we deﬁne x, y, z ∈ G by setting x = 2 2 2 s1s3s4, y = s2xs2, and z = s4ys4. Obviously x = y = z = 1; and deﬁne the subgroup N of G by setting N = hx, y, zi. We shall show that N is a nor-

mal elementary abelian subgroup of G; since G = G4N and |G4| = 48, this is enough. 17

(1) N is elementary abelian. First, note that in proving that xy = yx in the

tetrahedral case, the only identities that were used were s1s2s1 = s2s1s2 3 and (s2s4s3) = 1, which continue to be valid in both the octahedral and

icosahedral cases. Next, that x commutes with z is clear since z = s4yx4

and s4 commutes with x. The proof will be complete if we can show that

s2zs2 = z for then yz = zy follows by conjugating xz = zx by s2. We have

s2zs2 = s2s4s2(s1s3s4)s2s4s2

= s2s4s2(s4s1s3)s2s4s2

= (s2s4s2s4)s1s3s2s4s2 3 = (s2s4) s4s2s1s3s2s4s2 3 3 = s4s2s1s3s2s4s2(s2s4) (since (s2s4) is central)

= s4s2s1s3s4s2s4 = z. Therefore, N is elementary abelian, as claimed.

(2) N E G. In steps:

(a) s1xs1 = x, s1ys1 = xy, s1zs1 = xz. The ﬁrst statement is trivial. Next,

s1ys1 = s1s2(s1s3s4)s2s1

= s1s2(s4s3s1)s2s1

= s1s2s4s3(s1s2s1)

= s1s2s4s3(s2s1s2)

= s1(s2s4s3s2)s1s2 3 = s1(s3s4s2s3s4)s1s2 (since (s2s4s3) = 1)

= s1s3s4s2(s3s4s1)s2

= s1s3s4s2(s1s3s4)s2 = xy. Finally, 18

s1zs1 = s1s4s2s1s3s4s2(s4s1)

= s1s4s2(s1s3s4)s2(s1s4)

= s1s4s2(s3s4s1)s2s1s4

= s1s4s2s3s4(s1s2s1)s4

= s1s4s2s3s4(s2s1s2)s4

= s1(s4s2s3s4s2)s1s2s4 3 = s1(s3s2s4s3)s1s2s4 (since (s4s2s3) = 1)

= s1s3s2(s4s3s1)s2s4

= s1s3s2(s1s3s4)s2s4

= s1s3(s4s4)s2s1s3s4s2s4 = xz.

(b) s2xs2 = y, s2ys2 = x, s2zs2 = z. The ﬁrst two statements follow by

deﬁnition of x, y. That s2zs2 = z was already proved above.

s3ys3 = s3s2s1(s3s4s2s3) 3 = s3s2s1(s2s4s3s2s4) (since (s3s4s2) = 1)

= s3(s2s1s2)s4s3s2s4

= s3(s1s2s1)s4s3s2s4

= (s3s1)s2s1s4s3s2s4

= (s1s3)s4s4s2s1(s4s3)s2s4

= s1s3s4s4s2s1(s3s4)s2s4 = xz. Finally, 19

s3zs3 = s3s4s2s1s3s4s2(s4s3)

= s3s4s2s1s3s4s2(s3s4)

= s3s4s2s1(s3s4s2s3s4) 3 = s3s4s2s1(s2s3s4s2) (since (s3s4s2) = 1)

= s3s4(s2s1s2)s3s4s2

= s3s4(s1s2s1)s3s4s2

= (s3s4s1)s2s1s3s4s2

= (s1s3s4)s2s1s3s4s2 = xy.

(d) Since s4 ≡ s1s3 (modulo N), we conclude that N E G, as claimed.

Icosahedron. Here, the Coxeter-Petrie group G = hs1, s2, s3, s4i has Coxeter relations taken from the diagram

s1 @ 3 g @ @ 10 5 sg2 sg4

s3 g 3 5 together with the Petrie relations (s2s3s4) = (s2s1s4) = 1.

5 Lemma 3.4. The element (s2s4) is in the center of G.

5 Proof. To prove that (s2s4) is central in G, we use the known fact that 5 ∼ (s2s1s3) is central in G4 = W4; argue exactly is in Lemma 3.3.

