/^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 403-410 https://doi.org/10.26493/1855-3974.1749.84e (Also available at http://amc-journal.eu) Regular self-dual and self-Petrie-dual maps of arbitrary valency* Jay Fraser, Olivia Jeans The Open University, Milton Keynes, U.K. Jozef Siran The Open University, Milton Keynes, U.K. and Slovak University of Technology, Bratislava, Slovakia Received 13 July 2018, accepted 18 December 2018, published online 30 January 2019 Abstract The existence of a regular, self-dual and self-Petrie-dual map of any given even valency has been proved by D. Archdeacon, M. Conder and J. Siran (2014). In this paper we extend this result to any odd valency > 5. This is done using algebraic number theory and maps defined on the groups PSL(2,p) in the case of odd prime valency > 5 and valency 9, and using coverings for the remaining odd valencies. Keywords: Regular map, automorphism group, self-dual map, self-Petrie-dual map. Math. Subj. Class.: 05C25, 05C10 1 Introduction In this paper we consider regular maps (that is, cellular embeddings of graphs on closed surfaces) with the highest 'level of symmetry', which are, in addition, invariant under the operators of duality and Petrie duality. Regular maps have been addressed in a number of papers and we refer here to the latest survey [11] for a large number of details; here we just sum up the essentials needed for our purposes. *The third author thanks the Banff International Research Station (BIRS) for supporting his participation in the stimulating workshop 'Symmetries of Surfaces, Maps and Dessins' (September 2017) at which the statement of the problem addressed in this paper crystallised and attracted the attention of a number of colleagues, notably M. Conder who produced important initial examples. The same author also acknowledges support from the APVV Research Grants 15-0220 and 17-0428, and from the VEGA Research Grants 1/0026/16 and 1/0142/17. E-mail addresses: jay.fraser@open.ac.uk (Jay Fraser), olivia.jeans@open.ac.uk (Olivia Jeans), , jozef.siran@open.ac.uk (Jozef Siran) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 403 Ars Math. Contemp. 16 (2019) 331-348 From an algebraic point of view, a regular map M can be identified with a finite group G with three distinguished involutory generators x,y,z and relators (yz)k, (zx)£ and (xy)2 so that x and y commute; we will formally write M = (G; x, y, z) to encapsulate the situation. The pair (k, £) is the type of M, and we will assume throughout that k, £ > 3; the type is hyperbolic if 1/k + 1/£ < 1/2. Geometrically and topologically, elements of G may be identified with flags (which correspond to mutually incident vertex-edge-face triples) and left cosets of the subgroups (x, y), (y, z) and (z, x) represent edges, vertices and faces of the embedded graph, with incidence given by non-empty intersection of cosets. Moreover, left multiplication by elements of G on the cosets induce map automorphisms of M and, in fact, G is isomorphic to the (full) automorphism group Aut(M) of M. Conjugates of x, y and z, respectively, induce automorphisms that locally act on M as reflections along some edge, in some edge, and in an axis of some corner of M. Similarly, conjugates of r = yz and s = zx represent rotations about vertices and face centres of the map; in particular, every vertex has valency k and every face is bounded by a closed walk of length The map M is orientable (meaning that its underlying surface is orientable) if and only if G+ = (r, s) is a subgroup of G of index two, and non-orientable otherwise. Thus, in the non-orientable case, the entire group G can be generated by the two rotations r and s only, and the involutions x, y, z are then expressible in terms of r and s; in such a situation we also write M = (G; r, s). Every automorphism of a map, regarded as a permutation of flags that preserves incidence along and across edges and within corners, is completely determined by its action on a single flag. If the automorphism group is transitive (and hence regular) on flags, one may identify the group with the flag set and arrive at the description outlined above. But even then a map may still exhibit 'external symmetries' induced by invariance under the operators of