Orientador: Alexandre Ananin / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T08:38:27Z (GMT). No. of bitstreams: 1 Goncalves_EduardoCarvalhoBento_M.pdf: 2390910 bytes, checksum: fac0be229bed582ba6336532e52d2cd1 (MD5) Previous issue date: 2010 / Resumo: Primeiramente, apresentamos uma introdução à geometria hiperbólica plana que pode ser útil, inclusive, para um principiante. A seguir, utilizando o conceito de "terremoto simples", descrevemos explicitamente, em termos de algumas coordenadas naturais, o espaço de Teichmüller T Hn de superfícies hiperelípticas. Esta descrição resulta simples: T Hn é o espaço de determinadas (2n ? 6)-uplas de pontos no bordo ideal do plano hiperbólico. Partindo da descrição em questão, diversos resultados são apresentados, incluindo: um critério simples e efetivo que permite verificar se uma dada representação de um grupo de superfície no grupo de isometrias do plano hiperbólico é fiel e discreta; uma demonstração nova e elementar de um resultado de W. Goldman caracterizando as representações fiéis e discretas como aquelas que têm invariante de Toledo maximal; uma demonstração nova e elementar de um teorema de D. Toledo referente à rigidez de representações de grupos de superfície no grupo de isometrias holomorfas do espaço hiperbólico complexo / Abstract: First, we present an introduction to plane hyperbolic geometry, which may be useful even for a beginner. Next, using the concept of "simple earthquake", we explicitly describe, in terms of some natural coordinates, the Teichmüller space T Hn of hyperelliptic surfaces. This description turns out to be simple: T Hn is the space of certain (2n ? 6)-tuples of points in the ideal boundary of the hyperbolic plane. Based on the description in question, many results are presented, including: a simple and effective criterion which allows one to verify if a given representation of a surface group in the group of isometries of the hyperbolic plane is faithful and discrete; a new and elementary proof for a result of W. Goldman, which characterizes the faithful and discrete representations as being those which have maximal Toledo invariant; a new and elementary proof for a theorem of D. Toledo, relative to the rigidity of representations of surface groups in the group of holomorphic isometries of the complex hyperbolic space. key-words: Area, discreteness, representations, plane hyperbolic geometry, Teichmüller space, complex hyperbolic geometry / Mestrado / Geometria / Mestre em Matemática

Nesta dissertação, definimos a geometria hiperbólica usando o disco de Poincaré (D2) e o semiplano superior (H2) com as respectivas propriedades. Além disso, apresentamos algumas funções e relações importantes da geometria hiperbólica; conceituamos as superfícies de Riemann, analisando suas propriedades e representações; estudamos o espaço de Teichmüller com a devida decomposição em calças. Esses temas são ferramentas necessárias para atingir o objetivo da dissertação: definir as coordenadas de Fenchel Nielsen como um sistema de coordenadas locais do espaço de Teichmüller Tg. / In this dissertation, we defined the hyperbolic geometry using the Poincares disk (D2) and upper half-plane (H2) with its properties. Besides, we presented some functions and important relations of the hyperbolic geometry; we conceptualize the Riemann surfaces, analyzing its properties and representations; we studied the Teichmüller Space with proper decomposition pants. These themes are essential tools to reach the goal of the work: The definition of the Fenchel Nielsen coordenates as local coordinate system of the Teichmüller space Tg.