By analogy with the previous cases, deﬁne x, y, z, w, u ∈ G by setting x =

s1s3s4, y = s2xs2, z = s4ys4, w = s2zs2, u = s4ws4; again, each of these elements is either an involution or is the identity. Deﬁne the subgroup N of G by setting N = hx, y, z, w, ui. We shall show that N is a normal elementary abelian

subgroup of G; since G = G4N, and |G4| = 120, this is enough. 20

(1) N is elementary abelian. The proof that xy = yx is proved exactly as in the

tetrahedral and octahedral cases. Thus, upon conjugating by s4, it follows immediately that xz = zx.

Next, if we can show that s1ws1 = xu, then it will follow that xu is an involution and hence xu = ux. Proving this assertion takes some work.

xu = s1s3s2s4s2(s1s3s4)s2s4s2s4

= s1s3s2s4s2(s3s4s1)s2s4s2s4

= s1s3s2(s4s2s3s4)s1s2s4s2s4 3 = s1s3s2(s3s2s4s3s2)s1s2s4s2s4 (since (s4s2s3) = 1)

= s1s3s2s3s2(s4s3)s2s1s2s4s2s4

= s1s3s2s3s2(s3s4)s2s1s2s4s2s4

= s1(s3s2s3s2s3)s4s2s1s2s4s2s4 5 = s1(s2s3s2s3s2)s4s2s1s2s4s2s4 (since (s3s2) = 1)

= s1s2s3s2(s3s2s4)s2s1s2s4s2s4 3 = s1s2s3s2(s4s2s3s4s2s3)s2s1s2s4s2s4 (since (s3s2s4) = 1)

= s1s2s3s2s4s2(s3s4)s2s3s2s1s2s4s2s4

= s1s2s3s2s4s2(s4s3)s2s3s2s1s2s4s2s4

= s1s2s3s2s4(s2s4s3s2)s3s2s1s2s4s2s4

= s1s2s3s2s4(s3s4s2s3s4)s3s2s1s2s4s2s4

= s1s2s3s2s4s3s4s2(s3s4s3)s2s1s2s4s2s4

= s1s2(s3s2s4s3)s4s2s4s2s1s2s4s2s4

= s1s2(s4s2s3s4s2)s4s2s4s2s1s2s4s2s4 5 = s1s2s4s2s3s2s4s2s4(s4s2) s1s2s4s2s4 5 5 = s1s2s4s2s3s2s4s2s1s4s2s4s2s4(s4s2) (since (s4s2) is central)

= (s1s2s4s2s3)s2s4s2s1s2s4s2s4s2.

Since s1ws1 = s1s2s4s2s3s1s4s2s4s2s1, we will have proved that s1ws1 = xu as soon as we show that

s2s4s2s1s2s4s2s4s2 = s1s4s2s4s2s1.

Notice that this is an equation not involving the involution s3; that is, it 21

suﬃces to prove this identity within the parabolic subgroup G3. However,

we know that G3 is mapped isomorphically onto G4 via s1 7→ s1, s2 7→

s2, s4 7→ s1s3. Under this isomorphism, the above equation is equivalent to the equation

s2s3s2s3s2s1s3s2 = s3s2s1s3s2s1

5 in G4. Using the fact that (s2s3) = 1 (and so s2s3s2s3s2 = s3s2s3s2s3), the above equation is seen to be valid, completing the veriﬁcation that

s1ws1 = xu, and hence the veriﬁcation that xu = ux. Since u = s4ws4,

and since s4 commutes with x, we conclude that xw = wx. Therefore, x is in the center of N.

Next, if we conjugate the equation xw = wx by s2, we get yz = zy. Like-

wise, if we conjugate xz = xz by s2, we get yw = wy. Proving that yu = uy

will follow from the assertion that s2us2 = u, for then yu = uy is obtained

by conjugating the equation xu = ux by s2. To this end, we have

s2us2 = s2s4s2s4s2(s1s3s4)s2s4s2s4s2

= s2s4s2s4s2(s4s1s3)s2s4s2s4s2

5 = (s2s4) s4s2s4s2s1s3s2s4s2s4s2

5 = s4s2s4s2s1s3s2s4s2s4s2(s2s4)

= s4s2s4s2s1s3s4s2s4s2s4 = u.

Therefore, y is also in the center of N.

Note that zw = wz and zu = uz follow by conjugating the equations

yu = uy and yw = wy by s4. Therefore z is in the center of N. Finally,

since zu = uz, and since s2us2 = u, we may conjugate zu = uz by s2 and get wu = uw. This concludes the proof that N is abelian.