duality and Petrie-duality. The two operators are well known; informally, duality interchanges the roles of vertices and faces, and the Petrie dual of a map is formed by re-embedding its underlying graph so that the new faces are the left-right ('zig-zag') closed walks in the original map. A map is self-dual or self-Petrie-dual if it is isomorphic to its dual or Petrie dual, respectively. In the case of a regular map M = (G; x, y, z) as above, it is also well known (cf. [11]) that M is self-dual if and only if the group G admits an automorphism interchanging x with y and fixing z, and M is self-Petrie-dual if G has an automorphism interchanging x with xy and fixing y and z. In [1], regular maps that are both self-dual and self-Petrie-dual have been said to have trinity symmetry. The natural question regarding the existence of regular maps with trinity symmetry for any valency was raised more than four decades ago. In [15] it was suggested that the map M = (G; x, y, z) for the group G = (x, y, z; x2, y2, z2, (xy)2, (yz)2n, (zx)2n, (xyz)2n, (xzyzxyz)2) is a regular map with trinity symmetry, of valency 2n for every n > 1. This was eventually proved in [1] in a much more general form, including also invariance under the so-called hole operators that represent additional levels of 'external symmetries' not discussed here. However, the question remained almost completely open for odd-valent regular maps with trinity symmetry, as pointed out by the third author at the 2017 BIRS Workshop 'Symmetries of Surfaces, Maps and Dessins' [4, Part 4.7]. Note that such a map must necessarily be non-orientable because of self-Petrie-duality with Petrie walks of odd length. There is no such map of valency 3 since the only regular map of type (3,3) is the 2-skeleton of a tetrahedron. At the time of publication of the report [4] the only two sets of known examples of regular maps with trinity symmetry of odd valency k > 5 were those discovered computationally by M. Conder for 5 < k < 19 and the ones resulting from J. Fraser et al.: Regular self-dual and self-Petrie-dual maps of arbitrary valency 405 the work of G. Jones [7]. The method of Jones actually has potential to produce examples for infinitely many odd values of k but in [7] explicit examples have been given only for k = 15 (found also in [1] by a different method) and k = 455. Here we completely settle the problem by showing that for every odd k > 5 there exists a regular, self-dual and self-Petrie-dual map of valency k. Our strategy is to establish this result first for every prime k > 5 and for k = 9 by algebraic methods motivated by those used in [8], and applied to more detailed results of [6] on regular maps defined on linear fractional groups. We then extend this to non-prime odd values of k > 5 by an analogue of a covering tool from [1]. The paper is organised accordingly: in Sections 2 and 3 we review results on regular self-dual and self-Petrie-dual maps on linear fractional groups and develop the algebraic methods needed for our purposes, and in Section 4 we prove our general result and make a few concluding remarks. 2 Regular maps on linear fractional groups Classification of all orientably-regular maps with orientation-preserving automorphism group isomorphic to PSL(2, q) or PGL(2, q) follows from [9] and can be found in a somewhat more explicit form in [10]; the latter was re-interpreted and extended to regular maps (orientable or not) in [5]. Since we will be interested only in the special case of odd valency and face length, we just reproduce the corresponding part of the classification result here (the cases when one of the entries in the type of the map is even are more involved and we refer to [8] for details). Proposition 2.1. Let (k, t) = (5,5) be a hyperbolic pair with both entries odd and let p be an odd prime dividing neither k nor t. Let e = e(k, t) be the smallest positive integer j such that 2n | (pj — e„) for each n g {k, t} and some e„ g {+1, -1}, and let £n be a primitive 2n-th root of unity in GF(pe) if e„ = 1 or in GF(p2e) if e„ = —1. Further, let D = £2 + £-2 + el + e-2 = 0 and let k ~ Sfc R = ± £k 0 .0 C1 and S = ±(£k - C1) +e-%-1 -D i (Ù + £,-1)£k be elements of PSL(2,pe) if ek = 1 and of PSL(2,p2e) otherwise. Then, (a) the group