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-19T09:31:01Z (GMT). No. of bitstreams: 1 Alves_AlessandroFerreira_D.pdf: 1080224 bytes, checksum: 0748952c3176e9548151bec7e6d9c71d (MD5) Previous issue date: 2011 / Resumo: Para projetarmos um sistema de comunicação digital em espaços hiperbólicos é necessário estabelecer um procedimento sistemático de construção de reticulados como elemento base para a construção de constelações de sinais. De outra forma, em codificação de canal é de fundamental importância a caracterização das estruturas algébrica e geométrica associadas a canais discretos sem memória. Neste trabalho, apresentamos a caracterização geométrica de superfícies a partir dos possíveis emparelhamentos das arestas do polígono fundamental hiperbólico com 3 ? n ? 8 lados associado 'a superfície. Esse tratamento geométrico apresenta propriedades importantes na determinação dos reticulados hiperbólicos a serem utilizados no processo de construção de constelações de sinais, a partir de grupos fuchsianos aritméticos e da superfície de Riemann associada. Além disso, apresentamos como exemplo o desenvolvimento algébrico para a determinação dos geradores do grupo fuchsiano 'gama'8 associado ao polígono hiperbólico 'P IND. 8' / Abstract: In order to design a digital communication system in hyperbolic spaces is necessary to establish a systematic procedure of constructing lattices as the basic element for the construction of the signal constellations. On the other hand, in channel coding is of fundamental importance to characterize the geometric and algebraic structures associated with discrete memoryless channels. In this work, we present a geometric characterization of surfaces from the edges of the possible pairings of fundamental hyperbolic polygon with 3 ? n ? 8 sides associated with the surface. This treatment has geometric properties important in determining the hyperbolic lattices to be used in the construction of sets of signals derived from arithmetic Fuchsian groups and the associated Riemann surface / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Orientador: Reginaldo Palazzo Júnior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-19T10:50:06Z (GMT). No. of bitstreams: 1 Chaves_DanielPedroBezerra_D.pdf: 1159929 bytes, checksum: 06894c7e904c6209a690af3080f7cc32 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho apresentamos um método de descrição combinatorial para o fluxo geodesico sobre uma região hiperbólica compacta, tendo como objetivo associar a seqüências de codificação, parâmetros topologicos oriundos destas superfícies. Isto permite conjugar conceitos topologicos e combinatoriais oriundos das superfícies estudadas com conceitos de teoria da informação e codificação. Demonstramos como a propriedade de completude de um sistema dinâmico de eventos discretos invariantes no tempo se reflete na topologia do espaço de trajetórias do sistema, quando especificadas por seqüências bi-infinitas e descritas sobre um alfabeto finito. A mesma estrutura obtida pelo processo de codificação do fluxo geodesico, e a qual passamos a chamar de sistema simbólico fechado (ssf). Identificamos como um ssf pode ser caracterizado globalmente, através do seu conjunto de restrições irredutíveis, ou localmente, por conjuntos de restrições dependentes do contexto. Ambas derivadas de relações de ordem parcial. Disto determinamos métodos de representação do ssf. Através da relação entre os métodos de codificação aritmético e geométrico, propomos processos de codificação sobre superfícies hiperbólicas, determinando como as representações mínimas das seqüências código do fluxo geodesico podem ser construídas a partir das propriedades topológicas e combinatoriais da superfície / Abstract: In this work we present methods for a combinatorial description of the geodesic flow on a hyperbolic compact surface, with the intent of identifying how the topological parameters of the surface may be associated with discrete sequences. This approach allows to conjugate the topological and combinatorial properties of a surface with concepts of information theory and coding. We determine the intrinsic topological property of complete and time-invariant discrete dynamical systems whose trajectories are bi-infinite sequences over a finite alphabet. The same structure generated by the geodesic flow coding methods, that we call shift space. We show how a shift space can be completely characterized by the irreducible forbidden set and locally by the constraint sets, and how both can be obtained through partial order relations. As consequence of these results, some constructions to represent the shift spaces are proposed. Methods for coding source sequences on hyperbolic surfaces are proposed, based on T-piecewise and common-sets relations that exist between these methods. We conclude by specifying a construction procedure for presentations of arithmetic codes that is related with the topological and combinatorial properties of the hyperbolic surface / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Orientador: Reginaldo Palazzo Júnior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-18T16:43:26Z (GMT). No. of bitstreams: 1 Pilla_ElianeCristinaGeroli_M.pdf: 2007393 bytes, checksum: 2b95255e6d4fca123c23a039d1a083a5 (MD5) Previous issue date: 2005 / Resumo: O presente trabalho tem como meta principal construir constelações de sinais geometricamente uniformes no plano hiperbólico, visando considerá-las como alfabeto para geração de códigos de espaço de sinais, em particular os códigos de classes laterais generalizados. Para estabelecer estas constelações foi escolhido um conjunto de sinais geometricamente uniforme, constituído pelos centros dos octógonos da tesselação {8, 8}. Depois foi obtido um rotulamento para os elementos do grupo gerador dos conjuntos de sinais geometricamente uniformes em cada classe lateral. Finalmente, a partir do isomorfismo rótulo obtivemos um rotulamento isométrico para os elementos do conjunto de sinais / Abstract: Our goal in this work is to construct hyperbolic geometrically uniform signal constellations (more specifically g-torus) that are able to act as alphabets for ge neration of codes. To obtain these constellations we choose geometrically uniform signal sets consisting of the centers of the p-gons of tessellations of type {p, q}. From these constellations we obtain labelings for the elements of the generator group of the geometrically uniform signal sets in each coset. Finally, by the label isomorphism we obtain an isometric labeling for the elements of the signal set / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