(2) N E G. 22

(a) s1xs1 = x, s1ys1 = xy, s1zs1 = xz, s1ws1 = xu, s1us1 = xw. In fact the ﬁrst three equations follow exactly as in the octahedral case, since only relations that continue to be satisﬁed in the icosahedral case were used.

Next, note that the relations s1ws1 = xu, s1us1 = xw are equivalent

to each other in light of the fact that s1xs1 = x. Since we have already

noted that s1ws1 = xu, the second relation is valid, as well.

(b) s2xs2 = y, s2ys2 = x, s2zs2 = w, s2ws2 = z, s2us2 = u. We have

already proved that s2us2 = u; the remaining equations are obvious.

(c) s3xs3 = x, s3ys3 = xz, s3zs3 = xy, s3ws3 = xw, s3us3 = xu. The ﬁrst three relations were proved in the octahedral using identities valid

in the icosahedral case. Next, we have xw = s1s3s4w = s3s1us4 =

s3xws1s4 = xs3ws1s4. Therefore, w = s3ws1s4 from which it follows

that s3ws3 = ws1s3s4 = wx = xw. The proof that s3us3 = xu follows

by conjugating s3ws3 = xw by s4.

(d) Since s4 ≡ s1s3 (modulo N), we conclude that N E G.

We turn now to the determination of the structure of G(M), where M is a Platonic solid. As above, these will be taken up individually for the tetrahedron, ∼ octahedron and icosahedron. Note ﬁrst that the mapping G → G = G4 given by

s1 7→ a, s2 7→ b, s3 7→ c, s4 7→ ac, determines a surjective homomorphism. Therefore, we see that G has the structure of a semidirect product N o W , where W = G4 = hs1, s2, s3i. In each case, N will be an elementary abelian 2-group (and in fact will be the elementary abelian normal subgroup arising in the proof of Theorem 3.2), and hence can be regarded as a vector space over the binary ﬁeld F2. Therefore, the structure of N o W is determined by a linear representation of W on N, i.e., the multiplication in N o W is given by

(n, w) · (n0, w0) = (n + w(n0), ww0), n, n0 ∈ N, w, w0 ∈ W, 23

and where w(n0) is in terms of the representation of W on N.

In what follows, we shall, in each case, explicitly construct a semidirect product of the form N o W , where N is an elementary abelian 2-group and W is the monodromy group of the corresponding Platonic solid. Furthermore, we shall show that this group is a homomorphic image of the Coxeter-Petrie group G, thereby reversing the inequalities given in Theorem 3.2.

We now itemize the three cases.

Tetrahedron. Here, take     α  N =   | α, β ∈ F2 .  β 

We take the matrix representation of W on N to be that determined by       1 1 0 1 1 1 s1 7→   , s2 7→   , s3 7→   . 0 1 1 0 0 1

Map G → N o W via       0 1 si 7→   , si , i = 1, 2, 3, s4 7→   , s1s3 . 0 0

It is routine to check that this determines a surjective homomorphism. Since |G| ≤ 96, we conclude that this must be an isomorphism.

Octahedron. In this case, set     α       N =  β  | α, β, γ ∈ F2 .     γ 

In this case the matrix representation is given by       1 1 1 0 1 0 1 1 1             s1 7→  0 1 0  , s2 7→  1 0 0  , s3 7→  0 0 1  .       0 0 1 0 0 1 0 1 0 24

Map G → N o W via       0 1             si 7→  0  , si , i = 1, 2, 3, s4 7→  0  , s1s3 .       0 0 Again, this is checked to be a surjective homomorphism; comparing group orders shows it to be an isomorphism.

Icosahedron. In this case, set    α       β         N =  γ  | α, β, γ, δ, ∈ F2 ,        δ        on which W is represented by

      1 1 1 1 1 0 1 0 0 0 1 1 1 1 1        0 1 0 0 0   1 0 0 0 0   0 0 1 0 0              s1 7→  0 0 1 0 0  , s2 7→  0 0 0 1 0  , s3 7→  0 1 0 0 0  .              0 0 0 0 1   0 0 1 0 0   0 0 0 1 0        0 0 0 1 0 0 0 0 0 1 0 0 0 0 1

We map G → N o W by       0 1        0    0               si 7→  0  , si , i = 1, 2, 3 , s4 7→  0  , s1s3 .              0    0         0 0 Again, this can be checked to be an isomorphism.

