Hypermaps: constructions and operations

It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genusg. This conjecture can be rewritten in another form: given positive integers l,m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely manynonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.We prove that the conjecture is true, when two of the parameters l, m, n areequal, by showing how to construct those hypermaps, and we extend the resultto nonorientable hypermaps.A classification of all operations of finite order in oriented hypermaps isgiven, and a detailed study of one of these operations (the duality operation)is developed. Adapting the notion of chirality group, the duality group ofH can be defined as the minimal subgroup D(H) ≤¦ M on (H) such thatH = D (H) is a self-dual hypermap. We prove that for any positive integer d,we can find a hypermap of that duality index (the order of D (H) ), even whensome restrictions apply, and also that, for any positive integer k, we can find anon self-dual hypermap such that |Mon (H) | = d = k. We call this k the dualitycoindex of the hypermap. Links between duality index, type and genus of aorientably regular hypermap are explored.Finally, we generalize the duality operation for nonorientable regular hypermaps and we verify if the results about duality index, obtained for orientably regular hypermaps, are still valid

Enumerating Regular Objects associated with Suzuki Groups

We use the M\"obius function of the simple Suzuki group Sz(q) to enumerate regular objects such as maps, hypermaps, dessins d'enfants and surface coverings with automorphism groups isomorphic to Sz(q).Comment: 20 page

Hypermap operations of finite order

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups Out ∼= PGL2(Z) and Out + ∼= GL2(Z) of the groups = C2 ∗C2 ∗C2 and + = F2. We will consider the elements of finite order in these two groups, and the operations they induce

Exotic behaviour of infinite hypermaps

This is a survey of infinite hypermaps, and of how they can be constructed by using examples and techniques from combinatorial group theory, with particular emphasis on phenomena which have no analogues for finite hypermaps.<br/

Bicontactual hypermaps

Doutoramento em MatemáticaEsta tese dedica-se ao estudo de hipermapas regulares bicontactuais, hipermapas com a propriedade que cada hiperface contacta só com outras duas hiperfaces. Nos anos 70, S. Wilson classificou os mapas bicontactuais e, em 2003, Wilson e Breda d’Azevedo classificaram os hipermapas bicontactuais no caso não-orientável. Quando esta propriedade é transferida para hipermapas origina três tipos de bicontactualidade, atendendo ao modo como as duas hiperfaces aparecem à volta de uma hiperface fixa: edge-twin, vertextwin and alternate (dois deles são o dual um do outro). Um hipermapa topológico é um mergulho celular de um grafo conexo trivalente numa superfície compacta e conexa tal que as células são 3-coloridas. Ou de maneira mais simples, um hipermapa pode ser visto como um mapa bipartido. Um hipermapa orientado regular é um triplo ordenado consistindo num conjunto finito e dois geradores, que são permutações (involuções) do conjunto tal que o grupo gerado por eles, chamado o grupo de monodromia, actua regularmente no conjunto. Nesta tese, damos uma classificação de todos os hipermapas orientados regulares bicontactuais e, para completar, reclassificamos, usando o nosso método algébrico, os hipermapas não-orientáveis bicontactuais.This thesis is devoted to the study of bicontactual regular hypermaps, hypermaps with the property that each hyperface meets only two others. In the seventies, S. Wilson classified the bicontactual maps and, in 2003, Wilson and Breda d’Azevedo classified the bicontactual non-orientable hypermaps. When this property is transferred for hypermaps it gives rise to three types of bicontactuality, according as the two hyperfaces appear around a fixed hyperface: edge-twin, vertex-twin and alternate (two of which are dual of each other). A topological hypermap is a cellular embedding of a connected trivalent graph into a compact and connected surface such that the cells are 3-colored. Or simply, a hypermap can be seen as a bipartite map. A regular oriented-hypermap is an ordered triple, consisting of a finite set and two generators, which are permutations of the set, such that the group generate by them, called monodromy group, acts regularly on the set. In this thesis, we give a classification of all bicontactual regular orientedhypermaps and, for completion, we reclassify, using our algebraic method, the bicontactual non-orientable hypermaps

Classification of the regular oriented hypermaps with a prime number of hyperfaces

Regular oriented hypermaps are triples (G; a; b) consisting of a nite 2-generated group G and a pair a, b of generators of G, where the left cosets of ⟨a⟩, ⟨b⟩ and ⟨ab⟩ describe respectively the hyperfaces, hypervertices and hyperedges. They generalise regular oriented maps (triples with ab of order 2) and describe cellular embeddings of regular hypergraphs on orientable surfaces. In [5] we have classi ed the regular oriented hypermaps with prime number hyperfaces and with no non-trivial regular proper quotients with the same number of hyperfaces (i.e. primer hypermaps with prime number of hyperfaces), which generalises the classi cation of regular oriented maps with prime number of faces and underlying simple graph [13]. Now we classify the regular oriented hypermaps with a prime number of hyperfaces. As a result of this classi cation, we conclude that the regular oriented hypermaps with prime p hyperfaces have metacyclic automorphism groups and the chiral ones have cyclic chirality groups; of these the \canonical metacyclic" (i.e. those for which ⟨a⟩ is normal in G) have chirality index a divisor of n (the hyperface valency) and the non \canonical metacyclic" have chirality index p. We end the paper by counting, for each positive integer n and each prime p, the number of regular oriented hypermaps with p hyperfaces of valency n

A Whitney polynomial for hypermaps

We introduce a Whitney polynomial for hypermaps and use it to generalize the results connecting the circuit partition polynomial to the Martin polynomial and the results on several graph invariants