Deux méthodes de construction de représentations de groupes sont présentées. La première propose une stratégie essayant de déterminer les représentations de groupes libres de type fini à valeurs dans tout réseau de groupes de Lie réel. La seconde, après avoir revu une construction d'une surface hyperbolique complexe, c'est-à-dire le quotient du plan hyperbolique complexe par un réseau, et examiné soigneusement ses propriétés, produit une infinité de représentations non-conjuguées, à valeurs dans un réseau du groupe des isométries du plan hyperbolique complexe, de groupes fondamentaux de variétés hyperboliques fermées de dimension 3, obtenues comme des fibrés en surfaces sur le cercle. / Two construction methods of group representations are presented. The first one proposes a strategy to try to determine the representations of finitely generated free groups into any lattice in real Lie groups. The second, after reviewing a construction of a complex hyperbolic surface, that is the quotient of the complex hyperbolic plane by a lattice, and examining its properties carefully, yields infinitely many non-conjugate representations into a lattice in the group of isometries of the complex hyperbolic plane, of fundamental groups of closed hyperbolic 3-dimensional manifolds, obtained as surface bundles over the circle.

Grâce au théorème d'hyperbolisation, nous savons précisément quand une variété de dimension trois compacte admet une métrique hyperbolique. Par ailleurs, d'après le théorème de rigidité de Mostow, cette structure géométrique est unique. Cependant, trouver des liens pratiques entre la géométrie et la topologie est un problème difficile. La plupart des résultats décrits dans cette thèse visent à concrétiser ces liens. Toute géodésique fermée orientée dans une surface hyperbolique admet un relèvement canonique dans le fibré tangent unitaire de la surface, et on peut donc le voir comme un nœud dans une variété de dimension trois. Les extérieurs des nœuds ainsi construits admettent une structure hyperbolique. Cette thèse a pour objet d'estimer le volume des extérieurs des relèvements canoniques. Pour toute surface hyperbolique on construit une suite de géodésique sur la surface, tel que les extérieurs associées ne sont pas homéomorphes entre elles et dont la suite des volumes respectifs est bornée. Aussi on minore le volume de l'extérieur à l'aide d'un réel explicite qui décrit une relation entre la géodésique et une décomposition en pantalons de la surface. Ceci donne une méthode pour construire une suite de géodésiques dont les volumes des extérieurs associées sont minorées en termes de la longueur de la géodésique correspondant. Dans le cas particulier de la surface modulaire, on obtient des estimations du volume de l'extérieur en termes de la période de la fraction continue associée à la géodésique. / Due to the Hyperbolization Theorem, we know precisely when does a given compact three dimensional manifold admits a hyperbolic metric. Moreover, by the Mostow's Rigidity Theorem this geometric structure is unique. However, finding effective and computable connections between the geometry and topology is a challenging problem. Most of the results on this thesis fit into the theme of making the connections more concrete. To every oriented closed geodesic on a hyperbolic surface has a canonical lift on the unit tangent bundle of the surface, and we can see it as a knot in a three dimensional manifold. The knot complement given in this way has a hyperbolic structure. The objective of this thesis is to estimate the volume of the canonical lift complement. For every hyperbolic surface we give a sequence of geodesics on the surface, such that the knot complements associated are not homeomorphic with each other and the sequence of the corresponding volumes is bounded. We also give a lower bound of the volume of the canonical lift complement by an explicit real number which describes a relation between the geodesic and a pants decomposition of the surface. This give us a method to construct a sequence of geodesics where the volume of the associated knot complements is bounded from below in terms of the length of the corresponding geodesic. For the particular case of the modular surface, we obtain estimations for the volume of the canonical lift complement in terms of the period of the continuous fraction expansion of the corresponding geodesic.