For the tetrahedron and octahedron, there are alternative descriptions of the Coxeter-Petrie groups. 25

Theorem 3.5. Let G be the Coxeter-Petrie group of the tetrahedron. Then G ∼=

W (D4)/Z, where W (D4) is the Coxeter group of type D4 and Z is the center of

W (D4).

Proof. First of all, the Coxeter group W of type D4 has generators w1, w2, w3, w4 and relations taken from the Coxeter diagram:

w1 @ 3 g @ @ 3 3 wg2 wg4

w3 g

It is well known that W has order 192 and that its center Z(W ) is cyclic of order 2 [3, p. 141]. Furthermore, if i1, i2, i3, i4 is any permutation of 1, 2, 3, 4, then the element 3 (wi1 wi2 wi3 wi4 ) is the central involution in Z(W ). Using [1, pp. 66–68] this is easily deduced as follows. First of all, the Coxeter group W (Bn) of type Bn can be identiﬁed with the wreath product Z2 o Sn and hence acts on the set {±1, ±2,..., ±n}. With this identiﬁcation, the generators corresponding to the diagram

3 3...... 3 4

s1g sg2 sng−1 sgn correspond to the “signed permutations”

s1 = (1 2), s2 = (2 3), . . . , sn−1 = (n − 1 n), sn = (−n); here (i, i + 1) is the permutation ±i ↔ ±(i + 1), i = 1, 2, . . . , n − 1, and (−n) is the transposition ±n ↔ ∓n. The Coxeter group W (Dn) can be found inside W (Bn) as the subgroup of index 2 generated by (1 2), (2 3) ..., (n − 1 n) and (−(n − 1))(−n). In case n = 4, the generators can be identiﬁed thus:

w1 = (1 2)

@ 3 g @ w4 = (3 4)(−3)(−4) @ 3 3 w2g= (2 3) g

w3 = (3g 4) 26

The central involution of W (D4) is given by the permutation (−1)(−2)(−3)(−4); it is now a routine calculation to show that if i1, i2, i3, i4 is any permutation of 1, 2, 3, 4, 3 then the element (wi1 wi2 wi3 wi4 ) = (−1)(−2)(−3)(−4), and hence is the central involution in Z(W ).

Inside W/Z deﬁne the elements vi = wi, i = 1, 2, 3, v4 = w1w4 (read modulo Z).

Then one has that the elements v1, v2, v3, v4 satisfy the Coxeter relations depicted by

v1 @ 3 g @ @ 4 3 vg2 vg4

v3 g

3 3 together with the relations (v1v2v4) = 1 = (v3v2v4) . Clearly v1, v2, v3, v4 generate W/Z and so we get a surjective homomorphism G → W/Z. Comparing the group orders ﬁnishes the job.

Theorem 3.6. Let G be the Coxeter-Petrie group of the octahedron. Then G ∼=

W (D4) × Z2, where Z2 is a cyclic group of order 2.

Proof. As above, let W = W (D4) with relations taken from

w1 @ 3 g @ @ 3 3 wg2 wg4

w3 g

0 Let Z2 = hzi and let z be the nontrivial central element of W . In W × Z2, deﬁne 0 the elements v1 = w1, v2 = w2, v3 = w1w3z, v4 = w4zz . Then one shows that the elements v1, . . . , v4 satisfy the Coxeter relations depicted by

v1 @ 3 g @ @ 6 4 vg2 vg4

v3 g 27

4 3 and that (v1v2v4) = (v3v2v4) = 1. As W ×Z2 = hv1, v2, v3, v4i, we obtain a surjective homomorphism G → W × Z2. A comparison of group orders shows that this is an isomorphism.

4 Coxeter-Petrie Complexes Corresponding to Regular Aﬃne Maps

The aﬃne Coxeter groups of rank 3 are deﬁned by

2 r k l ∆(k, l) = hs1, s2, s3 | si = (s1s3) = (s1s2) = (s2s3) = 1i, where {k, l} = {3, 3}, (in which case r = 3), {4, 4}, or {3, 6} (in which case r = 2). The corresponding hypermaps are called the universal affine hypermaps; note that except when {k, l} = {3, 3}, these hypermaps are maps, the universal affine maps. We say that the map M is a regular affine map if it is of the form M ∼= M(∆(k, l)/K), where K E ∆(k, l), {k, l} = {4, 4} or {3, 6}, and where K ∩ P = 1 for all rank 2 parabolic subgroups P ≤ ∆(k, l). This ensures that M also has the same face and vertex valencies as M(∆(k, l)); put differently, this says that the orbit mapping M(∆(k, l)) → M(∆(k, l)/K) is unramified.

