A cellular embedding of a connected graph (also known as a map) on an orientable surface has trinity symmetry if it is isomorphic to both its dual and its Petrie dual. A map is regular if for any two incident vertex-edge pairs there is an automorphism of the map sending the first pair onto the second. Given a map M with all vertices of the same degree d, for any e relatively prime to d the power map M e is formed from M by replacing the cyclic rotation of edges at each vertex on the surface

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... the e th power of the rotation. A map is kaleidoscopic if all of its power maps are pairwise isomorphic. In this paper, we present a covering construction that gives infinite families of kaleidoscopic regular maps with trinity symmetry. 2 DAN ARCHDEACON, MARSTON CONDER, AND JOZEFŠIRÁŇ proving our results are developed in Sections 3 and 4. The actual proofs are in Sections 5 and 6, followed by concluding remarks in Section 7. 1.1. Maps and automorphisms. A map M is a cellular embedding of a connected graph or multigraph G on a surface. For the most part these surfaces will be connected and orientable, although we will make occasional mention of the disconnected or non-orientable cases. We can describe the embedding combinatorially in terms of rotations. To begin with, we will do this for embeddings on orientable surfaces. Fix an orientation on the surface, clockwise or anticlockwise, making the surface oriented. This orientation induces a cyclic permutation of the edge-ends incident with a vertex v, which we call the local rotation at v. A rotation is then any product of local rotations (over any/all vertices), which is then a permutation of the set E of all edge-ends of the underlying graph of the map. Let R be the product of all local rotations. This rotation contains all the information needed to recover the embedding. To see this, let I be the involutory permutation of the set E that swaps the two ends of each edge. Then the pair (R, I) of permutations of E completely determines the map. Indeed, orbits of the permutations I, R, and RI can be identified with edges, vertices, and face boundary walks of the map, and their mutual incidence is given by non-empty intersection of the orbits. We may thus identify an oriented map M with the corresponding permutation pair (R, I) acting transitively on the set of half-edges E, and write M = (E; R, I). We have seen that embeddings on oriented surfaces correspond to certain pairs of permutations, and vice versa. To understand how this may be extended to arbitrary surfaces, observe that every edge end of an embedded graph has two 'sides' on the supporting surface; these sides are usually called flags (or blades, see [6] ). This way, for every edge of a graph we may associate a set of four flags. Let F be the set of all flags of the embedded graph; observe that |F| = 2|E|. Let T be the involutory permutation of F that interchanges the two flags associated with each edge-end, and let L be the involutory permutation of F that interchanges the two flags appearing at the same side of each edge. Observe the important relation LT = T L. Finally, let C be the involutory permutation of F that interchanges every two flags forming a 'corner' (two edge-ends meeting at a vertex on the boundary of a face). If M = (E; R, I) is an oriented map, and C, L, T are the permutations of F defined as above, then the two products CT and T L are permutations of F that represent in a natural sense the effect of the two permutations R and I of edgeends, respectively. Furthermore, the permutation group generated by CT and T L has two orbits on F, with the two flags associated with each edge-end always lying in different orbits. We note that this description of a map by three involutions is suitable also for maps on non-orientable surfaces, corresponding to the situation where the permutation group generated by CT and T L has a single orbit on F. In any case, we sometimes use the notation M = (F; C, L, T ) if a representation of the map M is necessary in terms of the three involutions C, L, T acting on the flag set F. For more background on algebraic theory of maps, see the survey-type papers [10, 6] and the monograph [9]. A map isomorphism θ : M → M between two oriented maps M and M is an isomorphism of the underlying graphs that extends to a homeomorphism of the corresponding surfaces and preserves the set of face boundary walks. In algebraic terms, if M = (E; R, I) and M = (E ; R , I ), then a map isomorphism θ : M → M will be identified with a bijection E → E between the corresponding sets of TRINITY SYMMETRY AND KALEIDOSCOPIC REGULAR MAPS 3 edge-ends, such that θ(x R ) = (θ(x)) R and θ(x I ) = (θ(x)) I for every edge-end x ∈ E. Note that by connectedness, a map isomorphism θ : M → M is completely determined by the image of any particular edge-end. Let θ be a map isomorphism from M = (E; R, I) to itself. Then θ commutes with R and I in the sense explained before, and hence preserves the orientation of the supporting surface. We call any such θ an orientation-preserving automorphism of M . The family of all orientation-preserving automorphisms of M forms a group under composition of mappings, called the orientation-preserving automorphism group of M , and denoted by Aut + (M ). By the remark at the end of the previous paragraph we have |Aut + (M )| ≤ |E|, or, equivalently, |Aut + (M )| ≤ 2|E| where E denotes the edge set of the underlying graph of the map. We therefore have an upper bound on the number of orientation-preserving 'symmetries' of an oriented map. If the equality |Aut + (M )| = 2|E| is achieved, the map M is called orientablyregular. In that case, Aut + (M ) acts regularly on the edge-ends of M , and M has as much orientation-preserving symmetry as possible. Given any oriented map M = (E; R, I), we can form its oriented mate M −1 = (E; R −1 , I). In general, M and M −1 need not be isomorphic, but if they are, then the map M is called reflexible. In such a case, an isomorphism θ : M → M −1 with the property that θ(x R ) = (θx) R −1 and θ(x I ) = (θx) I for every x ∈ E is called an orientation-reversing automorphism of M . From this point on, orientationpreserving and orientation-reversing automorphisms (if any) will be simply called automorphisms, and the group of all automorphisms of a map M will be denoted by Aut(M ). If an oriented map M is reflexible, then Aut + (M ) is a subgroup of index two in Aut(M ), while if M is not reflexible, then Aut + (M ) = Aut(M ), and M is called chiral. In either case we have |Aut(M )| ≤ 2|E| = 4|E|. Oriented maps achieving the equality |Aut(M )| = 4|E| are called regular, since in that case Aut(M ) acts regularly on the sides (or flags) of M . Such maps have as much symmetry as possible. In the general setting, when a map is represented as M = (F; C, L, T ), a permutation f of F is an automorphism of M if and only if for every flag x ∈ F one has (x C ) f = (x f ) C , (x L ) f = (x f ) L , and (x T ) f = (x f ) T -that is, if and only if f commutes with all of the three involutions in the algebraic description of M . In what follows we will extend the concept of symmetry of a map in several ways, depending on certain operations on maps, which we will introduce next. 1.2. New maps from old. In this section we describe a number of methods of forming (possibly) new maps from a given primal map M . We will first consider operations that do not change the underlying graph of M . Powers of maps: Taking the oriented mate of a map has a natural generalization, obtained by replacing R −1 by any integral power R e of the rotation R, for e relatively prime to the degree of every vertex of the map. For an oriented map M , the degree of M is defined as the least common multiple of all vertex degrees of M . If M = (E; R, I) is an oriented map of degree d, and e is relatively prime to d, then the map M e = (E; R e , I) is called the e th power of M . In terms of flags, if M = (F; C, L, T ), then M e = (F; C e , L, T ) where C e = (CT ) e−1 C. The e th power of M has the same underlying graph as M , but, in general, the supporting surfaces of M and M e may be different.