Made available in DSpace on 2016-04-27T16:58:51Z (GMT). No. of bitstreams: 1 Marilia Valerio Rocha.pdf: 8276158 bytes, checksum: d505c41608ef75bd40e4dec17fd3873e (MD5) Previous issue date: 2009-02-12 / This dissertation had as its main objective to propose an environment computational to the learning of Hyperbolic Geometry in the training of teachers of mathematics. Based on the Theory of Didactical Situation developed by Guy Brousseau (1986) and studies with the Comprehension of Demonstrations from Raymond Duval (1993), a didactic sequence has been prepared on the subject. The present work is oriented by the question to what extent the dinamic geometry could interfere in the build of hyperbolic geometry s concepts, in the axiomatic study by the professor of mathematics and how this new knowledge could contribute to your formation? This research is founded on some assumptions of Didactic Engieneering, described for Artigue (1988). The relevance of this research is justified by the Nacional Curriculum Guidelines for the courses off Bachelor s Degree in Mathematics, and shortage of teaching-material for the study of this content. Aimed at responding the question of research and gather information that enable the improvement of this didactic proposal, a pilot project was implemented with students of the Professional Master s Degree in Mathematics Education given by PUC-SP University. The results showed that the use of dinamic geometry in the formation of concepts of Hiperbolic Geometry, in the inicial axiomatic proposal, is a resource that contribute to understing these concepts / Esta dissertação teve como principal objetivo propor um ambiente computacional ao aprendizado da Geometria Hiperbólica na formação do professor de Matemática. Com base na Teoria das Situações Didáticas desenvolvida por Guy Brousseau (1986) e nos estudos sobre a compreensão das demonstrações, de Raymond Duval (1993), foi elaborada uma seqüência didática sobre o tema. A presente pesquisa orienta-se pela questão Em que medida a geometria dinâmica pode interferir na construção dos conceitos da Geometria Hiperbólica, no estudo axiomático realizado pelo professor de Matemática e como esse novo conhecimento pode contribuir para sua formação? . É fundamentada em alguns pressupostos da Engenharia Didática, descrita por Artigue (1988). Entende-se que a relevância desta pesquisa justifica-se nas orientações das Diretrizes Curriculares Nacionais para os cursos de Matemática, Bacharelado e Licenciatura (DCN) e na escassez de material didático para o estudo desse conteúdo. Visando a responder à questão de pesquisa e colher informações que possibilitem a melhoria desta proposta didática, aplicou-se um projeto-piloto com alunos do curso de Mestrado Profissional em Ensino de Matemática, ministrado pela Pontifícia Universidade Católica (PUC-SP). Os resultados apontaram que a utilização da geometria dinâmica na formação dos conceitos da Geometria Hiperbólica, em uma proposta axiomática inicial, é um recurso que contribui para a interiorização desses conceitos

The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.

The investigation of the hidden metric space behind complex network topologies is a fervid topic in current network science and the hyperbolic space is one of the most studied, because it seems associated to the structural organization of many real complex systems. The popularity-similarity-optimization (PSO) model simulates how random geometric graphs grow in the hyperbolic space, generating realistic networks with clustering, small-worldness, scale-freeness and rich-clubness. However, it misses to reproduce an important feature of real complex networks, which is the community organization. The geometrical-preferential-attachment (GPA) model was recently developed in order to confer to the PSO also a soft community structure, which is obtained by forcing different angular regions of the hyperbolic disk to have a variable level of attractiveness. However, the number and size of the communities cannot be explicitly controlled in the GPA, which is a clear limitation for real applications. Here, we introduce the nonuniform PSO (nPSO) model. Differently from GPA, the nPSO generates synthetic networks in the hyperbolic space where heterogeneous angular node attractiveness is forced by sampling the angular coordinates from a tailored nonuniform probability distribution (for instance a mixture of Gaussians). The nPSO differs from GPA in other three aspects: it allows one to explicitly fix the number and size of communities; it allows one to tune their mixing property by means of the network temperature; it is efficient to generate networks with high clustering. Several tests on the detectability of the community structure in nPSO synthetic networks and wide investigations on their structural properties confirm that the nPSO is a valid and efficient model to generate realistic complex networks with communities.