We begin by elucidating the structure of the above rank-3 aﬃne Coxeter groups. First of all, we remark that in the context of Lie theory, the above groups are usually denoted as follows: Af2 = ∆(3, 3), Bf2 = ∆(4, 4), Gf2 = ∆(3, 6). Furthermore, it is well known that the above groups can each be exhibited as the semidirect product of the corresponding rank-2 Coxeter group with the corresponding “coroot lattice;” see [4, (6.5)]. That the coroot lattice can be replaced by a suitable ring of algebraic integers of the form Z[ω], where ω is a root of unity was already noticed by Jones and Singerman in [6, Section 7]. Here, we give here an approach that is both self-contained and more concrete. 28

Proposition 4.1. There is an isomorphism

∼ ∆(k, l) = Z[ω] o hω, τi, where (k, l) = (3, 3), (4, 4), or (3, 6), ω = e2πi/l, ω acts by left multiplication on Z[ω], ∼ and where τ acts as conjugation. (Note that hω, τi = D2l.)

Proof. In each case above we have a semidirect product of an additive group (Z[i], or Z[ζ] = Z[ω], ζ = e2πi/3, ω = e2πi/6) by a dihedral group. We write elements of this group as ordered pairs (a, g), remembering that the group multiplication is given by

(a, g) · (a0, g0) = (a + ga0, gg0).

Case 1. (k, l) = (4, 4). Inside the group W = Z[i] o hi, τi, deﬁne the elements

w1 = (1, −τ), w2 = (0, τi), w3 = (0, τ).

Claim. We have

(i) W = hw1, w2, w3i;

(ii) The mapping si 7→ wi, i = 1, 2, 3, determines an isomorphism

∼ α : ∆(4, 4) →= W.

Proof. (i) is entirely routine. For (ii), it is routine to check that w1, w2, w3 satisfy the Coxeter relations satisﬁed by s1, s2, s3 in ∆(4, 4). Therefore, the assignment si 7→ wi, i = 1, 2, 3, deﬁnes a surjective (by part (i)) homomorphism ∆(4, 4) → W .

Inside ∆ = ∆(4, 4) is the subgroup K = ha1 := s1s2s3s2, ai := s3s2s1s2i. Note that

(i) a1, ai have inﬁnite orders since their homomorphic images in W do. Indeed,

a direct calculation reveals that a1 7→ (1, 1) and that ai 7→ (i, 1) which have inﬁnite orders in Z[i] o hi, τi.

(ii) a1, ai commute (direct veriﬁcation); 29

(iii) α(K) = Z[i] × {1} ⊆ W ;

(iv) K is a rank 2 free abelian group and the restriction of α to K determines an isomorphism

α|K : K → Z[i].

−1 (v) K E ∆. Indeed, easy calculations reveal that s1a1s1 = a1 , s1ais1 = ai, s2a1s2 =

ai, s2ais2 = a1. Since s3 ∈ hs1, s2,Ki, this is enough.

From the above, we get a commutative diagram

- - - ∼ - 1 K ∆ ∆/K = hs1, s2i 1

∼= ∼=

? ? ? - - - ∼ - 1 Z[i] W W/Z[i] = hw1, w2i 1 By the Five Lemma, the middle vertical arrow is an isomorphism, proving (ii).

Case 2. (k, l) = (3, 3). Inside the group W = Z[ζ] o hi, ζi, ζ = e2πi/3, deﬁne the elements −1 w1 = (1, τ), w2 = (0, τζ), w3 = (0, τζ ).

Set K = ha1 = s1s2s3s2, aζ = s3s1s3s2i and prove the obvious analogs of (i)–(v) of the above claim, proving the result in this case.

Case 3. (k, l) = (3, 6). Inside the group W = Z[ω] o hi, ωi, ω = e2πi/6, deﬁne the elements

w1 = (1, −τ), w2 = (0, τω), w3 = (0, τ).

Set K = ha1 = s1s2s3s2s3s2, aω = s3s2s1s2s3s2i and prove the obvious analogs of (i)–(v) of the above claim, proving the result in this case, as well.

