On the two-point function of general planar maps and hypermaps

We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number of faces.Comment: 32 pages, 17 figure

Hipermapas 2-restritamente-regulares de baixo género

Doutoramento em MatemáticaNesta tese consideramos hipermapas com grande número de automorfismos em superfícies de baixo género, nomeadamente a esfera, o plano projectivo, o toro e o duplo toro. É conhecido o facto de que o número de automorfismos ou simetrias de um hipermapa H é limitado pelo seu número de flags, que, genericamente falando, são triplos vértice-aresta-face mutualmente incidentes. De facto, o número de automorfismos de H divide o número de flags de H. Hipermapas para os quais este limite é atingido são chamados regulares e estão classificados nas superfícies orientáveis até género 101 e em superfícies não-orientáveis até genero 202, usando computadores. Neste trabalho classificamos os hipermapas 2-restritamente-regulares na esfera, no plano projectivo, no toro e no duplo toro, isto é, hipermapas cujo número de automorfismos é igual a metade do número de flags, e calculamos os seus grupos quiralidade e índices de quiralidade, que podem ser vistos como medidas algébricas e numéricas de quanto H se distancia de ser regular. Estes hipermapas são uma generalização dos hipermapas quirais. Também introduzimos alguns métodos para construir hipermapas bipartidos. Duas destas construções têm um papel muito importante no nosso trabalho.This thesis deals with hypermaps having large automorphism group on surfaces of small genus, namely the sphere, the projective plane, the torus and the double torus. It is well-known that the number of automorphisms or symmetries of a hypermap H is bounded by its number of flags, which are, roughly speaking, incident triples vertex-edge-face. In fact, the number of automorphisms of H divides the number of flags of H. Hypermaps for which this upper bound is attained are called regular and have been classified on orientable surfaces up to genus 101 and on non-orientable surfaces up to genus 202, using computers. In this work we classify the 2-restrictedly-regular hypermaps on the sphere, the projective plane, the torus and the double torus, that is, hypermaps whose number of automorphism is equal to half the number of flags and compute their chirality groups and chirality indices, which may be regarded as algebraic and numerical measures of how far H deviates from being regular. These hypermaps are a generalization of chiral hypermaps. We also introduce some methods for constructing bipartite hypermaps. Two of those constructions will play an important role in our work

Unified bijections for planar hypermaps with general cycle-length constraints

We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for all classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider "well-charged maps", which are maps with an assignment of "charges" (real numbers) on vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints

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

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