Gk/ = {R, S} is isomorphic to PSL(2,pe), with R of order k and S of order £; (c) M = (Gk,e; R, S) is a regular map of type (k, £), which is non-orientable if and only if —D is a square in GF(pe). We note that if pe = ±1 (mod 10), the group PSL(2,pe) contains (up to conjugacy) two exceptional pairs R, S as above for (k, £) = (5, 5) with the property that {R, S} = A5; this case (omitted from [8, Theorem 2.2]) is addressed in [6]. However, this situation does not apply in what follows. Necessary and sufficient conditions for self-duality and self-Petrie-duality of the maps M = (Gk/; R, S) from Proposition 2.1 were established in [6]. As they are also quite complex we present here only a simple sufficient condition appearing as Corollary 4.3 in [6] which (in terms and notation of Proposition 2.1) can be re-stated as follows. Proposition 2.2. Let k > 5 be odd, and let p > 5 be a prime not dividing k. Further, let £ = k and let £ = = ^ be a primitive 2k-th root of unity in GF(pe) or in GF(p2e) 1 126 126 Ars Math. Contemp. 16 (2019) 445-463 for e = e(M) such that 3(£2 + £-2) + 2 = 0. Then, M = (GM; R, S) is a (non-orientable) self-dual and self-Petrie-dual regular map of valency k, with automorphism group isomorphic to PSL(2,pe). The condition 3(£2 +£-2)+2 = 0 is equivalent to 3(£+£-1)2 = 4 and for its fulfilment it is necessary that 3 be a square in GF(pe), p > 5. For e = 1, this holds if and only if p = ±1 (mod 12), and it is always the case if e > 2. But we can say more. Namely, the element Z = £2 in Proposition 2.2 is a primitive k-th root of unity in F = GF(pe) or F = GF(p2e), and the condition 3(Z + Z-1) + 2 = 0 represents a quadratic equation in the prime field Fp of F; it also says that Z + Z-1 € Fp. The last fact is equivalent to (Z + Z-1)p = Z + Z-1, which reduces to (Zp-1 - 1)(Zp+1 - 1) = 0 in F. It follows that either Z € Fp and p = 1 (mod 2k), or Z lies in a quadratic extension of Fp and p = -1 (mod 2k), and in both cases we have e =1 (recall that k is assumed to be odd). The bulk of Proposition 2.2 may now be restated in a form more suitable for our future use. Corollary 2.3. Let k > 5 be odd. Assume that there exists a prime p > 5 such that p = ±1 (mod 2k) and p = ±1 (mod 12), and a primitive k-th root of unity Z in a finite field of order p or p2 with the property that 3(Z + Z-1) +2 = 0. Then, there exists a non-orientable self-dual and self-Petrie-dual regular map ofvalency k with automorphism group PSL(2,p). 3 Algebraic preliminaries For any k > 3, let a be a primitive complex k-th root of unity; its minimal polynomial is the k-th cyclotomic polynomial. Let h = a + a-1 and let K = q(h) be the field obtained by adjoining h to the rationals. It is known [13, Proposition 2.16] that the ring o of algebraic integers of K is z(h). We will focus on the algebraic integer g = 3h + 2 € o. Observe that g = 0, for otherwise a would be a root of a quadratic polynomial over z, contrary to k > 3. Recall that the norm N(y) of an element y € O is defined as the product f]t at(y), where at denotes the injective homomorphism o ^ c into the field of complex numbers, uniquely determined by at (a) = at, and t ranges over all integers between 1 and (k -1)/2 that are relatively prime to k. It is well known that N(y) is an integer for any y € o, which is a consequence of the invariance of N(y) under the endomorphisms at. For the norm of our element g € o we thus have N (g) = f] t(3at(h) + 2), the product being taken over all t between 1 and (k - 1)/2, coprime to k. The y(k)/2 images at(h) appearing in this product are precisely the roots of the minimal polynomial ^(x) of degree y(k)/2 for h = a + a-1, see e.g. [8]. So, if ^(x) = f]t(x - 5 and k is a prime, then the norm N(g) is not equal to ±1, which means that g is then not a unit of the ring O. Indeed, let k > 5 be a prime, so that r = y(k)/2 = (k - 1)/2. By [12] we then also have ar-1 = 1, and the congruence (3.2) becomes N(g) = 2(k-1)/2 - 3 • 2(k-3)/2 = -2(fc-3)/2 (mod 9). It is easy to check that 2j = ±1 (mod 9) for a positive integer j if and only if j isamultiple of 3. This means that if k is prime, the norm N(g) can be congruent to ±1 (mod 9) only if (k - 3)/2 is amultiple of 3, giving a contradiction if k > 5. Further, from (3.1) with the help of ar = 1 and