Next, we shall determine those normal subgroups K E ∆(k, l), such that K triv- ially intersects each proper parabolic subgroup of ∆(k, l). The following was already pointed out in [6, p. 304]. 30

2πi/l Lemma 4.2. The normal subgroups K E Z[ω] o hω, τi, ω ∈ {e , l = 3, 4, 6}, not meeting any proper parabolic subgroups are of the form I × {1}, where I ⊆ Z[ω] is an ideal of Z[ω] which is closed under complex conjugation.

Proof. We proof this for ω = i, leaving the remaining (similar) cases to the reader. Let Z be the normal subgroup Z = Z[i] o {1} ⊆ W = Z[i] o hi, τi. If K 6⊆ Z, then K contains an element of the form α = (a, g) ∈ Z[i] o hi, τi, where 1 6= g ∈ hi, τi. Since K E W , it follows easily that, in fact, K must contain an element of the form α = (a, −1) ∈ W . From this, we infer that Z 2 = 2Z[i] o {1} = [α, Z] ⊆ K, from which it follows that K contains one of the elements (0, −1), (1, −1), (i, −1), −1 or (1 − i, −1). But (0, −1) = w2w3w2w3, (0, i) (i, −1)(0, i) = (1, −1) = w1w3, and

(1 − i, −1) = w1w2w1w2, proving the result.

Using the above, we can determine a presentation for the monodromy group of a regular affine map of type (4, 4). In the oriented case, this is already contained in [6, p. 304]. Thus, let M be a regular affine map of type (4, 4), i.e., one of the form M(∆(4, 4)/K), where as proved above, K can be identified with an ideal I ⊆ Z[i] invariant under complex conjugation. Since Z[i] is Euclidean with respect to the norm map, we have that Z[i] is a principal ideal domain. Therefore, we may write I in the form I = (x + yi)Z[i], where x, y ∈ Z.

The next two results are already contained in [3, Section 8.3].

Lemma 4.3. The ideal I = (x + yi)Z[i] is invariant under complex conjugation precisely when one of the following three conditions holds:

1. x = 0;

2. y = 0;

3. x = ±y. 31

As a result of Lemma 4.3, any ideal of Z[i] invariant under complex conjugation is of the form nZ[i] or n(1 + i)Z[i] for some non-negative integer n.

Corollary 4.3.1. The monodromy group of a ﬁnite regular aﬃne map of type (4, 4) has one of the following presentations:

1. G = ha, b, c | a2 = b2 = c2 = (ac)2 = (ab)4 = (bc)4 = (abcb)n = 1i, for some positive integer n;

2. G = ha, b, c | a2 = b2 = c2 = (ac)2 = (ab)4 = (bc)4 = (acbcab)n = 1i, for some positive integer n.

Proof. The regular aﬃne map M has the form M(∆(4, 4)/K), where, as shown n n n n above, K is one of the groups K1 = ha1 = (s1s2s3s2) , ai = (s3s2s1s2) i or K2 = n −1 n n h(a1ai) , (a1 ai) i. Therefore, K1 is the normal closure in ∆(4, 4) of a1 and K2 is the n n normal closure in ∆(4, 4) of (a1ai) = (s1s3s2s3s1s2) .

We turn now to the Petrie polygons in M. Recall that the length m of the Petrie polygons in M is the order of the glide reﬂection w1w2w3 (modulo I).

Proposition 4.4. If I is either nZ[i] or n(1+i)Z[i], n ∈ Z, and if K = I ohi, τi E W , then

m = o(w1w2w3) = 2n (modulo K).

Proof. A direct calculation reveals that

w1w2w3 = (1, −iτ).

From this it follows that for any non-negative integer k,

2k 2k+1 (w1w2w3) = (k(1 − i), 1), (w1w2w3) = (k + 1 − ki, −iτ) 6∈ K.

The result follows easily. 32

As a result of the above, we can itemize the possible presentations for the Coxeter- Petrie group corresponding to a regular affine type (4, 4) map. For any fixed positive integer n, the Coxeter-Petrie group G has generators s1, s2, s3, s4 having defining relations deduced from the Coxeter diagram:

s1 @ 4 g @ @ 2n 4 sg2 sg4

s3 g together with the additional relations

4 4 n n n (1) (s2s1s4) = (s2s3s4) = (s1s2s3s2) = (s1s2s1s4s2) = (s3s4s2s3s2) = 1, or

4 4 n (2) (s2s1s4) = (s2s3s4) = (s1s3s2s3s1s2) = 1.

