Antonio Breda d'Azevedo

Classification of regular maps of negative prime Euler characteristic

We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus where is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus where is a prime such that (mod ) and .

Constructions of Chiral Polytopes of Small Rank

An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.

Half-arc-transitive graphs and chiral hypermaps

A subgroup G of automorphisms of a graph X is said to be ½-arc-transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be ½-arc-transitive if Aut X is ½-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and ½-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and ½-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a ½-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G ≤ Aut X acting ½-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is ½-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated ½-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of ½-arc-transitive group action. Indeed, in last section we have applied general results to construct ½-arc-transitive graphs with cycle stabilisers of arbitrarily large orders.

Classification of primer hypermaps with a prime number of hyperfaces

A (face-)primer hypermap is a regular oriented hypermap with no regular proper quotients with the same number of hyperfaces. Primer hypermaps are then regular hypermaps whose automorphism groups induce faithful actions on their hyperfaces. In this paper we classify the primer hypermaps with a prime number of hyperfaces. This classification generalises an earlier dual classification by Du, Kwak and Nedela of regular oriented maps with a prime number of vertices and simple underlying graph.

Surfaces having no regular hypermaps

The central question of this paper is the Genus Question: For which N is it possible to draw a regular map or hypermap on the non-orientable surface of characteristic -N? We answer this question for all N from -1 to 50, and we display a body of theorems and techniques which can be used to settle the question for more complicated surfaces. These include: two ways to diagram an action of symmetry group, an equivalence relation on vertices (rotation centers in general), several applications of Sylow theory, and some non-Sylow observations on the size of the symmetry group.

Maps of Archimedean class and operations on dessins

In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin H applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from H ) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere.

Riemann surfaces and restrictively-marked hypermaps

If S is a compact Riemann surface of genus g > 1 then S has at most 84( g − 1) (orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of automorphisms of S and | G | > 24( g − 1) then G is the automorphism group of a regular oriented map (of genus g ) and if | G | > 12(g − 1) then G is the automorphism group of a regular oriented hypermap of genus g (Singerman). We generalise these results and prove that if | G | > g − 1 then G is the automorphism group of a regular restrictedly-marked hypermap of genus g . As a special case we also show that a marked finite transitive permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.

Regular pseudo-oriented maps and hypermaps of low genus

Pseudo-orientable maps were introduced by Wilson in 1976 to describe non-orientable regular maps for which it is possible to assign an orientation to each vertex in such a way that adjacent vertices have opposite orientations. This property extends naturally to non-orientable and orientable hypermaps. In this paper we classify the regular pseudo-oriented maps and hypermaps of characteristic ? ? - 3 . With the help of GAP (The GAP group, 2014) and its library of small groups, we extend the classification down to characteristic ? = - 16 (Tables?7-19 in the Appendix).

Classification of regular maps with prime number of faces and the asymptotic behaviour of their reflexible to chiral ratio

In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n , and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans (2014), has produced a nill asymptotic ratio.

Bicontactual Regular Hypermaps

Bicontactual regular maps (regular maps with the property that each face meets only two others) were introduced and classified by Wilson in 1985. This property generalizes to hypermaps (cellular embeddings of hypergraphs in compact surfaces) giving rise to three types of bicontactuality, namely, the edge-twin, the vertex-twin, and the alternate (the first two of which are the dual of each other). In 2003 Wilson and Breda classified (up to an isomorphism and duality) the bicontactual regular nonorientable hypermaps, leaving the orientable case open. In this paper we classify the bicontactual regular oriented hypermaps.