a0 = ±1 [12] it follows that if k > 5 is a prime, then N(g) is divisible neither by 2 nor by 3. We thus have: Lemma 3.1. If k > 5 is a prime, then N(g) = ±1; in particular, the non-zero element g GO is not a unit of the ring O. Moreover, for every prime factor p of N(g) one has p > 5. □ Consider now the field K' = Q(a), an extension of K of degree two. Let O' be the ring of algebraic integers of K'; it is well known [13, Theorem 2.6] that O' = Z(a), and, of course, [O' : O] = 2. The (integral) norm N'(z) of any z G O' is now the product at(z) taken over all injective homomorphism at: O' ^ C given by at(a) = a1 for t between 1 and k - 1 coprime to k, and again one has N'(z) G Z. The two norms, N on O and N' on O', are related by N'(y) = (N(y))2 for each y gO. We will keep assuming that k > 5 is an odd prime, and we let p > 5 be an arbitrary prime divisor of N(g), which exists by Lemma 3.1. We continue by considering the ideal (g,p) of O' = Z(a) generated by the elements g and p. Lemma 3.2. If k > 5 is a prime and ifp > 5 is a prime divisor of N(g), the ideal (g,p) is proper in the ring O'. Proof. Suppose that (g,p) = O', which means that 1 = Ag + Bp for some A, B G O'. Clearly A = 0, for otherwise 1 = N'(B)N'(p) = N'(B)pk-1 and so N'(B) would not be an integer. Now, 1 = N'(1) = N'(Ag + Bp) = f]a(Ag + Bp), where the product is being taken over all the y(k) = k - 1 embeddings a: O' ^ C. Expansion of this product gives N'(Ag + Bp) = N'(A)N'(g) + cp for some c G O'. Thus, cp G Z and so either c G Z or c = ±1/p. As p is a divisor of N'(g) = (N(g))2 and N' (A) is a non-zero integer, in either case it follows that N'(Ag + Bp) = 1, a contradiction. □ By Lemma 3.2, the ideal (g,p) is contained in some maximal ideal J = Jp of the ring O'. Since O' is a Dedekind domain, the ideal J has finite index in O' and so O'/J is a finite field F of characteristic p, that is, F = GF(pm) for some m > 1. Recalling our assumption of primality of k we show that the (multiplicative) order of the element a = a + J in the field F = O'/J is equal to k. Indeed, suppose this is not the case. Then, because of primality of k, the order of a in F would have to be one, meaning that a =1 in F. But then, since the element g = g + J is equal to zero in F, we would have 0 = g = 3(a + a -1) + 2 = 8 in F, a contradiction as p is odd. Observe also that k = p since no element in F has multiplicative order p. This way we have constructed a finite field F of characteristic p containing a primitive k-throot a of unity such that 3(a+a -1)+2 = 0. We now invoke the analysis immediately preceding Corollary 2.3 in Section 2, which fully applies to our situation. As the result we conclude that F is the prime field Fp if and only if a G Fp for p = 1 (mod 2k); otherwise 408 Ars Math. Contemp. 16(2019)445-463 F is a quadratic extension of Fp for p = —1 (mod 2k). In both cases we have p = ±1 (mod 12) because 3 has to be a square in Fp. Summing up, we have proved: Proposition 3.3. Let k > 5 be an odd prime and let a be a primitive complex k-th root of unity. Further, let g = 3(a + a-1) + 2 and let N(g) be the norm of g in the ring Z(a). Then, N (g) e {0, ±1}, every prime divisor p of N (g) satisfies p > 5, and p = ±1 (mod 2k) and p = ±1 (mod 12), and for every such p there is a finite field F of order p or p2 containing a primitive k-th root a of 1 such that g = 3(a + a -1 ) + 2 = 0 in F. □ 4 The main result To obtain a restricted version of our main result for prime valencies at least five we just need to put the pieces together. Indeed, taking Z = a in Proposition 3.3 and combining it with Proposition 2.2 and Corollary 2.3 immediately gives: Theorem 4.1. For every odd prime k > 5 there exists a prime p = ±1 (mod 2k) and p = ±1 (mod 12) such that PSL(2,p) is the automorphism group of a (non-orientable) regular, self-dual and self-Petrie-dual map of valency k. □ We know that there is no 3-valent regular map with trinity symmetry, but there is one of valency 32 that can be constructed by the machinery of Section 2 as follows. The element 2 is a primitive 9-th root of unity mod 73, and so is Z = 24 and its multiplicative inverse Z-1 = 25, with Z and Z-1 satisfying the condition 3(Z + Z-1) + 2 = 0 (mod 73). By Proposition 2.2 the group PSL(2,73) carries a self-dual