Note that in case (2) above, the remaining two relations, i.e., those obtained from n (s1s3s2s3s1s2) = 1 by replacing s1 by s3s4 and by replacing s3 by s1s4 both reduce to n the relation (s4s2s4s2) = 1, which is already speciﬁed among the Coxeter relations.

Finally, we summarize the relevant results for the ﬁnite regular aﬃne maps of type (3, 6). Namely, that these maps are organized into two families (corresponding to ideals of the form nZ[ω], and n(1+ω)Z[ω], where ω = e2πi/6), and having monodromy groups

1. G = ha, b, c | a2 = b2 = c2 = (ac)2 = (ab)3 = (bc)6 = (abcbcb)n = 1i, for some positive integer n;

2. G = ha, b, c | a2 = b2 = c2 = (ac)2 = (ab)3 = (bc)6 = (acbcbcabcb)n = 1i, for some positive integer n.

2 As for the Petrie polygons, we have that w1w2w3 = (1, −ωτ), (w1w2w3) = (1 − ω, 1), from which it follows that if K = nZ[ω] o hω, τi or if K = n(1 + ω)Z[ω] o hω, τi, then 33

o(w1w2w3) = 2n (modulo K).

Finally, for any ﬁxed positive integer n, the Coxeter-Petrie group G has generators s1, s2, s3, s4 having deﬁning relations deduced from the Coxeter diagram:

s1 @ 3 g @ @ 2n 6 sg2 sg4

s3 g together with the additional relations

6 3 n (1) (s2s1s4) = (s2s3s4) = (s1s2s3s2s3s2) = 1, or

6 3 n (2) (s2s1s4) = (s2s3s4) = (s1s3s2s3s2s3s1s2s3s2) = 1.

In case (1) above, there are, ostensibly, two additional relations needed, i.e., those n obtained from (s1s2s3s2s3s2) = 1 by replacing replacing s1 by s3s4 and by replacing n n s3 by s1s4 reduce to the relations 1 = (s1s2s1s4s2s1s4s2) = (s2s1s2s4s2s4s1s2) , and n n 1 = (s4s3s2s3s2s3s2) = (s2s3s4s2s4s2s3s2) = 1. However, as s2s1s2s4s2s4s1s2 is conjugate to s2s4s2s4, and as s2s3s4s2s4s2s3s2 is conjugate to s4s2s4s2, we see that these relations are unnecessary. Case (2) is a bit more subtle, as follows. We have

s1s3s2s3s2s3s1s2s3s2 = a1aω

= (s1s2s3s2s3s2)(s3s2s1s2s3s2)

= (s1s2s3s2s3s2)(s3s1s2s1s3s2)

s17→s3s4 7→ (s3s4s2s3s2s3s2)(s4s2s4s2)

= (s2s3s4s2s4s2s3s2)(s4s2s4s2)

Since a1, aω commute, so do the factors (s2s3s4s2s4s2s3s2) and (s4s2s4s2). As both are n conjugates of s4s2s4s2, we see that the relation obtained from (s1s3s2s3s2s3s1s2s3s2) = n 1 obtained via s1 7→ s3s4 is already implied by the Coxeter relation (s4s2s4s2) = 1. 34

n Similarly, the relation obtained from (s1s3s2s3s2s3s1s2s3s2) = 1 by s3 7→ s1s4 already n follows from the relation (s4s2s4s2) = 1.

Final Remark. While the Coxeter-Petrie groups of the Platonic maps are finite, this question remains unsettled for the Coxeter-Petrie groups of the regular affine maps. For very small values of the parameter n, the use of GAP (see [9]) gives finite group orders, but the results are far from being even suggestive. References

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3. H. S. M. Coxeter, W. O. J. Moser, Generators and relation for discrete groups. Fourth edition, Ergebnisse der Mathematik und ifrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980.

4. J. E. Humphries, Reﬂection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge, 1990.

5. G. A. Jones and J. S. Thornton, Operations on maps, and outer automorphisms, J. Comb. Theory, Ser. B 35, 1987, 93-103.

6. G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307.

7. R. Nedela and M. Skoviera,ˇ Regular embeddings of canonical double coverings of graphs, J. Comb. Theory, Series B, 67 (1996), 249-277;

8. M. Ronan, Lectures on Buildings, Boston: Academic Press, 1989.

9. Martin Schönert et al, GAP – Groups, Algorithms, and Programming. Lehrstuhl D fürMathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Ger- many, first edition, 1992.