and self-Petrie-dual regular map of valency 9. Based on Theorem 4.1 and the above remark we are now in position to prove a full version of our main result. As alluded to in the Introduction (Section 1), this will be done with the help of coverings, and more specifically using a non-orientable analogue of Theorem 2.1 of [1]. We state it here in a restricted version sufficient for our purpose. Theorem 4.2. If there is a non-orientable regular map of odd valency d > 5 with trinity symmetry and with automorphism group G, then for any odd integer n > 3 there is a non-orientable regular map of degree nd with trinity symmetry and automorphism group isomorphic to (Z„)1+IGI/4 x G. Sketch of a proof. As indicated, this result was proved in [1, Theorem 2.1] for orientable maps (and, in this category, in a much more general setting that included also external symmetries induced by hole operators). The parts of the proof in [1] that refer to regularity, self-duality and self-Petrie-duality apply almost word-by-word to the non-orientable case and we thus give only a sketch of the arguments here. We will assume familiarity with the theory of lifts of maps by corner voltage assignments as explained e.g. in [1, 2, 3]; a corner of a regular map M = (G; x, y, z) is any 2-subset of the form {g, gz} for g e G. Now let M = (G; x, y, z) be a regular map as in the statement. For odd n > 3 let IGI /2 H = Zn be the space of all |G|/2-tuples with entries from Zn and let E be the set of unit vectors (those with exactly one non-zero coordinate, equal to 1) in H. Define a corner voltage assignment a on flags of M - that is, on the elements of G - in the group H by assigning the |G|/2 two-element subsets {e, —e} for e e E to the |G|/2 corners {g, gz} for g e G in an arbitrary one-to-one fashion. By arguments in the proof of Theorem 2.1 in [1] that do not depend on orientability, the lift of the map M of type J. Fraser et al.: Regular self-dual and self-Petrie-dual maps of arbitrary valency 409 (d, d) by the voltage assignment a has n 1+|G|/4 components, each isomorphic to a regular map Ma = (Ga; xa, ya, zCT) of type (nd, nd) for the group Ga = (Zn)1+|G|/4 x G and suitable involutory generators xa,ya, za of Ga. Moreover, by the reasoning in the same proof (again applying also to non-orientable maps), trinity symmetry of M implies trinity symmetry of M. Note that both M and Mare non-orientable as their Petrie walks (of length d and nd) have odd length. □ Collecting our findings we arrive at the main result of this paper as a consequence of Theorem 4.1 and the remark following it, both in combination with Theorem 4.2. Theorem 4.3. For every odd d > 5 there exists a regular, self-dual and self-Petrie-dual map of valency d. □ A few remarks are in order. The reader may have observed that if the conclusion of Proposition 3.3 in Section 3 was valid for all odd k > 5 (and not just for prime k > 5), we would have a proof of our main result that would be independent on coverings and the resulting regular maps with trinity symmetry would have automorphism group isomorphic to PSL(2,p) for suitable primes depending on k. Research in this direction is currently being undertaken by the first two authors of this paper. Here we include a table of the first few values of N(g) for odd k between 5 and 29, with $(n) standing for the prime factorisation of n; observe that all the primes p in the prime factorization of |N(g) | satisfy p = ±1 (mod 2k) and p = ±1 (mod 12): k N (g) *(|N (g)|) 5 -11 prime 7 -13 prime 9 -73 prime 11 +263 prime 13 -131 prime 15 -239 prime 17 -4079 prime 19 + 15503 37 x 419 21 +5209 prime 23 -4093 prime 25 +56149 prime 27 -16417 prime 29 +3161869 59 x 53591 As noted earlier, existence of the regular maps for the first eight entries in this table was discovered by M. Conder, who also found such maps of valency 7 and 17 for the Janko simple groups J2 and J3. We conclude by noting that a strategy for proving Theorem 4.3 was also outlined by S. Wilson [14] by reducing the problem to a number-theoretic question related to Cheby-shev polynomials over finite fields. References [1] D. Archdeacon, M. Conder and J. Siran, Trinity symmetry and kaleidoscopic regular maps, Trans. Amer. 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